Petz recovery versus matrix reconstruction

Holzäpfel, Milan; Cramer, Marcus; Datta, Nilanjana; Plenio, Martin B.

doi:10.1063/1.5009658

The reconstruction of the state of a multipartite quantum mechanical system represents a fundamental task in quantum information science. At its most basic, it concerns a state of a bipartite quantum system whose subsystems are subjected to local operations. We compare two different methods for obtaining the original state from the state resulting from the action of these operations. The first method involves quantum operations called Petz recovery maps, acting locally on the two subsystems. The second method is called matrix (or state) reconstruction and involves local, linear maps that are not necessarily completely positive. Moreover, we compare the quantities on which the maps employed in the two methods depend. We show that any state that admits Petz recovery also admits state reconstruction. However, the latter is successful for a strictly larger set of states. We also compare these methods in the context of a finite spin chain. Here, the state of a finite spin chain is reconstructed from the reduced states of a few neighbouring spins. In this setting, state reconstruction is the same as the matrix product operator reconstruction proposed by Baumgratz et al. [Phys. Rev. Lett. 111, 020401 (2013)]. Finally, we generalize both these methods so that they employ long-range measurements instead of relying solely on short-range correlations embodied in such local reduced states. Long-range measurements enable the reconstruction of states which cannot be reconstructed from measurements of local few-body observables alone and hereby we improve existing methods for quantum state tomography of quantum many-body systems.

I. INTRODUCTION

Consider a bipartite quantum state ρ_XY that is transformed to a state $τ_{X^{'} Y^{'}}$ under the action of local quantum operations $N_{X} : X \to X^{'}$ and $N_{Y} : Y \to Y^{'}$ ⁠. These local operations could either correspond to (i) undesirable noise (resulting from unavoidable interactions of the quantum system XY with its environment) or they could correspond to (ii) local measurements made by an experimenter doing quantum state tomography. We are interested in determining the conditions under which the state $τ_{X^{'} Y^{'}}$ can be transformed back to the original state ρ_XY with maps that act locally on X′ and Y′. In the case (i), these would be the conditions under which the effect of the noise can be reversed, whereas in the case (ii), these would be the conditions under which reconstruction of the original state from the outcome of the experimenter’s chosen measurements is possible.

The question whether $τ_{X^{'} Y^{'}}$ can be transformed back to ρ_XY can be answered with different methods. If the transformation is to be achieved with quantum operations, an answer is provided by the Petz recovery map^2,3 under a condition on the mutual information of the two states. If general linear (not necessarily completely positive) maps are allowed in the transformation, one can use a matrix reconstruction method. This matrix reconstruction method is related to MPO (i.e., matrix product operator) reconstruction and the so-called pseudoskeleton (or CUR) matrix decompositions.^1,4,5 In either case, the construction of the maps that transform $τ_{X^{'} Y^{'}}$ into ρ_XY does not require complete information on ρ_XY if suitable maps $N_{X}$ and $N_{Y}$ are used. In this case, the transformation can be used for efficient quantum state tomography of ρ_XY with less measurements than necessary for standard quantum state tomography.

A fundamental quantity in quantum information theory is the quantum relative entropy D(ρ∥σ) between a state ρ and a positive semi-definite operator σ (see Sec. II C for its definition). It acts as a parent quantity for other entropic quantities arising in quantum information theory, e.g., von Neumann entropy, conditional entropy, and mutual information. When ρ and σ are both states, D(ρ∥σ) also has an operational interpretation as a measure of distinguishability between the two states.^6,7 One of its most important properties is its monotonicity under the joint action of a quantum operation (say, $N$ ⁠). This is also called the data processing inequality (DPI) and is given by

D (N (ρ) ‖ N (σ)) \leq D (ρ ‖ σ) .

The condition under which the above inequality is saturated was obtained by Petz⁸ and has found important applications in quantum information theory. Petz proved that equality in the DPI holds if and only if there exists a recovery map, given by a quantum operation $R$ which reverses the action of $N$ on both ρ and σ, i.e., $R (N (ρ)) = ρ$ and $R (N (σ)) = σ$ ⁠. Petz also obtained an explicit form of such a recovery map, which is often called the Petz recovery map. Petz’s condition on the equality in the DPI immediately yields a necessary and sufficient condition under which the conditional mutual information I(A:C|B) of a tripartite state ρ_ABC is zero,³ which in turn is the condition under which strong subadditivity (SSA) of the von Neumann entropy (arguably the most powerful entropic inequality in quantum information theory) is saturated. Petz’s result, when applied to the problem studied in this paper, implies that the original state ρ_XY can be recovered from the transformed state $τ_{X^{'} Y^{'}}$ if and only if the mutual information I(X:Y)_ρ of ρ_XY is equal to the mutual information $I {(X^{'} : Y^{'})}_{τ}$ of $τ_{X^{'} Y^{'}}$ ⁠.^3,9 Moreover, a valid recovery map is a tensor product of maps acting locally on X′ and Y′, each having the structure of a Petz recovery map. A detailed discussion of Petz’s result and of the quantities on which the Petz recovery maps depend is given in Sec. II C.

The data processing inequality of the relative entropy implies a DPI for the mutual information,

I {(X^{'} : Y^{'})}_{τ} \leq I {(X : Y)}_{ρ} .

The mutual information quantifies the amount of correlations that exist between the two subsystems of a bipartite quantum state. Another measure of such correlations is the operator Schmidt rank^10,11 of the state, which we denote as OSR(X:Y)_ρ for a bipartite state ρ_XY [see Eq. (8) for its definition].

In the following, we discuss the main results of this paper. We show that the operator Schmidt rank also satisfies a DPI

O S R {(X^{'} : Y^{'})}_{τ} \leq O S R {(X : Y)}_{ρ},

where $τ_{X^{'} Y^{'}}$ is the state obtained from ρ_XY via the local quantum operations $N_{X}$ and $N_{Y}$ ⁠, as discussed above. The DPI for the operator Schmidt rank is directly implied by the fact that the matrix rank satisfies rk(MN) ≤ rk(M)rk(N) for any two matrices M and N (see Corollary 11 for details). We show that $τ_{X^{'} Y^{'}}$ can be transformed into ρ_XY with local maps if and only if the DPI of the operator Schmidt rank is saturated. Our proof does not guarantee that the maps that transform $τ_{X^{'} Y^{'}}$ into ρ_XY are completely positive but it also does not require that ρ_XY and $τ_{X^{'} Y^{'}}$ are positive semidefinite or that $N_{X}$ and $N_{Y}$ are completely positive. The proof proceeds by transforming the reconstruction problem into a reconstruction problem for a general, rectangular matrix. Here, we provide an extension of the known pseudoskeleton decomposition,^4,5 which is also known as CUR decomposition and which can reconstruct a low-rank matrix from few of its rows and columns. Our method reconstructs a matrix M from the matrix products LM and MR if the rank of M equals the rank of LMR; L, M, and R are the general rectangular matrices.

We explore the relation between Petz recovery and state/MPO reconstruction for the case of 2, 4, and n parties. State/MPO reconstruction, when compared to Petz recovery, is shown to be possible for a strictly larger set of states but requires more information.

The state of an n-partite quantum system, such as n spins in a linear chain, can be represented as a matrix product operator (MPO) with MPO bond dimensions given by the operator Schmidt ranks OSR(1, …, k: k + 1, …, n) (between the sites 1, …, k and k + 1, …, n;^12,13). If the operator Schmidt ranks are all bounded by a constant D, the MPO representation is given in terms of ∼nD² complex numbers, which is much less than the number of entries of the density matrix of the n-partite quantum system. Baumgratz et al.¹ presented a condition under which an MPO representation of the state of an n-partite quantum system can be reconstructed from the reduced states of few neighbouring systems (MPO reconstruction). We will demonstrate that their work implies, for the case where the local operations $N_{X}$ and $N_{Y}$ are partial traces, that $τ_{X^{'} Y^{'}}$ can be transformed into ρ_XY if the two states have equal operator Schmidt rank.

The ability to reconstruct the state of an n-partite quantum system from reduced states of l < n systems, as provided, e.g., by MPO reconstruction, is advantageous for quantum state tomography of many-body systems. Standard quantum state tomography requires the expectation values of a number of observables which grow exponentially with n. If the full state can be reconstructed from l-body reduced states, then the number of observables grows exponentially with l but only linearly with the number of reduced states. MPO reconstruction uses the reduced states of blocks of l neighbouring sites on a linear chain. As the number of such blocks increases linearly with n, MPO reconstruction enables quantum state tomography with a number of observables which increases only linearly with n.

We call a method for quantum state tomography efficient if it requires only polynomially many (in n) sufficiently simple observables (more details on permitted observables are given in Sec. VI A, Remark 18). Here, we assume that exact expectation values are available. For a given method to be useful in practice, it is however necessary that the quantum state can be estimated up to a fixed estimation error using approximate expectation values from measurements on at most polynomially many (in n) copies of the state. In this paper, we discuss only the number of necessary observables but not the number of necessary copies of the state. Numerical simulations indicate that, e.g., MPO reconstruction and similar methods are efficient also in the number of necessary copies.^1,14–16

There are multipartite quantum states (e.g., states of a spin chain) that admit an efficient matrix product state (MPS) or MPO representation but which cannot be reconstructed from reduced states of a few of its parties (e.g., a few neighbouring sites of the spin chain). The n-qubit GHZ state is an example of such a state (Sec. V A). However, it has been shown that the GHZ state can be reconstructed from a number of observables linear in n, provided global observables (i.e., those which act on the whole system) are allowed.^14,15 The necessary observables are given by simple tensor products¹⁵ or simple tensor products and unitary control of few neighbouring sites.¹⁴ We generalize MPO reconstruction and a similar technique based on the Petz recovery map¹⁷ to use a certain class of long-range measurements which includes those just mentioned as special cases (Sec. VI). We represent a long-range measurement as a sequence of local quantum operations followed by the measurement of a local observable. However, a tensor product of single-party observables, whose expectation value can be obtained by a simple, sequential measurement of the single-party observables, already constitutes an allowed long-range measurement.

The example of the GHZ state shows that long-range measurements enable the recovery or reconstruction of a larger set of states than those obtained by local few-body observables. Our reconstruction and recovery methods provide a representation of the reconstructed state in terms of a sequence of local linear maps which is equivalent to an MPO representation. For methods based on the Petz recovery map, the local linear maps are quantum operations, and because of this, a PMPS (locally purified MPS) representation can be obtained (Ref. 18, Appendix A 5). A PMPS representation is advantageous because it can be computationally demanding to determine whether a given MPO representation represents a positive semidefinite operator,¹⁸ whereas a PMPS representation always represents a positive semidefinite operator. Our work on the reconstruction of spin chain states is partially based on similar ideas developed in the context of tensor train (TT) representations¹⁹ and there is also related work on Tucker and hierarchical Tucker representations.^5,20,21

The remainder of the paper is structured as follows: Sec. II introduces notation, definitions, MPS/MPO representations, and known results on the Petz recovery map. Section III shows how a low-rank matrix reconstruction technique enables bipartite state reconstruction, i.e., a transformation of $τ_{X^{'} Y^{'}}$ into ρ_XY. We also prove that approximate matrix reconstruction is possible if a low-rank matrix is perturbed by a small high-rank component (Sec. III B). We apply the Petz recovery map to the bipartite setting in Sec. IV and investigate the relation between Petz recovery and state reconstruction in Sec. V. Any state which admits Petz recovery is found to also admit state reconstruction. In Sec. VI, we discuss reconstruction of spin chain states with Petz recovery maps and state reconstruction. If reconstruction is performed with local reduced states (Sec. VI A), a known application of the Petz recovery map¹⁷ and the known MPO reconstruction technique¹ are obtained. In Sec. VI B, we reconstruct spin chain states from recursively defined long-range measurements. We show that successful recovery of a given spin chain state implies successful reconstruction both for local reduced states and for long-range measurements. The set of states which can be reconstructed with long-range measurements is seen to be strictly larger than the set of states which can be reconstructed with measurements on local reduced states. Long-range measurements were used in earlier work on the reconstruction of pure states¹⁴ and we show that our methods can recover or reconstruct these states if the same long-range measurements are used.

II. PRELIMINARIES

A. Notation and basic definitions

In this paper, all Hilbert spaces $H$ are finite-dimensional. We use capital letters A, B, C, … to denote quantum systems with Hilbert spaces $H_{A}$ ⁠, $H_{B}$ ⁠, $H_{C}$ ⁠, ..., and set $d_{A} = \dim (H_{A})$ ⁠. For notational simplicity, we often use A to denote both the system and its associated Hilbert space, when there is no cause for confusion. If n systems are involved, we denote their Hilbert spaces by $H_{1}$ ⁠, ..., $H_{n}$ and tensor products of the latter by $H_{1, \dots, n} = H_{1} \otimes \dots \otimes H_{n}$ ⁠.

We denote the set of linear maps from A to B by $B (A; B) = B (H_{A}; H_{B})$ and the set of linear operators on A by $B (A) \equiv B (A; A)$ ⁠. If tensor products are involved, we use the notation $B (A B; C D) \equiv B (A, B; C, D) \equiv B (A \otimes B; C \otimes D)$ ⁠. The trace of a linear operator $F \in B (A)$ (or a square matrix $F \in C^{m \times m}$ ⁠) is denoted by Tr(F). A quantum state (or density matrix) of a system A is a positive semi-definite operator $ρ \in B (A)$ with unit trace. Let $D (A)$ denote the set of quantum states in $B (A)$ ⁠. In the case of a pure quantum state $ρ = |ψ⟩ ⟨ψ|$ ⁠, $| ψ ⟩) \in H_{A}$ ⁠, we refer to both $ρ = |ψ⟩ ⟨ψ|$ and |ψ⟩ as the pure state. For any state $ρ \in D (A)$ ⁠, its von Neumann entropy is defined as S(A)_ρ = −Tr(ρ_A log (ρ_A)). In this paper, all logarithms are taken to base 2.

For any $F \in B (A)$ ⁠, let F* denote its Hermitian adjoint, supp(F) denote its support, rk(F) denote its rank, ∥F∥ denote its operator norm (largest singular value), and σ_min(F) denote its smallest non-zero singular value. A linear operator $F \in B (A)$ is an observable if it is Hermitian. The Hilbert-Schmidt inner product on $B (A)$ is denoted by

⟨ F, G ⟩ = T r (F^{*} G) \forall F, G \in B (A) .

(1)

The vector space $B (A)$ becomes a Hilbert space when equipped with this inner product.

The notation $B (B (A); B (B))$ denotes the set of linear maps from $B (A)$ to $B (B)$ ⁠. This includes the set of quantum operations (or superoperators) from A to B which are given by linear, completely positive, trace-preserving (CPTP) linear maps $N \in B (B (A); B (B))$ ⁠. We use the shorthand notation $N : A \to B$ to indicate such a quantum operation. Given any linear map $N \in B (B (A); B (B))$ ⁠, its Hermitian adjoint (with respect to the Hilbert-Schmidt inner product) is denoted by $N^{*} \in B (B (B); B (A))$ ⁠, i.e., $⟨ F, N (G) ⟩ = ⟨ N^{*} (F), G ⟩,$ for all $F \in B (B)$ ⁠, $G \in B (A)$ ⁠.

Since we are dealing with finite-dimensional Hilbert spaces, all linear operators and maps are represented by matrices. Given a matrix $M \in C^{m \times n}$ (or a linear map M), let M* denote its conjugate transpose matrix, $\bar{M}$ denotes its (element-wise) complex conjugate matrix, and M⁺ denotes its Moore–Penrose pseudoinverse. The following four properties of the pseudoinverse also define it uniquely:^22,23

M M^{+} M = M, M^{+} M M^{+} = M^{+}, {(M M^{+})}^{*} = M M^{+}, {(M^{+} M)}^{*} = M^{+} M .

(2)

Given a real number t ≥ 0, we define $M_{t}^{+} = {(M_{t})}^{+}$ where M_t is obtained from M by replacing its singular values that are smaller than or equal to t by zero.

For a system A, we choose an operator basis ${F_{i}^{(A)}}_{i = 1}^{d_{A}^{2}}$ which is orthonormal in the Hilbert–Schmidt inner product

F_{i}^{(A)} \in B (A), ⟨ F_{i}^{(A)}, F_{j}^{(A)} ⟩ = δ_{i j}, i, j \in {1, \dots, d_{A}^{2}} .

(3)

Given a basis element $F_{i}^{(A)}$ ⁠, we denote its dual element (in the Hilbert-Schmidt inner product) by ${\tilde{F}}_{i}^{(A)}$ ⁠,

{\tilde{F}}_{i}^{(A)} \in B (B (A); C) : G \mapsto {\tilde{F}}_{i}^{(A)} (G) = ⟨ F_{i}^{(A)}, G ⟩ .

(4)

The identity map, id, on $B (A)$ can then be expressed as

i d = \sum_{i = 1}^{d_{A}^{2}} F_{i}^{(A)} {\tilde{F}}_{i}^{(A)} .

(5)

This is nothing but the resolution of the identity operator for the Hilbert space $B (A)$ ⁠.

Consider a linear map $M \in B (B (Y); B (X))$ ⁠. Since the vector spaces $B (X Y)$ and $B (B (Y); B (X))$ have the same finite dimension, ${(d_{X} d_{Y})}^{2} = d_{X}^{2} d_{Y}^{2}$ ⁠, it is possible to define a bijective linear map between the two spaces. To do so, we define the components of $ρ \in B (X Y)$ and $M$ in terms of the operator bases from above

{[ρ]}_{i j} = ⟨ F_{i}^{(X)} \otimes F_{j}^{(Y)}, ρ ⟩, {[M]}_{i j} = ⟨ F_{i}^{(X)}, M (F_{j}^{(Y)}) ⟩ .

(6)

Given a linear operator $ρ \in B (X Y)$ ⁠, we define a linear map $M_{ρ}$ by

{[M_{ρ}]}_{i j} = {[ρ]}_{i j}, M_{ρ} \in B (B (Y); B (X)) .

(7)

We denote the matrix representation of $M_{ρ}$ in the operator basis chosen above by M_ρ. The maps $ρ \mapsto M_{ρ}$ and ρ ↦ M_ρ defined by Eq. (7) are of course bijective. Note that ρ can be represented by a matrix of size d_Xd_Y × d_Xd_Y, while $M_{ρ}$ can be represented by the matrix $M_{ρ} \in C^{d_{X}^{2} \times d_{Y}^{2}}$ ⁠. The transpose map $M^{⊺}$ is defined in the same operator basis, i.e., ${[M^{⊺}]}_{i j} = {[M]}_{j i}$ ⁠.

Given a linear operator $ρ \in B (X Y)$ ⁠, its operator Schmidt rank is given by

O S R {(X : Y)}_{ρ} = min \{r : ρ = \sum_{k = 1}^{r} G_{k}^{'} \otimes G_{k}^{″}, G_{k}^{'} \in B (X), G_{k}^{″} \in B (Y)\} .

(8)

The operator Schmidt rank is equal to

O S R {(X : Y)}_{ρ} = r k (M_{ρ}),

(9)

which can be shown as follows: The matrix representation M_ρ of $M_{ρ}$ can be written as

M_{ρ} = G H where G \in C^{d_{X}^{2} \times s}, H \in C^{s \times d_{Y}^{2}} and s = r k (M_{ρ}) = r k (M_{ρ}) .

(10)

Since the components of M_ρ and ρ are related by ${[M_{ρ}]}_{i j} = {[ρ]}_{i j}$ ⁠, we have

ρ = \sum_{i j} {[ρ]}_{i j} F_{i}^{(X)} \otimes F_{j}^{(Y)} = \sum_{k = 1}^{s} (\sum_{i} G_{i k} F_{i}^{(X)}) \otimes (\sum_{j} H_{k j} F_{j}^{(Y)}),

(11)

where G_ik and H_kj are the components of the matrices G and H. This shows that the operator Schmidt rank cannot exceed s = rk(M_ρ). Now suppose that the operator Schmidt rank was less than that, i.e., r = OSR(X:Y)_ρ < s = rk(M_ρ). Then, a decomposition of ρ as in Eq. (8) implies that

{[M]}_{i j} = {[ρ]}_{i j} = \sum_{k = 1}^{r} G_{i k} H_{k j}, G_{i k} = ⟨ F_{i}^{(X)}, G_{k}^{'} ⟩, H_{k j} = ⟨ F_{j}^{(Y)}, G_{k}^{″} ⟩,

(12)

i.e., that rk(M_ρ) ≤ r < rk(M_ρ). This contradiction shows that the operator Schmidt rank must equal $r k (M_{ρ}) = r k (M_{ρ})$ ⁠. The operator Schmidt rank is also equal to the (smallest possible) bond dimension of a matrix product operator (MPO) representation of the linear operator.¹² This is discussed in Sec. II B.

B. MPS, MPO, and PMPS representations

In this section, we introduce frequently used efficient representations of pure and mixed quantum states on n systems. We call a representation efficient if it describes a state with a number of parameters (i.e., complex numbers) which increases at most polynomially with n. The number of parameters of a particular representation of a state is accordingly given by the total number of entries of all involved vectors, matrices, and tensors. For example, a pure state $| ψ ⟩) \in C^{d^{n}}$ of n quantum systems of dimension d has dⁿ parameters and is not an efficient representation. To discuss whether a given representation is efficient or not, we use the following notation: For a function f(n), we write $f = O (p o l y (n))$ or $f (n) = O (p o l y (n))$ if there is a polynomial g(n) such that f(n) ≤ g(n). We write $f = O (\exp (n))$ if there are constants c₁ and c₂ such that f(n) ≤ c₁ exp (c₂n) holds for all n.

First, we introduce the matrix product state (MPS) representation (see, e.g., Ref. 13), which is also known as tensor train (TT) representation.²⁴ Consider n quantum systems of dimensions d₁, …, d_n respectively and let ${| ϕ_{i_{k}}^{(k)} ⟩}_{i_{k} = 1}^{d_{k}}$ be an orthonormal basis of the kth system. An MPS representation of a pure state on n systems is given by

⟨ ϕ_{i_{1}}^{(1)} \dots ϕ_{i_{n}}^{(n)} |) ψ ⟩ = G_{1} (i_{1}) G_{2} (i_{2}) \dots G_{n} (i_{n}),

(13)

where D₀ = D_n = 1, $G_{k} (i_{k}) \in C^{D_{k - 1} \times D_{k}}$ and i_k ∈ {1, …, d_k}. The condition D₀ = D_n = 1 ensures that G₁(i₁) and G_n(i_n) are row and column vectors, while the G_k(i_k) for k between 1 and n can be matrices. The matrix sizes D_k are called the bond dimensions of the representation. The maximal local dimension and the maximal bond dimension are indicated by d = max_kd_k and D = max_kD_k. For $d = O (p o l y (n))$ and $D = O (p o l y (n))$ ⁠, the total number of parameters of the MPS representation is $n d D^{2} = O (p o l y (n))$ and the representation is efficient. The bond dimension D_k of any MPS representation of |ψ⟩ is larger than or equal to the Schmidt rank of |ψ⟩ for the bipartition 1, …, k|k + 1, …, n and a representation with all bond dimensions equal to the corresponding Schmidt ranks can always be determined (see, for example, Ref. 13). We discuss the analogous property of the matrix product operator (MPO) representation in more detail.

A matrix product operator (MPO) representation^25,26 of a mixed state on n systems is given by

⟨ ϕ_{i_{1}}^{(1)} \dots ϕ_{i_{n}}^{(n)} | ρ | ϕ_{j_{1}}^{(1)} \dots ϕ_{j_{n}}^{(n)} ⟩ = G_{1} (i_{1}, j_{1}) G_{2} (i_{2}, j_{2}) \dots G_{n} (i_{n}, j_{n})

(14)

where D₀ = D_n = 1, $G_{k} (i_{k}, j_{k}) \in C^{D_{k - 1} \times D_{k}}$ and i_k, j_k ∈ {1, …, d_k}. Alternatively, an MPO representation may be given in terms of operator bases $F_{i_{k}}^{(k)}$ ⁠,

⟨ F_{i_{1}}^{(1)} \otimes \dots \otimes F_{i_{n}}^{(n)}, ρ ⟩ = G_{1} (i_{1}) G_{2} (i_{2}) \dots G_{n} (i_{n}),

(15)

where D₀ = D_n = 1, $G_{k} (i_{k}) \in C^{D_{k - 1} \times D_{k}}$ ⁠, and $i_{k} \in {1, \dots, d_{k}^{2}}$ ⁠. If the operator basis $F_{(i_{k}, j_{k})}^{(k)} = |ϕ_{i_{k}}^{(k)}⟩ ⟨ϕ_{j_{k}}^{(k)}|$ is used, Eq. (15) turns into Eq. (14). As before, we denote the maximal local and bond dimensions by d = max_kd_k and D = max_kD_k. The number of parameters of an MPO representation is at most nd²D² and it is an efficient representation if $d = O (p o l y (n))$ and $D = O (p o l y (n))$ ⁠. The operator Schmidt ranks of ρ provide lower bounds to the bond dimensions of any MPO representation of ρ,¹²

O S R {(1 \dots k : k + 1 \dots n)}_{ρ} \leq D_{k} .

(16)

This becomes clear if we rewrite Eq. (14) as follows:

ρ = \sum_{b_{0} \dots b_{n}} H_{1} (b_{0}, b_{1}) \otimes H_{2} (b_{1}, b_{2}) \otimes \dots \otimes H_{n} (b_{n - 1}, b_{n}),

(17)

where $H_{k} (b_{k - 1}, b_{k}) \in C^{d_{k} \times d_{k}}$ and ${[H_{k} (b_{k - 1}, b_{k})]}_{i_{k}, j_{k}} = {[G_{k} (i_{k}, j_{k})]}_{b_{k - 1}, b_{k}}$ ⁠. The sum runs over b_k ∈ {1, …, D_k}. It can also be shown that a representation with equality in Eq. (16) always exists (see, for example, Ref. 13). If the linear operator ρ represented by an MPO is a quantum state, it is desirable to ensure that ρ is positive semi-definite. However, deciding whether a given MPO represents a positive semi-definite operator is an NP-hard problem in the number of parameters of the representation,¹⁸ i.e., a numerical solution in polynomial (in n) time may not be obtained. As an alternative, one can use a PMPS (locally purified MPS) representation of the mixed state. A PMPS representation represents a positive semidefinite linear operator by definition. PMPS representations are also called evidently positive representations and they are introduced in Subsection 5 of the Appendix.

Suppose that a quantum state $ρ \in D (H_{1, \dots, n})$ was prepared via quantum operations $W_{k} : B (H_{k - 1}) \to B (H_{k - 1, k})$ ⁠, i.e.,

ρ = W_{n} W_{n - 1} \dots W_{3} W_{2} (σ),

(18)

where $σ \in D (H_{1})$ ⁠. Clearly, this is an efficient representation of the quantum state ρ as it is described by at most nd⁶ parameters. It is known that such a representation can be efficiently, i.e., with at most poly(n) computational time, converted into an MPO representation or a PMPS representation.^17,18 Subsection 5 of the Appendix provides the details of the conversion and of the PMPS (locally purified MPS) representation. The state recovery and reconstruction techniques presented in Sec. VI provide a representation of the reconstructed state which is similar to Eq. (18). Lemma 38 in the Appendix provides PMPS and MPO representations of the recovered state for techniques based on the Petz recovery map and an MPO representation of the reconstructed state for state reconstruction results.

The structure of MPO and PMPS representations was used in Ref. 27 in the analysis of translationally invariant (TI) states of infinite spin chains. MPOs and PMPSs correspond to finitely correlated states (FCS) and C*-FCS, respectively, where the infinite chain has been replaced by a finite chain and where the requirement of translational invariance has been dropped (Definitions 2.2 and 2.4 in Ref. 27). In classical probability theory, a similar structure, called a hidden Markov model (HMM), has been used already in 1957 (see Ref. 28 and references therein). For more information about other similar structures, refer to Ref. 18.

C. The Petz recovery map

The (quantum) relative entropy, for two quantum states $ρ, σ \in D (A)$ ⁠, was defined by Umegaki²⁹ as

D (ρ ‖ σ) = T r (ρ \log (ρ) - ρ \log (σ)),

(19)

if supp ρ ⊆ supp σ, and is set equal to +∞ otherwise. For a bipartite quantum state $ρ \equiv ρ_{A B} \in D (A B)$ ⁠, the mutual information between the subsystems A and B is defined in terms of the von Neumann entropies of ρ_AB and its reduced states ρ_A = Tr_Bρ_AB and ρ_B = Tr_Aρ_AB,

I {(A : B)}_{ρ} = S {(A)}_{ρ} + S {(B)}_{ρ} - S {(A B)}_{ρ} .

(20)

It can also be expressed in terms of the relative entropy as follows:

I {(A : B)}_{ρ} = D (ρ ‖ ρ_{A} \otimes ρ_{B}) .

(21)

The quantum conditional mutual information (QCMI) of a tripartite quantum state $ρ \in D (A B C)$ is given by

I (A : C | B) ≔ I (A : B C) - I (A : B) = I (A B : C) - I (B : C)

(22)

and is expressed in terms of the von Neumann entropy as follows:

I (A : C | B) = S (A B) + S (B C) - S (A B C) - S (B) .

(23)

As mentioned in the Introduction, a fundamental property of the quantum relative entropy is its monotonicity under quantum operations. This is given by the data processing inequality (DPI): for quantum states $ρ, σ \in D (A)$ and a quantum operation $N$ acting on $D (A)$ ⁠,

D (ρ ‖ σ) \geq D (N (ρ) ‖ N (σ)) .

(24)

For the choice ρ = ρ_ABC, σ = ρ_AB ⊗ ρ_C, and $N = {T r}_{A} \otimes {i d}_{B C}$ ⁠, the DPI (24) implies that the QCMI of a tripartite state ρ_ABC is always non-negative. Using the definition (23) of the QCMI, we further infer that

S (A B C) + S (B) \leq S (A B) + S (B C),

(25)

which is the well-known strong subadditivity (SSA) property of the von Neumann entropy.

Equality in the DPI (24) was first discussed by Petz.⁸ A necessary and sufficient condition for equality in the DPI (24) was derived by Petz² and Hayden et al.³ and is stated in the following theorem.

Theorem 1 (Petz recovery map).

Let

ρ, σ \in D (A)

be quantum states and

N : A \to A^{'}

be a quantum operation. The equality

D (N (ρ) ‖ N (σ)) = D (ρ ‖ σ)

(26)

holds if and only if there is a quantum operation

R : A^{'} \to A

which satisfies

ρ = R N (ρ) a n d σ = R N (σ) .

(27)

If the above condition is satisfied, the so-called Petz recovery map

R_{σ, N}^{P}

satisfies the two equations. On the support of

N (ρ)

and for

ω \in B (A^{'})

⁠, this map is given by

R_{σ, N}^{P} (ω) = σ^{1 / 2} N^{*} (N {(σ)}^{- 1 / 2} ω N {(σ)}^{- 1 / 2}) σ^{1 / 2},

(28)

where

N^{*}

is the Hermitian adjoint of

N

in the Hilbert–Schmidt inner product.

Further, Hayden et al. derived the following necessary and sufficient condition on the structure of tripartite states satisfying equality in the SSA (25) (see Theorem 6 in Ref. 3).

Theorem 2.

Let

ρ \equiv ρ_{A B C} \in D (A B C)

be a tripartite quantum state. The equality I(A:BC)_ρ = I(A:B)_ρ [which is equivalent to equality in the SSA (25)] holds if and only if there is a decomposition of

H_{B}

into

H_{B_{L_{j}}}

and

H_{B_{R_{j}}}

as

H_{B} = \underset{j}{\oplus} H_{B_{L_{j}}} \otimes H_{B_{R_{j}}} .

(29)

such that ρ can be written as

ρ_{A B C} = \underset{j}{\oplus} p_{j} ρ_{A B_{L_{j}}} \otimes ρ_{B_{R_{j}} C},

for a probability distribution {p_j}, and sets of quantum states

{ρ_{A B_{L_{j}}}}

and

{ρ_{B_{R_{j}} C}}

⁠.

III. RECONSTRUCTION OF BIPARTITE STATES SUBJECTED TO LOCAL OPERATIONS

Let $ρ_{X Y} \in D (X Y)$ be a bipartite state and let $τ_{X^{'} Y^{'}} \in D (X^{'} Y^{'})$ be a state obtained from ρ_XY by the action of local operations

τ_{X^{'} Y^{'}} = (N_{X} \otimes N_{Y}) (ρ_{X Y}),

(30)

where $N_{X} : X \to X^{'}$ and $N_{Y} : Y \to Y^{'}$ denote quantum operations (or more generally, linear maps). We are interested in the conditions under which the original state ρ_XY can be reconstructed from $τ_{X^{'} Y^{'}}$ with local maps, i.e., $ρ_{X Y} = (R_{X^{'}} \otimes R_{Y^{'}}) (τ_{X^{'} Y^{'}})$ with reconstruction maps $R_{X^{'}} : X^{'} \to X$ and $R_{Y^{'}} : Y^{'} \to Y$ ⁠.

Our reconstruction scheme is particularly useful for states ρ with low operator Schmidt rank because then a reconstruction of ρ can be achieved with fewer measurements than required for standard quantum state tomography, as discussed in Remarks 18 and 29 (see also Sec. V A). The operator Schmidt rank of ρ is equal to the rank of the matrix M_ρ [Eqs. (7) and (9); M_ρ has size $d_{X}^{2} \times d_{Y}^{2}$ ]. Hence, in Sec. III A, we first consider the more general problem of reconstruction of low-rank matrices (which are not necessarily states). Section III B discusses the stability of our matrix reconstruction technique and Sec. III C shows how it can be used to reconstruct a quantum state.

A. Reconstruction of low-rank matrices

Suppose that we want to obtain a matrix $M \in C^{m \times n}$ but we only know the entries of the matrix products LM and MR where L and R are r × m and n × s complex matrices. We refer to LM, MR, and LMR as the marginals of the matrix M. Proposition 3 states that M can indeed be obtained from LM and MR if the condition rk(LMR) = rk(M) holds. This rank condition implies r, s ≥ rk(M). If the rank of M is much smaller than its maximal value, min{m, n}, this provides a way to obtain M from LM and MR which, taken together, have much fewer entries than M. If the matrices L and R are restricted to submatrices of permutation matrices, the matrix products LM and MR comprise selected rows and columns of M. In this case, Proposition 3 provides a reconstruction of a low-rank matrix M from few rows and columns (cf. Ref. 4).

Proposition 3.

Let

L \in C^{r \times m}

⁠,

M \in C^{m \times n}

⁠, and

R \in C^{n \times s}

be matrices. Then

r k (L M R) = r k (M) \Leftrightarrow \exists X \in C^{s \times r} : M = M R X L M .

(31)

If the condition is satisfied, M = MR X LM holds for any matrix X with CXC = C, C = LMR. The Moore–Penrose pseudoinverse X = (LMR)⁺ has the required property CXC = C.

Furthermore, rk(LM) = rk(M) implies rk(LMR) = rk(MR).

Proof.

“⇒” of Eq. (): Assume that rk(LMR) = rk(M) holds. The property CXC = C, C = LMR implies that LMRXu = u holds for all u ∈ im(LMR). Let q = rk(M) = rk(LMR). Let u₁, …, u_q be a basis of im(LMR) and set

v_{i}

= Xu_i,

w_{i}

=

{M R v}_{i}

⁠. The

v_{i}

are linearly independent because

{L M R v}_{i}

= LMRXu_i = u_i. The

w_{i}

are linearly independent because

L w_{i}

=

{L M R v}_{i}

= u_i. The

w_{i}

are a linearly independent sequence of length q = rk(M) and they satisfy

w_{i}

∈ im(M), i.e., they are a basis of im(M). Now observe

M R X L w_{i} = M R X u_{i} = M R v_{i} = w_{i} .

(32)

As a consequence, MR X L maps any vector from im(M) to itself. Accordingly, (MR X L)M = M holds.

rk(LM) = rk(M) implies rk(LMR) = rk(MR): The equality rk(LM) = rk(M) implies M = M(LM)⁺ LM [use the “⇒” direction of Eq. (31) for R = $1$ ]. As a consequence, MR = M(LM)⁺ LMR and rk(MR) ≤ rk(LMR) hold. The converse inequality rk(LMR) ≤ rk(MR) always holds and we arrive at rk(LMR) = rk(MR).

“⇐” of Eq. (31): Assume that M = MR X LM holds for some matrix X. The equality M = MR X LM implies rk(M) ≤ rk(MR) and rk(M) ≤ rk(LM). The converse inequalities rk(MR) ≤ rk(M) and rk(ML) ≤ rk(M) always hold. As a consequence, we have rk(LM) = rk(M) and rk(MR) = rk(M). Above, we saw that the former equality implies rk(LMR) = rk(MR) which, together with the latter equality rk(MR) = rk(M), proves the theorem.

Remark 4.

A violation of the rank condition rk(LMR) = rk(M) does not in general imply that there is no method to obtain M from LM and MR. As a trivial example, consider L = $1$ and R = 0. Then, the rank condition is violated for all M ≠ 0, but M is obtained trivially from LM = M.□

Remark 5

(Related work). Proposition 3 states that M can be obtained from LM and MR if rk(LMR) = rk(M) holds. Special cases of Proposition 3 have appeared before in several places. If r = s = rk(M) and L and R select exactly r = rk(M) rows and columns of M, the decomposition M = MR(LMR)⁻¹LM is known as skeleton decomposition of M.⁴ Decompositions of the form M = MR X LM where L and R select rows and columns of M are known as pseudoskeleton/CUR decomposition of M and it has been recognized that the truncated Moore–Penrose pseudoinverse $X = {(L M R)}_{τ}^{+}$ may provide a good approximation if r = s < rk(M) and suitable rows, columns, and threshold τ are chosen;⁴ we come back to the case of approximately low rank in Sec. III B. The case r = s = rk(M), X = (LMR)⁺ is contained in the results on tensor decompositions by Caiafa and Cichocki.²¹ This matrix decomposition with X = (LMR)⁺, restricted L and R but general r, s ≥ rk(M) forms the basis of MPO reconstruction¹ which is discussed in Sec. VI.

B. Stability of the reconstruction under perturbation

Suppose that we have a matrix S which satisfies the rank condition

r k (L S R) = r k (S),

(33)

for given matrices L and R. We want to reconstruct the perturbed matrix

M = S + E, ϵ = ∥ E ∥ / ∥ S ∥

(34)

and ϵ = ∥E∥/∥S∥ quantifies the magnitude of the perturbation E relative to the unperturbed matrix S. In Theorem 6, we provide a reconstruction ${\overset{̌}{M}}_{τ}$ and show that it is close to M if the magnitude ϵ of the perturbation E is small enough. A bound on the distance in operator norm between the reconstruction ${\overset{̌}{M}}_{τ}$ and M is provided by

∥ {\overset{̌}{M}}_{τ} - M ∥ \leq ∥ {\overset{̌}{M}}_{τ} - S ∥ + ∥ S - M ∥ \leq ∥ {\overset{̌}{M}}_{τ} - S ∥ + ϵ ∥ S ∥

(35)

and Theorem 6 provides a bound on $∥ {\overset{̌}{M}}_{τ} - S ∥$ ⁠.

Recall that given a matrix M, we define $M_{τ}^{+} = {(M_{τ})}^{+}$ and M_τ is given by M with singular values smaller or equal to τ replaced by zero.

Theorem 6.

Let

M = S + E

where

M

⁠,

S

and

E

are matrices. Let rk(S) = rk(LSR), η = ∥L∥∥S∥∥R∥, γ = σ_min(LSR)/η, ϵ = ∥E∥/∥S∥. Let γ > 2ϵ and ϵ ≤ τ < γ − ϵ. Then

∥ {\overset{̌}{M}}_{τ} - S ∥ \leq \frac{∥ S ∥}{γ - ϵ} (\frac{4 ϵ}{γ} + 2 ϵ + ϵ^{2}), \leq \frac{7 ϵ ∥ S ∥}{γ (γ - ϵ)}, {\overset{̌}{M}}_{τ} = M R {(L M R)}_{η τ}^{+} L M .

Proof.

We prove the proposition for ∥L∥ = ∥S∥ = ∥R∥ = 1 (without loss of generality as explained in Subsection 2 of the Appendix). We have M = S + E, ϵ = ∥E∥, γ = σ_min(LSR), and LMR = LSR + LER with ∥LER∥ ≤ ϵ. We insert S = SR(LSR)⁺LS and use Lemma 7 (provided at the end of this subsection):

∥ M R {(L M R)}_{τ}^{+} L M - S ∥

(36)

\leq ∥ {(L M R)}_{τ}^{+} - {(L S R)}^{+} ∥ + 2 ∥ {(L M R)}_{τ}^{+} ∥ ϵ + ∥ {(L M R)}_{τ}^{+} ∥ ϵ^{2}

(37)

\leq \frac{4 ϵ}{γ (γ - ϵ)} + \frac{2 ϵ}{γ - ϵ} + \frac{ϵ^{2}}{γ - ϵ} \leq \frac{7 ϵ}{γ (γ - ϵ)} .

(38)

Note that by premise, we have ϵ < γ ≤ 1. As a consequence,

1 \leq \frac{1}{γ}

and ϵ ≤ 1 (which were used in the last equation) hold. This proves the theorem.

For the interpretation of the theorem, it is convenient to use the case with ∥L∥ = ∥S∥ = ∥R∥ = 1 and η = 1. Theorem 6 shows that the reconstruction ${\overset{̌}{M}}_{τ}$ reconstructs the low-rank component S of M = S + E up to a small error if the smallest singular value γ of the low-rank component LSR is much larger than the norm ϵ of the noise component. In addition, the threshold τ must be chosen larger than the noise norm ϵ but smaller than γ − ϵ. Subsection 1 of the Appendix discusses examples which show that the bound from Theorem 6 is optimal up to constants and that the reconstruction error can diverge as ϵ approaches zero if small singular values in LMR are not truncated.

Choosing a suitable threshold τ is equivalent to estimating the rank of the low rank contribution S. If the rank and support of S are known, the measurements L and R can be chosen such that LSR becomes invertible. For this special case, an upper bound on the reconstruction error has been given by Caiafa and Cichocki.²¹ Their bound also includes constants that depend on LSR and may diverge as γ approaches zero. In Subsection 3 of the Appendix, we generalize their approach to our more general setting and obtain a bound that is similar to Theorem 6.

The following lemma was used in the proof of Theorem 6:

Lemma 7.

Let

A, B, F \in C^{m \times n}

⁠. Let γ = σ_min(A), B = A + F with ∥F∥ ≤ ϵ. Let γ > 2ϵ and choose τ such that ϵ ≤ τ < γ − ϵ. Then σ_min(B_τ) ≥ γ − ϵ,

∥ B_{τ}^{+} ∥ \leq 1 / (γ - ϵ)

⁠, B_τ = B_ϵ, and ∥B − B_τ∥ ≤ ϵ. In addition

∥ B_{τ}^{+} - A^{+} ∥ \leq \frac{4 ϵ}{γ (γ - ϵ)} .

(39)

Proof.

The singular values of B = A + F satisfy (see, e.g., Ref. 30)

| σ_{i} (A + F) - σ_{i} (A) | \leq σ_{1} (F) \leq ϵ

and therefore, with r = rk(A), we obtain

σ_{1} (B) \geq \dots \geq σ_{r} (B) \geq γ - ϵ > τ \geq ϵ \geq σ_{r + 1} (B) \geq \dots \geq 0 .

This shows already everything except the inequality in Eq. (). To show the latter one, we use ∥X⁺ − A⁺∥ ≤2 ∥X⁺∥∥A⁺∥∥X − A∥.^30,31 Inserting ∥B_τ − A∥ ≤ ∥B_τ − B∥ + ∥B − A∥ shows the desired inequality.

C. Reconstruction of bipartite states

Let $ρ \in D (X Y)$ be a bipartite quantum state and let $τ \in D (X^{'} Y^{'})$ be a state obtained from it by the action of local operations

τ = (N_{X} \otimes N_{Y}) (ρ),

(40)

where $N_{X} : X \to X^{'}$ and $N_{Y} : Y \to Y^{'}$ denote quantum operations. We are interested in the conditions under which the original state ρ can be reconstructed from τ with local quantum operations, i.e., $ρ = (R_{X^{'}} \otimes R_{Y^{'}}) τ$ with $R_{X^{'}} : X^{'} \to X$ and $R_{Y^{'}} : Y^{'} \to Y$ ⁠. This question can be answered with the matrix decomposition from Sec. III A without using the positivity properties of ρ, $N_{X, Y}$ ⁠, and $R_{X^{'}, Y^{'}}$ ⁠. The result is provided by the following theorem:

Theorem 8.

Let

ρ \in B (X Y)

be a linear operator and let

N_{X} \in B (B (X); B (X^{'}))

⁠,

N_{Y} \in B (B (Y); B (Y^{'}))

be linear maps. Let

τ = (N_{X} \otimes N_{Y}) (ρ)

⁠. There are local linear maps

R_{X^{'}}

⁠,

R_{Y^{'}}

which reconstruct the original operator ρ from τ, i.e.,

ρ = (R_{X^{'}} \otimes R_{Y^{'}}) (τ),

(41)

if and only if the following equality holds:

O S R {(X^{'} : Y^{'})}_{τ} = O S R {(X : Y)}_{ρ} .

(42)

If the condition is satisfied, the following linear maps

R_{X^{'}} \in B (B (X^{'}); B (X))

and

R_{Y^{'}} \in B (B (Y^{'}); B (Y))

satisfy Eq. ():

R_{X^{'}} = R_{M_{ρ} N_{Y}^{⊺}, N_{X}}^{M}, R_{Y^{'}} = R_{M_{ρ}^{⊺} N_{X}^{⊺}, N_{Y}}^{M}, R_{L, N}^{M} = L {(N L)}^{+} .

(43)

The operators

(i d \otimes N_{Y}) (ρ)

and

(N_{X} \otimes i d) (ρ)

are sufficient to construct the two

R_{L, N}^{M}

maps if

N_{X}

and

N_{Y}

are known. The superscript M indicates that the reconstruction map is based on matrix reconstruction. If the condition is satisfied, the following equation also holds for

R_{X^{'}}

and

R_{Y^{'}}

from Eq. ():

ρ = (R_{X^{'}} N_{X} \otimes i d) (ρ) = (i d \otimes R_{Y^{'}} N_{Y}) (ρ) .

(44)

Remark 9.

If the rank condition (42) is satisfied and $σ^{(X Y^{'})} = (i d \otimes N_{Y}) (ρ)$ and $σ^{(X^{'} Y)} = (N_{X} \otimes i d) (ρ)$ are given, one can obtain ρ by computing the maps $R_{X^{'}}$ and $R_{Y^{'}}$ [Eq. (43)], $τ = N_{X} \otimes i d (σ^{(X^{'} Y)})$ and $ρ = (R_{X^{'}} \otimes R_{Y^{'}}) (τ)$ ⁠. More directly, one can also compute only the map $R_{X^{'}}$ followed by computing $ρ = (R_{X^{'}} \otimes i d) (σ^{(X^{'} Y)})$ [Eq. (44)]. The two options correspond to using either Eq. (45b) or (45c) to obtain $M_{ρ}$ ⁠.□

Proof (of Theorem 8).

The operator τ is given by $τ = (N_{X} \otimes N_{Y}) (ρ)$ ⁠, therefore $O S R {(X^{'} : Y^{'})}_{τ} \leq O S R {(X : Y)}_{ρ}$ always holds (Corollary 11). Let Eq. (41) hold. Again by Corollary 11, the converse inequality $O S R {(X : Y)}_{ρ} \leq O S R {(X^{'} : Y^{'})}_{τ}$ also holds. As a consequence, the two operator Schmidt ranks must be equal.

Let the rank condition () hold. Using Lemma 10 and

τ = (N_{X} \otimes N_{Y}) (ρ)

⁠, we obtain

{[(R_{X^{'}} \otimes R_{Y^{'}}) (τ)]}_{i j} = {[R_{X^{'}} M_{τ} R_{Y^{'}}^{⊺}]}_{i j}

(45a)

= {[M_{ρ} N_{Y}^{⊺} {(N_{X} M_{ρ} N_{Y}^{⊺})}^{+} (N_{X} M_{ρ} N_{Y}^{⊺}) {(N_{X} M_{ρ} N_{Y}^{⊺})}^{+} N_{X} M_{ρ}]}_{i j}

(45b)

= {[M_{ρ} N_{Y}^{⊺} {(N_{X} M_{ρ} N_{Y}^{⊺})}^{+} N_{X} M_{ρ}]}_{i j}

(45c)

= {[M_{ρ}]}_{i j} .

(45d)

In Eqs. () and (), we used Lemma 10 and inserted the maps from Eq. (). In Eq. (), we used the property A⁺AA⁺ = A⁺ of the Moore–Penrose pseudoinverse. In Eq. (), we applied the matrix reconstruction result from Proposition 3. The therefor needed rank condition

r k (N_{X} M_{ρ} N_{Y}^{⊺}) = r k (M_{ρ})

is equivalent to the rank condition () because of

τ = (N_{X} \otimes N_{Y}) (ρ)

and Lemma 10. This shows that Eq. () holds if Eq. () is assumed and the maps from Eq. () are inserted. Equation () can be shown by omitting

R_{Y^{'}}

(or

R_{X^{'}}

⁠) from left-hand side of Eq. (). This finishes the proof of the theorem.

The remainder of the section provides the ingredients used in the preceding proof. It also provides a data processing inequality (DPI) for the operator Schmidt rank which is used below.

Lemma 10.

Let

ρ \in B (X Y)

be a linear operator and let

N_{X} : B (X) \to B (X^{'})

⁠,

N_{Y} : B (Y) \to B (Y^{'})

⁠, be linear maps. Set

τ = (N_{X} \otimes N_{Y}) (ρ)

⁠. Then

{[M_{τ}]}_{i j} = {[N_{X} M_{ρ} N_{Y}^{⊺}]}_{i j},

(46)

where

M_{τ} \in B (B (Y^{'}); B (X^{'}))

and

M_{ρ} \in B (B (Y); B (X))

are the linear maps with the same entries as τ and ρ [as defined in Eq. ()].

Proof.

The basic idea of the proof is shown in Fig. 1. It is given by

{[M_{τ}]}_{i j} = {[τ]}_{i j} = {[(N_{X} \otimes N_{Y}) (ρ)]}_{i j} \overset{(*)}{=} \sum_{k l} {[N_{X}]}_{i k} {[N_{Y}]}_{j l} {[ρ]}_{k l} = {[N_{X} M_{ρ} N_{Y}^{⊺}]}_{i j} .

(47)

In the following, we provide a formal derivation of the equation marked with an asterisk. First, note that

[(F_{k}^{(X)} {\tilde{F}}_{k}^{(X)}) \otimes (F_{l}^{(Y)} {\tilde{F}}_{l}^{(Y)})] (ρ) = (F_{k}^{(X)} \otimes F_{l}^{(Y)}) [({\tilde{F}}_{k}^{(X)} \otimes {\tilde{F}}_{l}^{(Y)}) (ρ)]

(48a)

= (F_{k}^{(X)} \otimes F_{l}^{(Y)}) {[ρ]}_{k l} .

(48b)

The proof of the equation marked with an asterisk involves several basic steps:

{[(N_{X} \otimes N_{Y}) (ρ)]}_{i j} = ⟨ F_{i}^{(X^{'})} \otimes F_{j}^{(Y^{'})}, (N_{X} \otimes N_{Y}) (ρ) ⟩

(49a)

= \sum_{k l} ⟨ F_{i}^{(X^{'})} \otimes F_{j}^{(Y^{'})}, (N_{X} \otimes N_{Y}) ([(F_{k}^{(X)} {\tilde{F}}_{k}^{(X)}) \otimes (F_{l}^{(Y)} {\tilde{F}}_{l}^{(Y)})] (ρ)) ⟩

(49b)

= \sum_{k l} ⟨ F_{i}^{(X^{'})}, N_{X} (F_{k}^{(X)}) ⟩ ⟨ F_{j}^{(Y^{'})}, N_{Y} (F_{l}^{(Y)}) ⟩ {[ρ]}_{k l}

(49c)

= \sum_{k l} {[N_{X}]}_{i k} {[N_{Y}]}_{j l} {[ρ]}_{k l} .

(49d)

This completes the proof of the lemma.

In the Introduction, we saw that the operator Schmidt rank is given by $O S R {(X : Y)}_{ρ} = r k (M_{ρ})$ where $M_{ρ} \in B (B (Y); B (X))$ is a linear map. As corollary from Lemma 10, we obtain the monotonicity of the operator Schmidt rank under local maps, i.e., a data processing inequality.

FIG. 1.

View large Download slide

Proof sketch for Lemma 10. The symbol “≅” indicates that two objects have the same entries, e.g., $ρ ≅ M_{ρ}$ because ${[ρ]}_{k l} = {[M_{ρ}]}_{k l}$ ⁠.

Corollary 11.

Let ρ,

N_{X}

⁠,

N_{Y}

and

τ = (N_{X} \otimes N_{Y}) (ρ)

as in Lemma 10. Then,

O S R {(X^{'} : Y^{'})}_{τ} \leq O S R {(X : Y)}_{ρ}

(50)

and

O S R {(X^{'} : Y^{'})}_{τ} \leq min {r k (N_{X}), r k (N_{Y})} .

(51)

Proof.

Use the property $O S R {(X : Y)}_{ρ} = r k (M_{ρ})$ [Eq. (9)], the identity $r k (M_{τ}) = r k (N_{X} M_{ρ} N_{Y}^{⊺})$ (Lemma 10) and the rank inequality rk(AB) ≤ min{rk(A), rk(B)} for arbitrary matrices or linear maps A and B.

IV. PETZ RECOVERY OF BIPARTITE STATES SUBJECTED TO LOCAL QUANTUM OPERATIONS

In Sec. III, we considered a linear operator $ρ \in B (X Y)$ subjected to local linear maps $N_{X} \in B (B (X); B (X^{'}))$ and $N_{Y} \in B (B (Y); B (Y^{'}))$ ⁠,

τ = (N_{X} \otimes N_{Y}) (ρ) .

(52)

In Sec. III C, we presented a condition under which ρ can be reconstructed from τ via local linear maps. Here, we discuss the same question for a bipartite quantum state $ρ \equiv ρ_{X Y} \in D (X Y)$ and quantum operations $N_{X}$ and $N_{Y}$ ⁠. The answer is obtained by restricting Theorem 1 to the bipartite setting, i.e., by inserting ρ = ρ_XY, σ = ρ_X ⊗ ρ_Y, and $N = N_{X} \otimes N_{Y}$ ⁠.⁹

Corollary 12 (Bipartite Petz recovery map⁹).

Let

ρ \in B (X Y)

be a quantum state and

N_{X} : X \to X^{'}

⁠,

N_{Y} : Y \to Y^{'}

be quantum operations. Set

τ = (N_{X} \otimes N_{Y}) (ρ)

⁠. The equality

I {(X^{'} : Y^{'})}_{τ} = I {(X : Y)}_{ρ}

(53)

holds if and only if there are quantum operations

R_{X^{'}} : X^{'} \to X

and

R_{Y^{'}} : Y^{'} \to Y

which satisfy

ρ = (R_{X^{'}} \otimes R_{Y^{'}}) (τ) .

(54)

If the condition is satisfied, the two Petz recovery maps

R_{X^{'}} = R_{ρ_{X}, N_{X}}^{P}

and

R_{Y^{'}} = R_{ρ_{Y}, N_{Y}}^{P}

satisfy the equation.

In Sec. V, we explore the relation between bipartite state reconstruction (Theorem 8) and bipartite Petz recovery (Corollary 12).

V. COMPARISON OF PETZ RECOVERY AND STATE RECONSTRUCTION

In this section, we compare Petz recovery with state reconstruction for a bipartite quantum state $ρ \in D (X Y)$ subject to local quantum operations $N_{X} : X \to X^{'}$ and $N_{Y} : Y \to Y^{'}$ ⁠,

τ = (N_{X} \otimes N_{Y}) (ρ) .

(55)

The reconstruction is to be achieved via local linear maps

ρ = (R_{X^{'}} \otimes R_{Y^{'}}) (τ) .

(56)

State reconstruction and the Petz recovery map both provide maps $R_{X^{'}}$ and $R_{Y^{'}}$ under the assumption of different conditions on ρ and τ (Theorem 8 and Corollary 12). There is the following evident relation between state reconstruction and Petz recovery.

Theorem 13.

Let

ρ \in D (X Y)

a quantum state and let

N_{X} : X \to X^{'}

⁠,

N_{Y} : Y \to Y^{'}

quantum operations. Let

τ = (N_{X} \otimes N_{Y}) (ρ)

⁠. Then

I {(X^{'} : Y^{'})}_{τ} = I {(X : Y)}_{ρ} \Rightarrow O S R {(X^{'} : Y^{'})}_{τ} = O S R {(X : Y)}_{ρ} .

(57)

The converse implication does not hold. The premise of the implication (57) is equivalent to Petz recovery being possible, i.e., there are CPTP maps

R_{X^{'}}

and

R_{Y^{'}}

which recover ρ from τ (Corollary 12),

ρ = (R_{X^{'}} \otimes R_{Y^{'}}) (τ) .

(58)

CPTP maps which recover ρ in this way are given by the Petz recovery maps

R_{X^{'}} = R_{ρ_{X}, N_{X}}^{P}

and

R_{Y^{'}} = R_{ρ_{Y}, N_{Y}}^{P}

(see Theorem 1). The conclusion of the implication (57) is equivalent to state reconstruction being possible, i.e., there are linear maps

R_{X^{'}}

and

R_{Y^{'}}

which reconstruct ρ from τ (Theorem 8),

ρ = (R_{X^{'}} \otimes R_{Y^{'}}) (τ) .

(59)

Linear maps which recover ρ in this way are given by the reconstruction maps

R_{X^{'}} = R_{M_{ρ} N_{Y}^{⊺}, N_{X}}^{M}

and

R_{Y^{'}} = R_{M_{ρ}^{⊺} N_{X}^{⊺}, N_{Y}}^{M}

(see Theorem 8).

Proof.

The premise of Eq. (57) implies that the CPTP maps from Eq. (58) exist (Corollary 12). These CPTP maps are linear maps that satisfy Eq. (59), which in turn implies that the conclusion of Eq. (57) holds (Theorem 8).

A counterexample for the converse implication of Eq. (57) will be provided in Sec. V A.

Remark 14.

Suppose that the conclusion of Eq. (57) holds while its premise does not hold. If both linear maps $R_{X^{'}}$ and $R_{Y^{'}}$ were CPTP, the equality $I {(X^{'} : Y^{'})}_{ρ} \leq I {(X : Y)}_{τ}$ would be implied by Eq. (59) and $I {(X^{'} : Y^{'})}_{ρ} = I {(X : Y)}_{τ}$ would follow [since the converse inequality always holds because of $τ = (N_{X} \otimes N_{Y}) ρ$ ]. This would contradict our assumption; i.e., at least one of $R_{X^{'}}$ and $R_{Y^{'}}$ is not CPTP. For example, the reconstruction maps for the W state on four qubits (Sec. V A) are non-positive.□

Theorem 13 implies that any state which admits Petz recovery also admits state reconstruction. Table I compares the quantities on which the state reconstruction maps and the Petz recovery maps depend: Both methods require knowledge of the quantum operations $N_{X}$ and $N_{Y}$ ⁠. The marginal states ρ_X and ρ_Y are sufficient for computing the Petz recovery maps. To compute the state reconstruction maps, the states $(i d \otimes N_{Y}) (ρ) \in D (X Y^{'})$ and $(N_{X} \otimes i d) (ρ) \in D (X^{'} Y)$ are needed. The marginals ρ_X, ρ_Y, $τ_{X^{'}} \equiv N_{X} (ρ_{X})$ ⁠, and $τ_{Y^{'}} \equiv N_{Y} (ρ_{Y})$ (which are also needed for the recovery maps) can be inferred from the states $(i d \otimes N_{Y}) (ρ)$ and $(N_{X} \otimes i d) (ρ)$ ⁠. However, in addition, these states contain correlations between the systems X and Y′ and between X′ and Y, respectively. This means that state reconstruction requires more input data for a reconstruction of ρ than state recovery. On the other hand, Theorem 13 and the examples in Subsection V A show that state reconstruction is successful for a strictly larger set of states than Petz recovery.

TABLE I.

Necessary input data for a recovery or reconstruction of a quantum state $ρ \in D (X Y)$ from $τ = (N_{X} \otimes N_{Y}) (ρ)$ ⁠. In both cases, ρ is obtained from $ρ = (R_{X^{'}} \otimes R_{Y^{'}}) (τ)$ ⁠.

Method	$R_{X^{'}}$ depends on	$R_{Y^{'}}$ depends on
Petz recovery	ρ_X, $N_{X}$	ρ_Y, $N_{Y}$
Reconstruction	$(i d \otimes N_{Y}) (ρ)$ ⁠, $N_{X}$	$(N_{X} \otimes i d) (ρ)$ ⁠, $N_{Y}$

A. Comparison for four-partite systems

After the comparison of state reconstruction and Petz recovery for bipartite quantum systems, we apply this result to the more specific case of quantum systems which comprise four subsystems. Specifically, we consider four systems A, B, C, and D. We insert the partial trace Tr_A: AB → B for $N_{X} : X \to X^{'}$ and the partial trace Tr_D: CD → C for $N_{Y} : Y \to Y^{'}$ in Theorem 13. Accordingly, reconstruction of $ρ \in D (A B C D)$ from its reduced state ρ_BC = Tr_AD(ρ) is achieved with maps $R_{B} \in B (B (B); B (A B))$ and $R_{C} \in B (B (C); B (C D))$ as

ρ = (R_{B} \otimes R_{C}) (ρ_{B C}) .

(60)

A straightforward application of Theorem 13 provides the implication

I {(B : C)}_{ρ} = I {(A B : C D)}_{ρ} \Rightarrow O S R {(B : C)}_{ρ} = O S R {(A B : C D)}_{ρ} .

(61)

Since the W state on four qubits satisfies the conclusion of the last equation but not its premise, it constitutes a counterexample for the converse implication. In addition, it provides a counterexample for the converse implication of Eq. (57) in Theorem 13. On n qubits, the W state is given by

W_{n} = |w_{n}⟩ ⟨w_{n}|, | w_{n} ⟩) = \frac{1}{\sqrt{n}} [| 10 \dots 0 ⟩) + | 010 \dots 0 ⟩) + \dots + | 0 \dots 01 ⟩)]

(62)

and the operator Schmidt rank and mutual information values of the four-qubit W state W_ABCD are provided in Table III on p. 16.

Above, we presented one possible application of bipartite Petz recovery (Corollary 12) to a quadripartite system. It turns out that Petz recovery can be applied to a quadripartite system in three different ways. The first row of Table II corresponds to the application of state reconstruction and Petz recovery to a quadripartite system as presented above. Rows two and three of Table II present two different ways to apply Petz recovery to a quadripartite system. In total, we have one possible application of state reconstruction and three possible applications of Petz recovery and for each application, there is a condition for successful reconstruction/recovery. These conditions read as follows:

TABLE II.

Possible applications of state reconstruction and Petz recovery (Theorem 13) to a quadripartite system. Corresponding conditions for successful recovery are stated in Lemma 15.

X → X′	Y → Y′	$N_{X}$	$N_{Y}$	Condition
AB → B	CD → C	Tr_A	Tr_D	(C1) and (C2)
AB → B	CD → CD	Tr_A	id	(C3)
ABC → BC	D → D	Tr_A	id	(C4)

X → X′	Y → Y′	$N_{X}$	$N_{Y}$	Condition
AB → B	CD → C	Tr_A	Tr_D	(C1) and (C2)
AB → B	CD → CD	Tr_A	id	(C3)
ABC → BC	D → D	Tr_A	id	(C4)

Lemma 15.

Given a quantum state

ρ \in D (A B C D)

⁠, consider the equations

O S R {(B : C)}_{ρ} = O S R {(A B : C D)}_{ρ},

(C1)

I {(B : C)}_{ρ} = I {(A B : C D)}_{ρ},

(C2)

I {(A : B)}_{ρ} = I {(A : B C D)}_{ρ},

(C3)

I {(A : B C)}_{ρ} = I {(A : B C D)}_{ρ} .

(C4)

The following implications hold but the converse implications do not:

(C2) \Rightarrow (C1), (C2) \Rightarrow (C3), (C3) \Rightarrow (C4) .

(63)

Proof.

The implication (C2) ⇒ (C1) follows from Theorem 13 with the substitutions given in the first row of Table II.

Equation (C2) ⇒ Eq. (C3): The inequality I(B : C)_ρ ≤ I(B : CD)_ρ ≤ I(AB : CD)_ρ always holds, therefore I(B : C)_ρ = I(AB : CD)_ρ implies I(B : CD)_ρ = I(AB : CD)_ρ. The latter can be written with the conditional mutual information as I(A : CD|B) = 0 [Eq. (22)]. The CMI in turn is also equal to I(A : CD|B) = I(A : BCD) − I(A : B), which shows the desired equality I(A : B) = I(A : BCD).

Equation (C3) ⇒ Eq. (C4): The inequality I(A : B)_ρ ≤ I(A : BC)_ρ ≤ I(A : BCD)_ρ always holds, therefore I(A : B)_ρ = I(A : BCD)_ρ implies I(A : BC)_ρ = I(A : BCD)_ρ.

Table III contains states which show that the converse implications do not hold. The states are constructed from the n-qubit W state from Eq. () and the GHZ and classical GHZ states on n qubits

{G H Z}_{n} = |{GHZ}_{n}⟩ ⟨{GHZ}_{n}|, | {GHZ}_{n} ⟩) = \frac{1}{\sqrt{2}} [| 0 \dots 0 ⟩) + | 1 \dots 1 ⟩)],

(64a)

{c G H Z}_{n} = \frac{1}{2} [|0 \dots 0⟩ ⟨0 \dots 0| + |1 \dots 1⟩ ⟨1 \dots 1|] .

(64b)

The values of the operator Schmidt rank and mutual information given in Table III show that the converse implications do not hold.

The relations between Eqs. (C1)–(C4) from Lemma 15 are illustrated in Fig. 2. The figure also shows which of the conditions are satisfied by the example states from Table III. For example, the W state W_ABCD on four qubits does not satisfy (C2)–(C4). We can understand that W_ABCD cannot satisfy (C4) by considering the following known result: If (C4) holds, then Theorem 2 tells us that the reduced state ρ_AD must be a separable state. However, the reduced state Tr_BC(W_ABCD) has a non-positive semidefinite partial transpose and therefore is inseparable, i.e., entangled:^32,33 The entanglement in the reduced state on AD mandates that Eq. (C4) is not satisfied.

Figure 2 illustrates that reconstruction and the different applications of Petz recovery work for different subsets of all quadripartite states but one should not forget that they also require different reduced states of ρ in order to recover $ρ \in D (A B C D)$ ⁠. Table IV shows the necessary reduced states for each case. In all four cases, the full state $ρ \in D (A B C D)$ can be reconstructed from marginal states on only two or three of the systems. Each scheme enables quantum state tomography with incomplete information (i.e., the necessary marginals) if the corresponding condition is assumed to hold. Each scheme also relies on the fact that correlations as measured by the operator Schmidt rank or the mutual information are less than maximal; this restriction is imposed by the conditions (C1)–(C4).

Table III shows that the state

ρ \in D (A B C D)

can be obtained from ρ_ABC and ρ_BCD with reconstruction under () but recovery under () requires only the marginal states on AB, BC, and CD. This prompts the question whether smaller marginal states are sufficient to reconstruct a state under the reconstruction condition (). For example, one could hope to obtain ρ from ρ_AB and ρ_BCD but the following two states σ₊ and σ₋ show that this is not possible:

σ_{\pm} = \frac{1}{32} (21 \otimes 1 \otimes 1 + (1 \pm Z) \otimes Z \otimes Z) \otimes 1, 1 = (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}), Z = (\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}) .

(65)

The states σ₊ and σ₋ have the same marginals on AB and BCD but they do satisfy ().³⁴ As a consequence, ρ_AB and ρ_BCD are not sufficient to obtain ρ under () and it is now apparent that ρ_ABC and ρ_BCD are necessary to reconstruct a state under that condition.³⁵

TABLE III.

Log-operator Schmidt rank κ(X:Y) = log₂(OSR(X:Y)) and mutual information^a I(X:Y) of the states shown in Fig. 2. Equation (C1)–(C4) are defined in Lemma 15.

	κ(B:C)	κ(AB:CD)	I(B:C)	I(AB:CD)	I(A:B)	I(A:BC)	I(A:BCD)	C1	C2	C3	C4
cGHZ_ABCD	1	1	1	1	1	1	1	✓	✓	✓	✓
ρ_A ⊗ W_BCD	2	2	≈0.92	≈1.84	0	0	0	✓	–	✓	✓
W_ABC ⊗ ρ_D	2	2	≈0.92	≈1.84	≈0.92	≈1.84	≈1.84	✓	–	–	✓
W_ABCD	2	2	≈0.62	2	≈0.62	1	≈1.62	✓	–	–	–
ρ_A ⊗ GHZ_BCD	1	2	1	2	0	0	0	–	–	✓	✓
GHZ_ABC ⊗ ρ_D	1	2	1	2	1	2	2	–	–	–	✓
GHZ_ABCD	1	2	1	2	1	1	2	–	–	–	–

	κ(B:C)	κ(AB:CD)	I(B:C)	I(AB:CD)	I(A:B)	I(A:BC)	I(A:BCD)	C1	C2	C3	C4
cGHZ_ABCD	1	1	1	1	1	1	1	✓	✓	✓	✓
ρ_A ⊗ W_BCD	2	2	≈0.92	≈1.84	0	0	0	✓	–	✓	✓
W_ABC ⊗ ρ_D	2	2	≈0.92	≈1.84	≈0.92	≈1.84	≈1.84	✓	–	–	✓
W_ABCD	2	2	≈0.62	2	≈0.62	1	≈1.62	✓	–	–	–
ρ_A ⊗ GHZ_BCD	1	2	1	2	0	0	0	–	–	✓	✓
GHZ_ABC ⊗ ρ_D	1	2	1	2	1	2	2	–	–	–	✓
GHZ_ABCD	1	2	1	2	1	1	2	–	–	–	–

^a

The exact values of the numerical constants are $2 \log (3) - \frac{4}{3} \approx 1.84$ ⁠, $4 - \frac{3}{2} \log (3) \approx 1.62$ ⁠, $\log (3) - \frac{2}{3} \approx 0.92$ ⁠, and $3 - \frac{3}{2} \log (3) \approx 0.62$ ⁠.

FIG. 2.

View large Download slide

State reconstruction vs. Petz recovery for a quadripartite system. The figure shows the relations between conditions (C1)–(C4) from Lemma 15 and several states whose position indicates which of the conditions are satisfied by each state, e.g., W_ABCD satisfies (C1) but it does not satisfy (C2)–(C4). ρ_A and ρ_D denote arbitrary states while the W, GHZ and classical GHZ states are defined in Eqs. (62) and (64).

TABLE IV.

Properties of the four reconstruction/recovery settings considered in Fig. 2. The last three columns Input, Domain/Range, and Depends on refer to the reconstruction map(s). See Table II for further details.

Method	Cond.	Input	Domain/Range	Depends on
Reconstruction	(C1)	ρ_BC	B → AB, C → CD	ρ_ABC, ρ_BCD
Recovery	(C2)	ρ_BC	B → AB, C → CD	ρ_AB, ρ_CD
Recovery	(C3)	ρ_BCD	B → AB	ρ_AB
Recovery	(C4)	ρ_BCD	BC → ABC	ρ_ABC

Method	Cond.	Input	Domain/Range	Depends on
Reconstruction	(C1)	ρ_BC	B → AB, C → CD	ρ_ABC, ρ_BCD
Recovery	(C2)	ρ_BC	B → AB, C → CD	ρ_AB, ρ_CD
Recovery	(C3)	ρ_BCD	B → AB	ρ_AB
Recovery	(C4)	ρ_BCD	BC → ABC	ρ_ABC

VI. EFFICIENT RECONSTRUCTION OF STATES ON SPIN CHAINS VIA RECURSIVELY DEFINED MEASUREMENTS

Under suitable conditions, the state of a linear spin chain with n spins can be reconstructed from marginal states of few neighbouring spins with the Petz recovery map¹⁷ or with state reconstruction.¹ In Sec. VI A, we explore the relation between Petz recovery and state reconstruction in that setting. In Sec. VI B, we generalize both techniques to use long-range measurements instead of or in addition to short-ranged correlations found in marginal states of few neighbouring spins. This allows for the efficient recovery/reconstruction of a larger set of states, as is explained in the following.

a. Motivation for long-range measurements. Consider the following quantum states on n qubits:

{G H Z}_{α, n} = |{GHZ}_{α, n}⟩ ⟨{GHZ}_{α, n}|, | {GHZ}_{α, n} ⟩) = \frac{1}{\sqrt{2}} [| 0 \dots 0 ⟩) + e^{i α} | 1 \dots 1 ⟩)],

(66a)

{c G H Z}_{n} = \frac{1}{2} [|0 \dots 0⟩ ⟨0 \dots 0| + |1 \dots 1⟩ ⟨1 \dots 1|] .

(66b)

All states from the set $S_{n} = {{c G H Z}_{n}} \cup {{G H Z}_{α, n} : α \in R}$ have the same reduced state cGHZ_k on k < n qubits. No recovery or reconstruction method which receives only local reduced states as input can distinguish between the states from the set S_n and this is also the reason why no method could recover or reconstruct the four-qubit state |GHZ_0,4⟩ = |GHZ₄⟩ in Sec. V A. Note that the pure states |GHZ_α,n⟩ can be represented as an MPS with bond dimension two (because they are the superposition of two pure product states) and that all states from the set S_n can be represented as an MPO with bond dimension at most four (because they are the sum of at most four tensor product operators).¹³

We call an MPS representation efficient if its bond dimension is at most $D = O (p o l y (n))$ and a we call a tomography scheme efficient if expectation values of at most poly(n) simple observables are needed; a possible definition of a simple observable is provided in Remark 18. Standard quantum state tomography is not efficient because it requires ∼exp(n) expectation values. By contrast, it has been shown that any pure state which admits an efficient MPS representation can be determined efficiently from observables with a simple structure.³⁶ The tomography scheme from Ref. 14 is efficient for the states |GHZ_α,n⟩ but recovery/reconstruction methods based on local reduced states must fail for these states. In Sec. VI B, we extend both Petz recovery and state reconstruction in a way which allows the long-range measurements from Ref. 14 to be used and thus the states |GHZ_α,n⟩ to be reconstructed successfully. What is more, we show that there are mixed states that cannot be reconstructed from local reduced states but can be reconstructed from long-range measurements (Remark 31). This shows that Petz recovery and state reconstruction with long-range measurements can reconstruct more states than prior techniques (recovery/reconstruction from local reduced states and the tomography scheme from Ref. 14). Furthermore, state reconstruction can reconstruct any MPO of bond dimension D from ∼nD² expectation values of global tensor product observables, as has been shown in related prior work.¹⁹ We build upon that to show that successful, efficient Petz recovery with long-range measurements implies that efficient state reconstruction with long-range measurements is also possible (Theorem 32).

b. Prior work: MPO reconstruction.¹ Many physically interesting quantum states $ρ \in D (H_{1, \dots, n})$ can be represented efficiently, i.e., with poly(n) parameters, via an MPO representation.¹ However, standard quantum state tomography requires ∼exp(n) different expectation values in ρ to reconstruct ρ, even if ρ admits such an efficient MPO representation. As an improvement over that, it has been shown³⁷ that almost all states with an MPO representation of bond dimension D can be reconstructed from their reduced states on ∼log(D) neighbouring spins if a suitable reconstruction scheme is used.¹ We refer to this reconstruction scheme as MPO reconstruction and we rederive it in Theorem 17 as a consequence of our result on bipartite state reconstruction (Theorem 8).

c. Prior work: Cross approximation of tensor trains.¹⁹ Our generalization of state reconstruction to long-range measurements in Theorem 27 can be used to construct an MPO representation of the quantum state (Remark 29). An MPO representation of a quantum state $ρ \in D (H_{1, \dots, n})$ is exactly the same as a tensor train representation of ρ if the operator ρ is regarded as a vector from the tensor product vector space $B (H_{1}) \otimes \dots \otimes B (H_{n})$ ⁠. Reference 19 provides a means to reconstruct a tensor of low tensor train rank (i.e., an MPO of low bond dimension) from few entries. This procedure is called tensor train cross approximation. When applied to quantum states, tensor train cross approximation allows for the reconstruction of a quantum state from the expectation values of few tensor product observables. Theorem 27 is more general because it admits more general measurements; e.g., it also permits the measurements introduced in Ref. 14 (cf. Remarks 23 and 33).

d. Prior work: Markov entropy decomposition.¹⁷ The strong subadditivity (SSA) property of the von Neumann entropy of a tripartite state ρ ≡ ρ_ABC [cf. (25) of Sec. II C] can be expressed in terms of the conditional entropy S(A|B)_ρ = S(AB)_ρ − S(B)_ρ,

S {(A | B C)}_{ρ} \leq S {(A | B)}_{ρ} .

(67)

If we choose arbitrary subsets $M_{k} \subset {1, \dots, k}$ ⁠, the entropy S(ρ) = S(1, …, n)_ρ can be rewritten and upper-bounded as follows:

S {(1, \dots, n)}_{ρ} = S {(12)}_{ρ} + \sum_{k = 2}^{n - 1} S {(k + 1 (| 1, \dots, k)}_{ρ},

(68)

\leq S {(12)}_{ρ} + \sum_{k = 2}^{n - 1} S {(k + 1 (| M_{k})}_{ρ} = : S_{M} (ρ) .

(69)

In the second step, we applied Eq. (67) n − 2 times. The sets $M_{k}$ are called Markov shields and the upper bound S_M(ρ) is called the Markov entropy.¹⁷ In the following, we consider the particular choice $M_{k} = {k}$ ⁠. In that case, the conditional entropies $S {(k + 1 | M_{k})}_{ρ} = S {(k + 1 | k)}_{ρ}$ depend only on the reduced state ρ_k,k+1. As a consequence, the Markov entropy S_M(ρ) is an upper bound on S(ρ) which depends only on the nearest-neighbour reduced states ρ_k,k+1 (k ∈ {1, …, n − 1}). For a nearest-neighbour Hamiltonian $H = \sum_{k = 1}^{n - 1} h_{k, k + 1}$ ⁠, the energy $E = T r (ρ H) = \sum_{k = 1}^{n - 1} T r (ρ_{k, k + 1} h_{k, k + 1})$ is determined by the same reduced states ρ_k,k+1. Therefore, lower bounds to the free energy F = E − TS of a thermal state at temperature T can be found with a variational algorithm which only uses the reduced states ρ_k,k+1.¹⁷

Equation (69) was obtained by applying Eq. (67) n − 2 times for A = {k + 1}, B = {k} and C = {1, …, k − 1} (k ∈ {2, …, n − 1}). These inequalities are equivalent to the following inequalities [because Eq. (67) is equivalent to I(B : C) ≤ I(AB : C)]:

I {(1, \dots, k - 1 : k)}_{ρ} \leq I {(1, \dots, k - 1 : k, k + 1)}_{ρ}, (k \in {2, \dots, n - 1}) .

(70)

If equality holds in Eq. (70) or, equivalently, in Eq. (69), the global state ρ can be obtained from the reduced states ρ_k,k+1 (k ∈ {1, …, n − 1}) via Petz recovery maps (see the supplementary material and main text of Ref. 17). We state this known result in Theorem 16 and show that these conditions imply that MPO reconstruction (as stated in Theorem 17) is possible (Theorem 19).

A. Reconstruction of states from local marginal states

The state of the spin chain is $ρ \in D (H_{1, \dots, n})$ where $H_{1 \dots n} = H_{1} \otimes \dots H_{n}$ is the tensor product of the single-spin Hilbert spaces. For each k ∈ {1, …, n}, we partition the spins on the chain into two parts

X_{k} = {1 \dots k} and {k + 1 \dots n} .

(71)

The marginal states $ρ_{X_{k}} = {T r}_{k + 1 \dots n} (ρ)$ ⁠, for k ∈ {2, …, n}, can be defined recursively via

ρ_{X_{k}} = ({i d}_{X_{k}} \otimes {T r}_{k + 1}) (ρ_{X_{k + 1}}) (k \in {2, \dots, n - 1}), ρ_{X_{n}} = ρ .

(72)

Each partial trace Tr_k+1 is a local CPTP map. If the partial trace Tr_k+1 does not decrease the mutual information between {1, …, k − 1} and {k, k + 1} for all k ∈ {2, …, n − 1}, then the n-spin state ρ can be recovered from marginal states of two neighbouring spins.¹⁷

Theorem 16.

Let

ρ \in D (H_{1, \dots, n})

be a quantum state. If the conditions

I {(X_{k - 1} : k)}_{ρ} = I {(X_{k - 1} : k, k + 1)}_{ρ} (k \in {2, \dots, n - 1})

(73)

are satisfied, then the marginal states

ρ_{X_{k}} = {T r}_{k + 1 \dots n} (ρ)

are given by

ρ_{X_{k + 1}} = ({i d}_{X_{k - 1}} \otimes R_{k}) (ρ_{X_{k}}), ρ = ρ_{X_{n}} = R_{n - 1} R_{n - 2} \dots R_{3} R_{2} (ρ_{12}),

(74)

where k ∈ {2, …, n − 1}. The recovery maps

R_{k} : B (H_{k}) \to B (H_{k, k + 1})

are given by Petz recovery maps

R_{k} = R_{ρ_{k, k + 1}, {T r}_{k + 1}}^{P}

(Theorem 1). In the above, ρ_ij denotes the reduced state of ρ on sites i and j.

Proof.

For k ∈ {2, …, n − 1}, apply Corollary 12 with X = X′ = X_k−1, $N_{X} = {i d}_{X}$ ⁠, $Y = H_{k, k + 1}$ ⁠, $Y^{'} = H_{k}$ ⁠, and $N_{Y} (\cdot) = {T r}_{k + 1} (\cdot)$ ⁠.

In a similar way, if the partial traces do not decrease certain operator Schmidt ranks, their actions can be reverted with state reconstruction.^1,19

Theorem 17.

Let

ρ \in B (H_{1, \dots, n})

be a linear operator. If the conditions

O S R {(k - 1 : k)}_{ρ} = O S R {(X_{k - 1} : k, k + 1)}_{ρ} (k \in {3, \dots, n - 1})

(75)

are satisfied, the marginal states

ρ_{X_{k}} = {T r}_{k + 1 \dots n} (ρ)

are given by

ρ_{X_{k + 1}} = (i d \otimes R_{k}) (ρ_{X_{k}}), ρ = R_{n - 1} R_{n - 2} \dots R_{4} R_{3} (ρ_{123}),

(76)

where k ∈ {3, …, n − 1} and

R_{k} \in B (B (H_{k}); B (H_{k, k + 1}))

⁠. The maps are given by

R_{k} = R_{L_{k}, {T r}_{k + 1}}^{M}

(Theorem 8),

L_{k} = M_{σ_{k}}^{⊺}

with σ_k = ρ_k−1,k,k+1, and

M_{σ_{k}} \in B (B (H_{k, k + 1}); B (H_{k - 1}))

[Eq. ()].

Proof.

For k ∈ {3, …, n − 1}, apply Theorem 8 with X = X_k−1, X′ = {k − 1}, $N_{X} (\cdot) = {T r}_{X_{k - 2}} (\cdot)$ ⁠, $Y = H_{k, k + 1}$ ⁠, $Y^{'} = H_{k}$ ⁠, $N_{Y} (\cdot) = {T r}_{k + 1} (\cdot)$ ⁠. Recall that $σ_{k} = ρ_{k - 1, k, k + 1} = ({T r}_{X_{k - 2}} \otimes i d) (ρ_{X_{k + 1}})$ implies $M_{σ_{k}} = ({T r}_{X_{k - 2}}) M_{ρ_{X_{k + 1}}}$ where $M_{σ_{k}} \in B (B (H_{k, k + 1}); B (H_{k - 1}))$ and $M_{ρ_{X_{k + 1}}} \in B (B (H_{k, k + 1}); B (X_{k - 1}))$ (Lemma 10). Therefore, the reconstruction map is given by $R_{k} = R_{L_{k}, {T r}_{k + 1}}^{M}$ with $L_{k} = M_{ρ_{X_{k + 1}}}^{⊺} {[{T r}_{X_{k - 2}}]}^{⊺} = M_{σ_{k}}^{⊺}$ ⁠.

The result from Theorem 17 has been obtained previously in Ref. 1 under the name reconstruction of quantum states or MPO reconstruction. For a discussion of further related work,¹⁹ see Remark 34.

Remark 18

(Efficient recovery/reconstruction). We call a recovery or reconstruction method to obtain $ρ \in D (H_{1, \dots, n})$ efficient if it satisfies the following conditions. The method provides an efficient representation of ρ (cf. Sec. II B). This representation of ρ can be constructed from suitable input data in at most poly(n) computational time. As a consequence, the size of the input data may be at most poly(n) [i.e., at most poly(n) complex numbers]. The necessary input data may be obtained from at most poly(n) different tensor product expectation values, i.e., expectation values of the form $T r [(A_{1} \otimes \dots \otimes A_{n} \otimes A^{'}) N (ρ \otimes ρ^{'})]$ where $A_{k} \in B (H_{k})$ ⁠, $A^{'} \in B (H_{Y^{'}})$ ⁠, $ρ^{'} \in D (H_{Y^{'}})$ ⁠, and Y′ is an ancilla system of dimension $d_{Y^{'}} = O (p o l y (n))$ ⁠.³⁸ The quantum operation $N$ is constructed from at most poly(n) quantum operations whose input and output dimension is at most poly(n). This severely restricts the available measurements because the number of two-qubit gates required to implement an arbitrary n-qubit unitary is exponential in n.³⁹

Standard quantum state tomography is not efficient because it fails to satisfy any of these criteria. For example, in quantum state tomography, ∼exp(n) expectation values are required in order to determine ρ.

Clearly, Theorems 16 and 17 satisfy all of these criteria because efficient representations are provided and the necessary input data consist only of two- and three-spin marginals of ρ. Lemma 38 also provides efficient MPO and PMPS representations for Theorem 16 and an efficient MPO representation for Theorem 17.□

Note that the operator Schmidt rank condition (75) is different from the mutual information condition (73) in that it contains {k − 1} instead of X_k−1 on the very left. If the partial trace ${T r}_{X_{k - 2}}$ which maps X_k−1 onto {k − 1} was left out, the state $ρ_{X_{k + 1}}$ would be needed to construct $R_{k}$ ⁠. Construction of $R_{n - 1}$ would need $ρ_{X_{n}} = ρ$ and the reconstruction would be neither efficient nor useful. Despite this difference, we show that the premise of state recovery (Theorem 16) implies the premise of state reconstruction (Theorem 17).

Theorem 19.

Let

ρ \in D (H_{1, \dots, n})

be a quantum state. The conditions

I {(X_{k - 1} : k)}_{ρ} = I {(X_{k - 1} : k, k + 1)}_{ρ} (k \in {2, \dots, n - 1})

(77)

imply

O S R {(k - 1 : k)}_{ρ} = O S R {(X_{k - 1} : k, k + 1)}_{ρ} (k \in {3, \dots, n - 1}) .

(78)

In other words, if the state ρ can be recovered with Petz recovery from the marginals ρ_k,k+1 (k ∈ {1, …, n − 1}), then it can also be reconstructed with state reconstruction from the marginals ρ_k−1,k,k+1 (k ∈ {2, …, n − 1}).

Proof.

Equation () implies [k ∈ {2, …, n − 1}, use Eq. ()]

0 = I (X_{k - 1} : k, k + 1) - I (X_{k - 1} : k) = I (X_{k - 1} : k + 1 | k)

(79a)

= I (X_{k - 1}, k : k + 1) - I (k : k + 1)

(79b)

= I (X_{k} : k + 1) - I (k : k + 1) .

(79c)

We shift the index of the last equation by one and obtain, for k ∈ {3, …, n − 1},

I (k - 1 : k) = I (X_{k - 1} : k) = I (X_{k - 1} : k, k + 1) .

(80)

This mutual information equality implies the corresponding operator Schmidt rank equality (Theorem 13).

Remark 20.

For n = 4, Theorem 19 reduces to “(C2) implies Eq. (C1)” from Lemma 15 if one takes into account that “I(B : C) = I(AB : CD)” (C2) is equivalent to “I(A : B) = I(A : BC) and I(AB : C) = I(AB : CD)” (see the following Lemma 21).

Lemma 21.

I(B : C) = I(AB : CD) holds if and only if I(A : B) = I(A : BC) and I(AB : C) = I(AB : CD).

Proof.

“⇒”: Let I(B : C) = I(AB : CD) hold. We have I(B : C) ≤ I(AB : C) ≤ I(AB : CD) = I(B : C), which implies that I(B : C) = I(AB : C) = I(AB : CD). As a consequence,

0 = I (A B : C) - I (B : C) = I (A : C | B) = I (A : B C) - I (A : B)

(81)

also holds [Eq. ()]. This already shows the proposed conclusion.

“⇐”: Let I(A : B) = I(A : BC) and I(AB : C) = I(AB : CD) hold. The former equality implies Eq. (81) and this shows that I(B : C) = I(AB : C) = I(AB : CD) holds.

B. Long-ranged measurements

In this subsection, we generalize recovery and reconstruction to use certain long-range measurements as input and show that successful recovery implies that successful reconstruction is also possible.

1. Recovery from long-ranged measurements

Recovery and reconstruction of a spin chain state from few-body marginals required that correlations (as measured by the mutual information or the operator Schmidt rank) do not decrease under the following partial traces (Fig. 3):

ρ_{X_{k}} = ({i d}_{X_{k}} \otimes {T r}_{k + 1}) (ρ_{X_{k + 1}}) (k \in {2, \dots, n - 1}), ρ_{X_{n}} = ρ .

(82)

In order to incorporate long-range measurements, we introduce ancillary systems $Y_{k}^{'}$ (k ∈ {0, …, n}, $d_{Y_{0}^{'}} = d_{Y_{n}^{'}} = 1$ ⁠), quantum operations $T_{k} : B (k, Y_{k}^{'}) \to B (Y_{k - 1}^{'})$ and define $ρ_{k} \in D (X_{k}, Y_{k}^{'})$ via (Fig. 3)

ρ_{k - 1} = ({i d}_{X_{k - 1}} \otimes T_{k}) (ρ_{k}) (k \in {2, \dots, n}), ρ_{n} = ρ .

(83)

The relation between long-range measurements and the maps $T_{k}$ is explained Remark 22. If the mutual information $I {(X_{k - 1} : k, Y_{k}^{'})}_{ρ_{k}}$ does not decrease when $T_{k}$ is applied, then Theorem 24 provides a reconstruction of ρ from the states $σ_{k} = {T r}_{X_{k - 1}} (ρ_{k}) \in D (k, Y_{k}^{'})$ (details are specified in the theorem). Before we state the theorem, we explain that measurements on σ_k correspond to recursively defined long-range measurements on ρ and we observe that suitable ancilla systems $Y_{k}^{'}$ and operations $T_{k}$ can be determined for any pure MPS.

FIG. 3.

View large Download slide

Left: Recursive definition of the reduced states $ρ_{X_{k}}$ on X_k = {1, …, k} [Eq. (82)]. Middle: Spaces on which the recursively defined states ρ_k act [Eqs. (83) and (86)]. As n = 5, we have $Y_{5}^{'} = C$ ⁠. Right: Recursive definition of the long-ranged observable G from the local observable F (Remark 22).

Remark 22

(Recursively defined long-range measurements). In Theorem 24 below, the state ρ is reconstructed from the states

σ_{k} = {T r}_{X_{k - 1}} (ρ_{k}) \in D (k, Y_{k}^{'})

(k ∈ {1, …, n}). The states σ_k can be reconstructed from the expectation values Tr(F_iσ_k) of a set of observables

{F_{i} \in B (k, Y_{k}^{'})}_{i}

which is complete, i.e., spans the full vector space

B (k, Y_{k}^{'})

⁠.⁴⁰ For simplicity, we drop the index i and denote a possible observable by

F \in B (k, Y_{k}^{'})

⁠. The expectation value Tr(Fσ_k) corresponds to the following expectation value in ρ (Fig. 3):

T r (F σ_{k}) = T r (F (T_{k + 1} \dots T_{n}) (ρ)) = T r ((T_{n}^{*} \dots T_{k + 1}^{*}) (F) ρ) = T r (G ρ),

(84a)

G = (T_{n}^{*} \dots T_{k + 1}^{*}) (F) \in B (H_{k, \dots, n}) .

(84b)

Here,

T_{k}^{*} : B (Y_{k - 1}^{'}) \to B (k, Y_{k}^{'})

are adjoint superoperators (Sec. II A) and we used that F is Hermitian, that the channels

T_{k}

map Hermitian operators onto Hermitian operators (because they are completely positive) and that

d_{Y_{n}^{'}} = 1

⁠. The observable F describes a measurement on σ_k and the recursively defined observable G which acts on

H_{k, \dots, n}

describes a measurement on ρ. Equation () hence demonstrates that measurements on

σ_{k} \in D (k, Y_{k}^{'})

correspond to recursively defined long-range measurements on ρ.□

Remark 23

(Example: Pure matrix product states). Suppose that we fix l ≥ 1 and set

H_{Y_{k}^{'}} = H_{k + 1, \dots, k + l}

⁠. In this case,

ρ_{k} \in D (X_{k}, Y_{k}^{'}) = D (X_{k + l})

[cf. Eq. ()]. We define

T_{k} (\cdot) = {T r}_{k + l} (U_{k} \cdot U_{k}^{*})

where

U_{k} \in B (H_{k, \dots, k + l})

are unitary operators. Suppose further that the unitaries have the property that they transform ρ_k into

(1 \otimes U_{k}) ρ_{k} (1 \otimes U_{k}^{*}) = ρ_{k}^{'} \otimes |ϕ_{k}⟩ ⟨ϕ_{k}|,

(85)

where

ρ_{k}^{'} \in D (H_{1, \dots, k + l - 1})

and

| ϕ_{k} ⟩) \in H_{k + l}

are states. Then,

ρ_{k - 1} = T (ρ_{k}) = ρ_{k}^{'}

and the action of

T_{k}

on ρ_k can be reversed with

M_{k} (\cdot) = U_{k}^{*} (\cdot \otimes |ϕ_{k}⟩ ⟨ϕ_{k}|) U_{k}

⁠, i.e.,

M_{k} (ρ_{k - 1}) = ρ_{k}

⁠. As a consequence, the mutual information

I {(X_{k - 1} : k, Y_{k}^{'})}_{ρ_{k}}

does not decrease if

T_{k}

is applied (Corollary 12) and we can apply Theorem 24 to reconstruct ρ. If ρ is a pure state which has an MPS representation of bond dimension D, then unitaries that satisfy Eq. () exist if l =

⌈)

log_d(D)

(⌉

where d = max_kd_k is the maximal dimension of a single spin.¹⁴ In this case, Theorem 24 provides an efficient reconstruction if

D = O (p o l y (n))

(cf. Remark 26). As a consequence, any state that can be reconstructed with the pure-state reconstruction scheme based on unitary operations from Ref. 14 can also be reconstructed with Theorem 24 if the same unitaries are used.

Theorem 24.

Let

ρ \in D (H_{1, \dots, n})

a quantum state. Let

Y_{k}^{'}

(k ∈ {1, …, n}) be ancilla systems with

\dim (Y_{n}^{'}) = 1

and choose quantum operations

T_{k} : B (k, Y_{k}^{'}) \to B (Y_{k - 1}^{'})

(k ∈ {2, …, n}). Define

ρ_{k} \in D (X_{k}, Y_{k}^{'})

(k ∈ {1, …, n}) recursively via

ρ_{k - 1} = (i d \otimes T_{k}) (ρ_{k}) (k \in {2, \dots, n}), ρ_{n} = ρ .

(86)

If the conditions

I {(X_{k - 1} : Y_{k - 1}^{'})}_{ρ_{k - 1}} = I {(X_{k - 1} : k, Y_{k}^{'})}_{ρ_{k}}, k \in {2, \dots, n},

(87)

are satisfied, then

ρ_{k} = ({i d \otimes R}_{k}) (ρ_{k - 1}) (k \in {2, \dots, n}), ρ = ρ_{n} = R_{n} R_{n - 1} \dots R_{3} R_{2} (ρ_{1}),

(88)

where the recovery maps

R_{k} : B (Y_{k - 1}^{'}) \to B (k, Y_{k}^{'})

are given by Petz recovery maps

R_{k} = R_{σ_{k}, T_{k}}^{P}

(Theorem 1) with

σ_{k} = {T r}_{X_{k - 1}} (ρ_{k}) \in D (k, Y_{k}^{'})

⁠.

Proof.

For k ∈ {2, …, n}, apply Corollary 12 with X = X′ = X_k−1, $N_{X} = i d$ ⁠, $Y = H_{k, Y_{k}^{'}}$ ⁠, $Y^{'} = Y_{k - 1}^{'}$ ⁠, and $N_{Y} = T_{k}$ ⁠.

Remark 25.

Theorem 24 provides Theorem 16 by restricting to the special case $H_{Y_{k}^{'}} = H_{k + 1}$ (k ∈ {1, …, n − 1}), $T_{k} (\cdot) = {T r}_{k + 1} (\cdot)$ (k ∈ {2, …, n − 1}), $T_{n} = i d$ and using Eq. (88) only for k ∈ {2, …, n − 1}.□

Remark 26.

Denote by $d_{Y^{'}} = {max}_{k} \dim (Y_{k}^{'})$ the maximal dimension of any ancillary system. If $d_{Y^{'}} = O (p o l y (n))$ ⁠, the recovery scheme from Theorem 24 is efficient (it satisfies all conditions from Remark 18). Lemma 38 provides efficient PMPS and MPO representations of ρ.□

2. Reconstruction from long-ranged measurements

State reconstruction can be generalized similarly but it requires that additional ancillary systems $X_{k}^{'}$ and linear maps $U_{k}$ are introduced (Fig. 4):

FIG. 4.

View large Download slide

Left: Spaces on which the operators ρ_k, σ_k, and τ_k act (Theorem 27). As n = 5, we have $Y_{5}^{'} = C$ ⁠. Right: Spaces on which the operators F and G act (Remark 30).

Theorem 27.

Let

ρ \in B (H_{1, \dots, n})

be a linear operator. Let

Y_{k}^{'}

(k ∈ {0, …, n}) and

X_{k}^{'}

(k ∈ {0, …, n − 1}) ancilla systems with

\dim (Y_{0}^{'}) = \dim (Y_{n}^{'}) = \dim (X_{0}^{'}) = 1

⁠. Choose linear maps

T_{k} \in B (B (k, Y_{k}^{'}); B (Y_{k - 1}^{'}))

(k ∈ {2, …, n}) and

U_{k} \in B (B (X_{k - 1}); B (X_{k - 1}^{'}))

(k ∈ {1, …, n},

U_{1} = 1

). As before [Eq. ()], we define

ρ_{k} \in B (X_{k}, Y_{k}^{'})

via

ρ_{k - 1} = (i d \otimes T_{k}) (ρ_{k}) (k \in {2, \dots, n}), ρ_{n} = ρ .

(89)

In addition, we define (Fig. 4)

σ_{k} = (U_{k} \otimes i d) (ρ_{k}) \in B (X_{k - 1}^{'}, k, Y_{k}^{'}) (k \in \{) 1, \dots, n),

(90)

τ_{k} = (i d \otimes T_{k}) (σ_{k}) \in B (X_{k - 1}^{'}, Y_{k - 1}^{'}) (k \in {2, \dots, n}) .

(91)

If the conditions

O S R {(X_{k - 1}^{'} : Y_{k - 1}^{'})}_{τ_{k}} = O S R {(X_{k - 1} : k, Y_{k}^{'})}_{ρ_{k}} (k \in {2, \dots, n})

(92)

are satisfied, there are linear maps

R_{k} \in B (B (Y_{k - 1}^{'}); B (k, Y_{k}^{'}))

(k ∈ {2, …, n}) such that

ρ_{k} = (i d \otimes R_{k}) (ρ_{k - 1}) (k \in {2, \dots, n}), ρ = R_{n} R_{n - 1} \dots R_{3} R_{2} (ρ_{1}) .

(93)

The maps are given by

R_{k} = R_{L_{k}, T_{k}}^{M}

(Theorem 8) where

L_{k} = M_{σ_{k}}^{⊺}

and

M_{σ_{k}} \in B (B (k, Y_{k}^{'}); B (X_{k - 1}^{'}))

[Eq. ()].

Proof.

For k ∈ {2, …, n}, apply Theorem 8 with X = X_k−1, $X^{'} = X_{k - 1}^{'}$ ⁠, $N_{X} = U_{k}$ ⁠, $Y = H_{k, Y_{k}^{'}}$ ⁠, $Y^{'} = Y_{k - 1}^{'}$ ⁠, and $N_{Y} = T_{k}$ ⁠. The equality $ρ = (i d \otimes R_{Y^{'}}) (i d \otimes N_{Y}) (ρ)$ from the theorem becomes $ρ_{k} = (i d \otimes R_{k}) (i d \otimes T_{k}) (ρ_{k})$ ⁠. Recall that Eq. (90) implies $M_{σ_{k}} = U_{k} M_{ρ_{k}}$ where $M_{σ_{k}} \in B (B (k, Y_{k}^{'}); B (X_{k - 1}^{'}))$ and $M_{ρ_{k}} \in B (B (k, Y_{k}^{'}); B (X_{k - 1}))$ (Lemma 10). Therefore, the reconstruction map is given by $R_{k} = R_{L_{k}, T_{k}}^{M}$ with $L_{k} = M_{ρ_{k}}^{⊺} U_{k}^{⊺} = M_{σ_{k}}^{⊺}$ ⁠.

Remark 28.

Theorem 27 provides Theorem 17 by restricting to the special case $Y_{k}^{'} = H_{k + 1}$ ⁠, $X_{k}^{'} = H_{k}$ ⁠, $T_{k} = {T r}_{k + 1}$ ⁠, $U_{k} = {T r}_{X_{k - 2}}$ and using Eq. (93) only for k ∈ {3, …, n − 1}.□

Remark 29.

Denote by $d_{Y^{'}} = {max}_{k} \dim (Y_{k}^{'})$ and $d_{X^{'}} = {max}_{k} \dim (X_{k}^{'})$ the maximal dimensions. If $d_{Y^{'}} = O (p o l y (n))$ and $d_{X^{'}} = O (p o l y (n))$ ⁠, the reconstruction scheme from Theorem 27 is efficient (it satisfies all conditions from Remark 18). Lemma 38 provides an efficient MPO representation of ρ.

Efficient reconstruction implies that a given state can be reconstructed from a number of expectation values which grow polynomially instead of exponentially with n. This improvement can only be achieved if the to-be-reconstructed state is not a completely general quantum state of n systems. In the following, we show that the condition for efficient reconstruction in particular implies that the operator Schmidt ranks of the state are restricted to growing polynomially (instead of exponentially) with n.

For $k = ⌈ \frac{n}{2} ⌉$ ⁠, the maximal value of the operator Schmidt rank $O S R {(X_{k} : k + 1 \dots n)}_{ρ}$ is ${(min {d_{1} \dots d_{k}, d_{k + 1} \dots d_{n}})}^{2} = O (\exp (n))$ and it is assumed, e.g., for maximally entangled pure states. Suppose that ρ can be reconstructed efficiently. The equality $ρ = ({i d}_{X_{k}} \otimes R_{n} \dots R_{k + 1}) (ρ_{k})$ [Eq. (93)] implies $O S R {(X_{k} : k + 1 \dots n)}_{ρ} \leq r k (R_{k + 1})$ (Corollary 11). The rank of $R_{k + 1}$ is, in turn, upper bounded by $r k (R_{k + 1}) \leq d_{Y^{'}}^{2} = O (p o l y (n))$ ⁠, i.e., the operator Schmidt rank of ρ is at most $O S R {(X_{k} : k + 1 \dots n)}_{ρ} = O (p o l y (n))$ ⁠. In conclusion, any state that admits an efficient reconstruction with Theorem 27 has a small operator Schmidt rank in the sense that it does not grow exponentially but only polynomially with the number of spins n.

Remark 30

(Recursively defined long-range measurements). In Theorem 27, ρ is reconstructed from

σ_{k} \in B (X_{k - 1}^{'}, k, Y_{k}^{'})

(k ∈ {1, …, n}, noting that σ₁ = ρ₁). As above (Remark 22), measurements on σ_k correspond to recursively defined long-range measurements on ρ (Fig. 4)

T r (F σ_{k}) = T r (F (U_{k} \otimes (T_{n}, \dots, T_{k + 1})) (ρ)) = T r (G ρ), F \in B (X_{k - 1}^{'}, k, Y_{k}^{'}),

(94a)

G = {[[U_{k}^{*} \otimes (T_{k + 1}^{*}, \dots, T_{n}^{*})] (F^{*})]}^{*} \in B (H_{1, \dots, n}) .

(94b)

If the superoperators involved are quantum operations and the operator F is Hermitian, G is Hermitian as well and there is a correspondence between observables on σ_k and recursively defined long-ranged observables on ρ. Otherwise, the correspondence holds only in an abstract sense between operators

F \in B (X_{k - 1}^{'}, k, Y_{k}^{'})

and

G \in B (H_{1, \dots, n})

⁠.□

Remark 31

(Mixed state which requires long-range measurements). Remark 23 showed that any pure MPS can be recovered with Theorem 24 if the unitary operations from Ref. 14 are used. Below, we show that recovery with Theorem 24 implies that reconstruction with Theorem 27 is also possible (see Theorem 32). The following simple mixed state shows that Theorems 24 and 27 can recover more than pure matrix product states and more than recovery or reconstruction from local reduced states (Theorems 16 and 17): The state

ρ = \frac{1}{2} [|0 \dots 0⟩ ⟨0 \dots 0| + |100 \dots 01⟩ ⟨100 \dots 01|] \in D (H_{1 \dots n})

(95)

does not admit recovery or reconstruction from local reduced states because it turns into a product state if the first or last site is traced out; this unavoidably reduces the mutual information from non-zero to zero and the operator Schmidt rank from larger than one to one. The state admits recovery or reconstruction via Theorems 24 and 27 if the following definitions are used: Assuming uniform local dimensions d = d_k (k ∈ {1, …, n}), set

H_{Y_{k}^{'}} = H_{k + 1}

⁠,

T_{k} = {T r}_{k + 1} {S W A P}_{k, k + 1}

(k ∈ {2, …, n − 1}),

T_{n} = 1_{n}

⁠,

X_{k - 1}^{'} = H_{k - 1}

⁠,

U_{k} = {T r}_{1, \dots, k - 2} {S W A P}_{1, k - 1}

(k ∈ {2, …, n}), and

U_{1} = 1

⁠.⁴¹ With these definitions, the states ρ_k used in Theorems 24 and 27 are given by ρ_n = ρ,

ρ_{k} = \frac{1}{2} [|0 \dots 0⟩ ⟨0 \dots 0| + |100 \dots 01⟩ ⟨100 \dots 01|] \in D (H_{1 \dots k + 1}),

(96a)

where k ∈ {1, …, n − 1}. The states σ_k and τ_k used in Theorem 27 are given by σ₁ = ρ₁,

σ_{n} = ρ_{1} \in D (H_{n - 1, n})

⁠,

σ_{k} = \frac{1}{2} [|000⟩ ⟨000| + |101⟩ ⟨101|] \in D (H_{k - 1, k, k + 1}) (k \in {2 \dots n - 1}),

(96b)

τ_{k} = \frac{1}{2} [|00⟩ ⟨00| + |11⟩ ⟨11|] \in D (H_{k - 1, k}) (k \in {2 \dots n}) .

(96c)

These states show that the conditions from Theorems 24 and 27 are satisfied. □

3. Recovery vs. reconstruction for long-ranged measurements

In this section, we show that the conditions for state recovery (Theorem 24) imply that state reconstruction (Theorem 27) is also possible. The premise of Theorem 24 implies the premise of Theorem 27 for $U_{k} = i d$ ⁠. However, Theorem 27 does not provide a useful reconstruction with $U_{k} = i d$ because the necessary input σ_n for the construction of $R_{n}$ would be σ_n = ρ. In Theorem 19, we used the symmetry of the conditional mutual information to work around this but this is no longer possible because $T_{k}$ was introduced. Note that Eq. (87) implies the same equality for operator Schmidt ranks and that Eq. (88) provides MPO representations of the ρ_k (Lemma 38). It is well known that maps $U_{k}$ suitable for Theorem 27 can be obtained directly from the matrices of the MPO representation after the matrices have been transformed into a suitable orthogonal (mixed-canonical) form (Refs. 13 and 24; see also Remark 35 in the Appendix). The maps $U_{k}$ obtained in this way are given by partial isometries on the vector space of linear operators. Such a map is not guaranteed to be completely positive or trace preserving, i.e., it does not represent a quantum operation and it may not allow for an efficient implementation in a given quantum experiment. An alternative construction has been put forward in Ref. 19: Here, maps $U_{k}$ are provided whose matrix representation is given by a submatrix of a permutation matrix in a product basis of $B (X_{k}) = B (H_{1}) \otimes \dots \otimes B (H_{k})$ ⁠.⁴² We use this result to prove that efficient recovery implies efficient reconstruction in Theorem 32. Remark 33 discusses advantages and disadvantages of the two different choices for $U_{k}$ mentioned in this paragraph.

Theorem 32.

Let the premise of Theorem 24 hold. Set

\dim (Y_{0}^{'}) = 1

and

X_{k}^{'} = Y_{k}^{'}

⁠. There are linear maps

U_{k} \in B (B (X_{k - 1}); B (X_{k - 1}^{'}))

(k ∈ {2, …, n}) such that

O S R {(X_{k - 1}^{'} : Y_{k - 1}^{'})}_{τ_{k}} = O S R {(X_{k - 1} : k, Y_{k}^{'})}_{ρ_{k}} (k \in {2, \dots, n})

(97)

holds where

τ_{k} = (U_{k} \otimes {i d}_{Y_{k - 1}^{'}}) (ρ_{k - 1})

is the same operator as in . Choose operator bases

F_{i_{k}}^{(k)}

⁠,

F_{j_{k}}^{(X_{k}^{'})}

⁠, and

F_{l_{k}}^{(Y_{k}^{'})}

⁠. There is an efficient algorithm to choose suitable maps

U_{k}

and they can be chosen such that their matrix representation [Eq. ()] is a submatrix of a permutation matrix. In this case, the resulting reconstruction is efficient (in the sense of Remark 18) if recovery is efficient and if the operator bases are chosen such that they contain only Hermitian operators.

Proof.

Lemma 38 provides an MPO representation of the states ρ_k from Eq. (88). It is well known that maps $U_{k}$ can be chosen recursively such that $O S R {(X_{k - 1}^{'} : Y_{k - 1}^{'})}_{τ_{k}} = O S R {(X_{k - 1} : Y_{k - 1}^{'})}_{ρ_{k - 1}}$ holds if an MPO representation of ρ_k−1 is given.^13,24 As $O S R {(X_{k - 1} : Y_{k - 1}^{'})}_{ρ_{k - 1}} = O S R {(X_{k - 1} : k, Y_{k}^{'})}_{ρ_{k}}$ is implied by (87) (Theorem 13), it is clear that Eq. (97) holds as well. It was also recognized that the maps $U_{k}$ can be chosen such that their matrix representation is a submatrix of a permutation matrix.¹⁹ We provide a self-contained description of the corresponding procedure in Lemma 36.

Let U_k the matrix representation of

U_{k}

and suppose that U_k is a submatrix of a permutation matrix. Denote by

f_{k} (j) = {(i_{1}, \dots, i_{k - 1}) : {[U_{k}]}_{j, (i_{1} \dots i_{k - 1})} = 1}

the set of columns with a non-zero entry in a given row of U_k (where

i_{j} \in {1, \dots, d_{j}^{2}}

and

j \in {1, \dots, d_{X_{k - 1}^{'}}^{2}}

⁠). The matrix elements of

σ_{k} = (U_{k} \otimes i d) (ρ_{k})

are given by [insert an identity map () into Eq. ()]

⟨ F_{j}^{(X_{k - 1}^{'})} \otimes F_{i_{k}}^{(k)} \otimes F_{l}^{(Y_{k}^{'})}, σ_{k} ⟩

(98)

= \sum_{i_{1} \dots i_{k - 1}} {[U_{k}]}_{j, i_{1} \dots i_{k - 1}} ⟨ F_{i_{1}}^{(1)} \otimes \dots \otimes F_{i_{k - 1}}^{(k - 1)} \otimes F_{i_{k}}^{(k)} \otimes F_{l}^{(Y_{k}^{'})}, ρ_{k} ⟩ = \{\begin{matrix} 0, & if f_{k} (j) = \emptyset, \\ ⟨ F_{i_{1}}^{(1)} \otimes \dots \otimes F_{i_{k}}^{(k)} \otimes F_{l}^{(Y_{k}^{'})}, ρ_{k} ⟩, & with {(i_{1}, \dots, i_{k - 1})} = f_{k} (j) . \end{matrix})

(99)

Here, we used that |f_k(j)| ≤ 1 because U_k is a submatrix of a permutation matrix. The last equation shows that σ_k, which needs to be known for reconstruction of ρ, can be determined from at most

{(d_{X_{k}^{'}} d_{k} d_{Y_{k}^{'}})}^{2}

tensor product expectation values in ρ_k. The structure of the given expectation values is permitted for efficient reconstruction (Remark 18) and the number of expectation values is at most poly(n) if recovery is efficient. Furthermore, efficient recovery implies that the MPO representation of the ρ_k and the procedures to determine

U_{k}

and f_k(j) are efficient as well (Ref. 19; for details see Lemmata 36 and 37). This finishes the proof of the theorem.

Remark 33.

The singular values of $M_{σ_{k}} \in B (B (k, Y_{k}^{'}); B (X_{k - 1}^{'}))$ equal those of $M_{ρ_{k}} \in B (B (k, Y_{k}^{'}); B (X_{k - 1}))$ if the maps $U_{k}$ are suitable partial isometries on the vector space of linear operators (cf. Remark 35). For reconstruction stability (Theorem 6), this is the optimal case (if the maps $T_{k}$ are predefined). If the maps $U_{k}$ are restricted to submatrices of permutation matrices, the singular values of $M_{σ_{k}}$ are smaller than or equal to those of $M_{ρ_{k}}$ (because $U_{k}$ has a unit operator norm). If the smallest non-zero singular value decreases, then stability of the reconstruction is reduced (Theorem 6; cf. Refs. 19 and 43). In the worst case, the smallest non-zero singular value decreases by a factor exponential in n because of the recursive construction of the $U_{k}$ ⁠.⁴³ However, empirical results show that this worst-case behaviour is usually not observed in practice.^1,19,43,44

If the maps $T_{k}$ are not predefined, the singular values of $M_{σ_{k}}$ equal those of $M_{ρ} \in B (B (H_{k \dots n}); B (X_{k - 1}))$ if the maps $U_{k}$ and $T_{k}$ are suitable partial isometries on the vector space of linear operators (Remark 35). In this case, Theorem 32 allows reconstruction of an arbitrary MPO (or matrix product state/tensor train) with optimal reconstruction stability. However, it remains an open question whether this can be fully exploited, e.g., in the reconstruction of quantum states as the necessary measurements may not allow for an efficient implementation if the maps $U_{k}$ and $T_{k}$ are general partial isometries on the vector space of linear operators.

The situation is different if the state ρ is a pure matrix product state. Here, partial isometries that act on the Hilbert spaces themselves can be obtained (Ref. 14, cf. Remarks 23 and 35). These partial isometries can be implemented via unitary control of the quantum system and they have the property that they preserve the singular values of $M_{ρ}$ ⁠. This also shows that the tomography scheme for pure matrix product states based on local unitary operations and proposed in Ref. 14 provides maps $T_{k}$ and $U_{k}$ for state recovery and reconstruction with optimal stability.□

Remark 34

(Related work). Note that nowhere in the proof of Theorems 8 and 27 did we use the fact that ρ is a linear operator on $H_{1, \dots, n}$ ⁠. The theorems apply equally well to arbitrary vectors $| ψ ⟩) \in H_{1, \dots, n}$ on tensor product vector spaces $H_{1 \dots n} = H_{1} \otimes \dots \otimes H_{n}$ ⁠. The components of ⟨i₁ … i_n|ψ⟩ of |ψ⟩ in a product basis define a tensor $t \in C^{d_{1} \times \dots \times d_{n}}$ (i.e., an array with n indices).

A result similar to Theorem 27 has been obtained before in the context of tensor train representations.^19,43,44 Their result is formulated for a tensor with n indices, i.e., replace $B (H_{k})$ by $H_{k}$ ⁠, $B (Y_{k}^{'})$ by $Y_{k}^{'}$ ⁠, $O S R {(X_{k - 1}^{'} : Y_{k - 1}^{'})}_{τ_{k}}$ by $r k (M_{τ_{k}})$ ⁠, etc. They restrict to $\dim (X_{k - 1}^{'}) = \dim (Y_{k - 1}^{'}) = r k (M_{τ_{k}})$ ⁠. In this case, the pseudoinverse in the reconstruction maps $R_{k}$ (defined in Theorem 27) is just the regular inverse (cf. Remark 5). They also restrict $U_{k}$ and $T_{k}$ to submatrices of permutation matrices in a fixed product basis. In addition, they provide an algorithm which attempts to determine suitable maps $U_{k}$ and $T_{k}$ incrementally and efficiently. Similar work has been carried out for the Tucker and hierarchical Tucker tensor representations^5,20,21 and the relation between this work and the matrix reconstruction from Sec. III A will be explored elsewhere.⁴⁵□

ACKNOWLEDGMENTS

We acknowledge discussions with Oliver Marty. Work in Ulm was supported by an Alexander von Humboldt Professorship, the ERC Synergy grant BioQ, the EU projects QUCHIP and EQUAM, and the US Army Research Office Grant No. W91-1NF-14-1-0133. Work in Hannover was supported by the DFG through SFB 1227 (DQ-mat) and the RTG 1991, the ERC grants QFTCMPS and SIQS, and the cluster of excellence EXC201 Quantum Engineering and Space-Time Research.

APPENDIX:ADDITIONAL RESULTS AND BACKGROUND INFORMATION

1. Optimality of the stability bound

The following examples show that the bound from Theorem 6 is optimal up to constants and that the reconstruction error can diverge as ϵ approaches zero if small singular values in LMR are not truncated.

Theorem 6 provides an upper bound on the reconstruction error of a reconstructible matrix S (the signal) which is perturbed by some error matrix E. The following example shows that the upper bound from the theorem is optimal up to constants:

L = (\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix}), R = (\begin{matrix} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{matrix}), S = (\begin{matrix} 0 & Δ & 1 \\ Δ & 0 & 0 \\ 1 & 0 & 0 \end{matrix}), E = η ϵ (\begin{matrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{matrix}) .

The eigenvalues of S are $\pm \sqrt{1 + Δ^{2}}$ so $η = ∥ S ∥ = \sqrt{1 + Δ^{2}}$ approaches unity as Δ → 0. For simplicity, we might assume that using η ≈ 1 is sufficient for the discussion of this example but we keep η > 1. We have ∥E∥/η = ϵ. Suppose that we choose ϵ and τ such that $0 < ϵ \leq τ < \frac{1}{3 \sqrt{2}}$ ⁠. Set c = τ + 2ϵ, i.e., $0 < c < \frac{1}{\sqrt{2}}$ ⁠, and set $Δ = c / \sqrt{1 - c^{2}}$ ⁠. Then $Δ$ < 1 and $η < \sqrt{2}$ ⁠. In addition, we obtain

(τ + 2 ϵ) η = c \sqrt{1 + Δ^{2}} = c \sqrt{1 - \frac{c^{2}}{1 - c^{2}}} = Δ .

The eigenvalues of LSR are ±Δ. This provides us γ = Δ/η = τ + 2ϵ. Therefore, the condition ϵ ≤ τ < γ − ϵ is automatically satisfied and, as a consequence, 2ϵ < γ holds as well. LER can change the eigenvalues of LSR at most by ϵ (cf. proof of Lemma 7), so no truncation occurs. In this case, the reconstruction error has exactly the scaling from the theorem

∥ M R {(L M R)}_{τ}^{+} L M - S ∥ = ‖(\begin{matrix} 0 & 0 & 0 \\ 0 & η ϵ & 0 \\ 0 & 0 & - \frac{η ϵ}{Δ^{2}} \end{matrix})‖ = max \{η ϵ, \frac{η ϵ}{Δ^{2}}\} \geq \frac{η ϵ}{Δ^{2}} \geq \frac{ϵ}{9 \sqrt{2} {(γ - ϵ)}^{2}} .

Note that the conditions from above imply $Δ^{2} / η = η γ^{2} < \sqrt{2} γ^{2}$ and that γ = τ + 2ϵ ≥ 3ϵ. The latter implies −ϵ ≥ −γ/3 and 3(γ − ϵ) ≥ 2γ ≥ γ. Combining the relations provides the bound $Δ^{2} / η < \sqrt{2} γ^{2} \leq 9 \sqrt{2} {(γ - ϵ)}^{2}$ used above.

One may ask whether thresholds τ outside the interval permitted by the theorem reconstruct M successfully. In this example, a threshold that is large enough to produce a different reconstruction will replace at least one of the two singular values of the reconstruction by zero. As the two singular values of S are equal, the reconstruction error will be at least ∥S∥ in this case, i.e., larger thresholds do not provide a successful reconstruction in the sense that the error in operator norm is significantly smaller than ∥S∥. In this example, neither smaller nor larger thresholds (than the ones permitted by Theorem 6) provide an improved reconstruction: Smaller thresholds do not change the reconstruction at all because thresholding does not reduce the rank of LMR in this example. However, the following example shows that thresholding is in general necessary to obtain an error that satisfies the bound from Theorem 6. We keep L and R from above and choose

S = (\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}), E = ϵ (\begin{matrix} 0 & 0 & 0 \\ 0 & ϵ^{2} & 1 - ϵ^{2} \\ 0 & 1 - ϵ^{2} & ϵ^{2} \end{matrix}) .

We have η = ∥S∥ = 1, γ = 1 and the eigenvalues of E/ϵ are 1 and −1 + 2ϵ² such that ∥E∥ = ϵ; we choose $ϵ < \frac{1}{2}$ such that choosing a τ from ϵ ≤ τ < 1 − ϵ is permitted. The eigenvalues of LMR are 1 and ϵ³. We obtain (using $ϵ \leq \frac{1}{2}$ ⁠)

∥ M R {(L M R)}^{+} L M - S ∥ = ‖(\begin{matrix} 0 & 0 & 0 \\ 0 & ϵ^{3} & ϵ (1 - ϵ^{2}) \\ 0 & ϵ (1 - ϵ^{2}) & \frac{{(1 - ϵ^{2})}^{2}}{ϵ} \end{matrix})‖ \geq \frac{{(1 - ϵ)}^{2}}{ϵ} \geq \frac{1 - 2 ϵ^{2}}{ϵ} \geq \frac{1}{2 ϵ} .

Without truncating small singular values, the error diverges as ϵ → 0, i.e., it does not satisfy the bound from Theorem 6. Here, the effect of E is completely erased by truncation:

{(L M R)}_{τ}^{+} = (\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}), ∥ M R {(L M R)}_{τ}^{+} L M - S ∥ = 0 .

2. The stability bound for matrices with non-unit operator norm

In this section, we provide an argument that extends the proof of Theorem 6 from matrices S with unit operator norm to matrices S with arbitrary operator norm. Suppose that the matrix M is the sum of a signal S and a noise contribution E, M = S + E. The signal satisfies rk(S) = rk(LSR), but we only know the strength ∥E∥ of the noise. Suppose that for ∥S∥ = ∥L∥ = ∥R∥ = 1, we obtain some error bound of the form

∥ M R {(L M R)}_{τ}^{+} L M - S ∥ \leq f (ϵ, γ, τ), ϵ = ∥ E ∥, γ = σ_{min} (L M R) .

(A1)

We can obtain an error bound for M′ = S′ + E′ where S′, L′, and R′ have arbitrary norms as follows: Set M = M′/∥S′∥, S = S/∥S′∥, E = E′/∥S′∥, L = L′/∥ L′∥, R = R′/∥R′∥. With these definitions, we have

∥ M R {(L M R)}_{τ}^{+} L M - S ∥ = \frac{∥ M^{'} R^{'} {(L^{'} M^{'} R^{'})}_{τ^{'}}^{+} L^{'} M^{'} - S^{'} ∥}{∥ S^{'} ∥},

(A2)

where τ′ = ∥L∥∥R∥∥S∥τ. Therefore, the bound from the last but one equation implies

∥ M^{'} R^{'} {(L^{'} M^{'} R^{'})}_{τ^{'}}^{+} L^{'} M^{'} - S^{'} ∥ \leq ∥ S^{'} ∥ f (ϵ, γ, τ),

(A3)

ϵ = \frac{∥ E^{'} ∥}{∥ S^{'} ∥}, γ = \frac{σ_{min} (L^{'} M^{'} R^{'})}{∥ L^{'} ∥ ∥ R^{'} ∥ ∥ S^{'} ∥}, τ = \frac{τ^{'}}{∥ L^{'} ∥ ∥ R^{'} ∥ ∥ S^{'} ∥} .

(A4)

In proofs, we assume ∥S∥ = ∥L∥ = ∥R∥ = 1 and we use ϵ = ∥E∥.

3. Alternative proof of the stability bound

In this section, we obtain a bound similar to the one from Theorem 6 using the ansatz by Caiafa and Cichocki.²¹

As above, we use M = S + E, ∥S∥ = ∥L∥ = ∥R∥ = 1 and rk(S) = rk(LSR).

Note that rk(S) = rk(LSR) implies rk(S) = rk(LS) = rk(SR) = rk(LSR). We use the matrix reconstruction Proposition 3 several times, sometimes with L or R replaced by the identity matrix. The proposition, e.g., provides S = S(LS)⁺ LS. Using that identity, we obtain the following two equalities:

S = S {(L S)}^{+} L (M - E),

(A5)

M = S + E = S {(L S)}^{+} L M + (1 - S {(L S)}^{+} L) E .

(A6)

In the same way, we obtain

S = (M - E) R {(S R)}^{+} S,

(A7)

M = S + E = M R {(S R)}^{+} S + E (1 - R {(S R)}^{+} S) .

(A8)

We will also use

L M R {(L M R)}_{τ}^{+} L M R = {(L M R)}_{τ} = [{(L M R)}_{τ} - L M R] + L S R + L E R .

(A9)

We decompose into three parts

M R {(L M R)}_{τ}^{+} L M = (A) + (B) + (C) .

(A10)

We insert Eq. (A6) for M at the beginning of the expression and Eq. (A8) for M on the end of the expression. In the following equations, spaces separate factors which come from different equations. In part (A) below, we insert Eq. (A9):

\begin{matrix} (A) & = S {(L S)}^{+} L M R {(L M R)}_{τ}^{+} L M R {(S R)}^{+} S \\ = S {(L S)}^{+} [{(L M R)}_{τ} - L M R] {(S R)}^{+} S \\ + S {(L S)}^{+} L S R {(S R)}^{+} S \\ + S {(L S)}^{+} L E R {(S R)}^{+} S, \end{matrix}

(A11a)

\begin{matrix} (B) & = S {(L S)}^{+} L M R {(L M R)}_{τ}^{+} L E (1 - R {(S R)}^{+} S) \\ + (1 - S {(L S)}^{+} L) E R {(L M R)}_{τ}^{+} L M R {(S R)}^{+} S, \end{matrix}

(A11b)

(C) = (1 - S {(L S)}^{+} L) E R {(L M R)}_{τ}^{+} L E (1 - R {(S R)}^{+} S) .

(A11c)

The expression in Eq. (A11a) is equal to S. We use the relation $∥ A A_{τ}^{+} ∥ = ∥ A_{τ} A_{τ}^{+} ∥ \leq 1$ and obtain the following bound:

\begin{align} ∥ M R {(L M R)}_{τ}^{+} L M - S ∥ \\ \leq ∥ {(L S)}^{+} ∥ ∥ {(S R)}^{+} ∥ (∥ {(L M R)}_{τ} - L M R ∥ + ϵ) \\ + ∥ {(L S)}^{+} ∥ ∥ 1 - R {(S R)}^{+} S ∥ ϵ \\ + ∥ {(S R)}^{+} ∥ ∥ 1 - S {(L S)}^{+} L ∥ ϵ \\ + ∥ 1 - S {(L S)}^{+} L ∥ ∥ 1 - R {(S R)}^{+} S ∥ ∥ {(L M R)}_{τ}^{+} ∥ ϵ^{2} . \end{align}

(A12)

This bound has been given by Caiafa and Cichocki²¹ for the case that L and R have exactly r = rk(S) rows and columns (such that the matrix LSR is invertible). They proceed by defining constants a, b, and c which are independent of the threshold τ and of noise strength ϵ = ∥E∥ and obtain a bound of the form aτ + bϵ + cϵ²/τ.

We continue by analyzing how all terms in the last equation depend on L, R, and S. This will provide a bound similar to that of Theorem 6.

Because LSR, LS, and SR have all rank r = rk(S), the relation σ_min(LSR) = σ_r(LSR) holds for these three matrices. We obtain

γ = σ_{min} (L S R) = σ_{r} (L S R) \leq σ_{r} (L S) σ_{1} (R) \leq σ_{r} (L S) = σ_{min} (L S),

(A13)

where the first inequality is provided by Ref. 46 (Theorem 3.3.16, p. 178). This provides

∥ S {(L S)}^{+} L ∥ \leq ∥ {(L S)}^{+} ∥ = \frac{1}{σ_{min} (L S)} \leq \frac{1}{γ}

(A14)

and the same bound applies to ∥R(SR)⁺S∥. Note that γ ≤ ∥S∥ = 1.Using 1 ≤ 1/γ, we obtain

∥ 1 - S {(L S)}^{+} L ∥ \leq \frac{2}{γ}, ∥ 1 - R {(S R)}^{+} S ∥ \leq \frac{2}{γ}

(A15)

and

∥ M R {(L M R)}_{τ}^{+} L M - S ∥ \leq \frac{∥ {(L M R)}_{τ} - L M R ∥ + 5 ϵ + ∥ {(L M R)}_{τ}^{+} ∥ ϵ^{2}}{γ^{2}} .

(A16)

The inequality holds for arbitrary values of γ, τ, and ϵ.

Now, we assume ϵ ≤ τ < γ − ϵ and use bounds from Lemma 7. This provides

∥ M R {(L M R)}_{τ}^{+} L M - S ∥ \leq \frac{6 ϵ}{γ^{2}} + \frac{ϵ^{2}}{γ^{2} (γ - ϵ)} \leq \frac{7 ϵ}{γ^{2}} .

(A17)

Without the assumption ϵ ≤ τ < γ − ϵ, we obtain

∥ M R {(L M R)}_{τ}^{+} L M - S ∥ \leq \frac{τ + 5 ϵ}{γ^{2}} + \frac{ϵ^{2}}{γ^{2} τ} .

(A18)

This is again the bound aτ + bϵ + cϵ²/τ, but with a = 1/γ², b = 5/γ², and c = 1/γ². If we select ϵ = τ in this bound, we obtain exactly Eq. (A17). As ${(L M R)}_{τ}^{+}$ is the same operator for all τ ∈ [ϵ, γ − ϵ), the bound holds not only for ϵ = τ but also for all τ ∈ [ϵ, γ − ϵ): We obtain Eq. (A17) again.

4. Known results on matrix product representations

This section reviews known results on matrix product state/tensor train representations used in Sec. VI B 3. It also provides full formal details for the results which were used.

Given a tensor $t \in C^{d_{1} \times \dots \times d_{n}}$ ⁠, a matrix product representation of the tensor is given by

{[t]}_{i_{1}, \dots, i_{n}} = G_{1} (i_{1}) \dots G_{m} (i_{m}) H_{m} G_{m + 1} (i_{m + 1}) \dots G_{n} (i_{n}) (i_{k} \in {1, \dots, d_{k}}),

(A19)

where D₀ = D_n = 1, $H_{m} \in C^{D_{m} \times D_{m}}$ ⁠, $G_{k} (i_{k}) \in C^{D_{k - 1} \times D_{k}}$ ⁠, i_k ∈ {1, …, d_k}, and m ∈ {1, …, n − 1}. For simplicity, $H_{m} = 1_{D_{m}}$ may be used. The G_k are called the cores of the representation while the matrices G_k(i_k) give the representation its name. The left and right unfoldings of the cores are given by

G_{k}^{L} \in C^{D_{k - 1} d_{k} \times D_{k}} G_{k}^{R} \in C^{D_{k - 1} \times d_{k} D_{k}}

(A20)

and they have the same entries as G_k, e.g., ${[G_{k}^{L}]}_{b_{k - 1} i_{k}, b_{k}} = {[G_{k} (i_{k})]}_{b_{k - 1}, b_{k}}$ ⁠. This notation is partially inspired by Ref. 48. The left and right interface matrices are given by

G_{\leq k} = (G_{\leq k - 1} \otimes 1_{d_{k}}) G_{k}^{L} \in C^{d_{1} \dots d_{k} \times D_{k}} (k \in {1 \dots m}),

(A21)

G_{> k} = (1_{d_{k + 1}} \otimes G_{> k + 1}) {(G_{k + 1}^{R})}^{⊺} \in C^{d_{k + 1} \dots d_{n} \times D_{k}} (k \in {m \dots n - 1}),

(A22)

where G_≤0 = 1 and G_>n = 1. The unfolding t_k is the d₁, …, d_k × d_k+1, …, d_n matrix with the same entries as t and it can be written as

t_{m} = G_{\leq m} H_{m} {(G_{> m})}^{⊺} .

(A23)

It is well known that a singular value decomposition of the unfolding t_k can be obtained efficiently:^13,24

Remark 35.

Fix m ∈ {1, …, n − 1}, let t have a matrix product representation as in Eq. () with positive-semidefinite diagonal H_m and assume orthogonal cores,⁴⁷ i.e.,

{(G_{k}^{L})}^{*} G_{k}^{L} = 1_{D_{k}} (k \leq m) {(G_{k}^{R})}^{*} G_{k}^{R} = 1_{D_{k}} (k > m),

(A24a)

{(G_{\leq k})}^{*} G_{\leq k} = 1_{D_{k}} (k \leq m) {(G_{> k})}^{*} G_{> k} = 1_{D_{k}} (k \geq m) .

(A24b)

An arbitrary matrix product representation can be efficiently converted into such an orthogonal representation.^13,24 Then

t_{m} = G_{\leq m} H_{m} {(G_{> m})}^{⊺}

(A25)

is a singular value decomposition of the unfolding matrix t_m. Let

U_{\leq m} = {(G_{\leq m})}^{*}

and

V_{> m} = {(G_{> m})}^{*}

⁠. Then

U_{\leq m} t_{m} {(V_{> m})}^{⊺} = H_{m}

(A26)

has the same singular values as t_m.□

The following lemma provides an efficient, incremental construction of matrices U_≤k and V_>k such that the matrix $U_{\leq m} t_{m} {(V_{> m})}^{⊺}$ has the same rank as t_m. More general matrices are permitted than in Eq. (A26) and the rank is preserved [Eq. (A30c)] but the singular values of $U_{\leq m} t_{m} {(V_{> m})}^{⊺}$ can differ from those of t_m. The proof of Lemmata 36 and 37 has been sketched in Ref. 19. In the premise of the following Lemma, it is possible to choose U_k and V_k as submatrices of permutation matrices (the case considered in Ref. 19), but the actual proof is independent of this choice.

Lemma 36.

Assume a matrix product representation of t as in Eq. () with H_m =

1

⁠. In the following, m ∈ {1, …, n − 1} is fixed, j ∈ {1, …, m} and k ∈ {m, …, n − 1}. Choose matrices

U_{j} \in C^{D_{j} \times D_{j - 1} d_{j}}

and

V_{k + 1} \in C^{D_{k} \times d_{k + 1} D_{k + 1}}

⁠. Set U_≤0 = 1, V_>n = 1 and

U_{\leq j} = U_{j} (U_{\leq j - 1} \otimes 1_{d_{j}}) \in C^{D_{j} \times d_{1} \dots d_{j}},

(A27a)

V_{> k} = V_{k + 1} (1_{d_{k + 1}} \otimes V_{> k + 1}) \in C^{D_{k} \times d_{k + 1} \dots d_{n}} .

(A27b)

In addition, set

Ũ_{j} = [(U_{\leq j - 1} G_{\leq j - 1}) \otimes 1_{d_{j}}] G_{k}^{L} \in C^{D_{j - 1} d_{j} \times D_{j}},

(A28a)

{\tilde{V}}_{k} = [1_{d_{k + 1}} \otimes (V_{> k + 1} G_{> k + 1})] {(G_{k + 1}^{R})}^{⊺} \in C^{d_{k + 1} D_{k + 1} \times D_{k}} .

(A28b)

If the rank equalities

r k (U_{j} Ũ_{j}) = r k (Ũ_{j})

(A29a)

r k (V_{k} {\tilde{V}}_{k}) = r k ({\tilde{V}}_{k})

(A29b)

hold, then the following rank equalities hold as well,

r k (U_{\leq j} G_{\leq j}) = r k (G_{\leq j}),

(A30a)

r k (V_{> k} G_{> k}) = r k (G_{> k}),

(A30b)

r k (U_{\leq m} t_{m} {(V_{> m})}^{⊺}) = r k (t_{m}) .

(A30c)

Proof.

Note that

U_{\leq j} G_{\leq j} = U_{j} Ũ_{j}, V_{> k} G_{> k} = V_{k} {\tilde{V}}_{k} .

(A31)

The matrices

Ũ_{k}

and Ṽ_k can be computed efficiently by using Eq. () to compute U_≤jG_≤j and V_>kG_>k. If we refer to Proposition 3 in the remainder of the proof, we use the fact that rk(LM) = rk(M) implies rk(LMR) = rk(MR) for three matrices L, M, and R.

For j = 1, we have U_≤j = U_j and

Ũ_{j} = G_{j}^{L} = G_{\leq j}

⁠, i.e., () implies () for j = 1. Suppose that Eq. () holds for some j − 1 ∈ {1, …, m − 1}, i.e.,

r k (U_{\leq j - 1} G_{\leq j - 1}) = r k (G_{\leq j - 1}),

(A32)

which implies

r k ((U_{\leq j - 1} G_{\leq j - 1}) \otimes 1_{d_{j}}) = r k (G_{\leq j - 1} \otimes 1_{d_{j}}) .

(A33)

This in turn implies (Proposition 3)

r k (Ũ_{j}) = r k ([(U_{\leq j - 1} G_{\leq j - 1}) \otimes 1_{d_{j}}] G_{j}^{L}) = r k ([G_{\leq j - 1} \otimes 1_{d_{j}}] G_{j}^{L}) = r k (G_{\leq j}) .

(A34)

Then

r k (U_{\leq j} G_{\leq j}) = r k (U_{j} Ũ_{j}) = r k (Ũ_{j}) = r k (G_{\leq j}),

(A35)

where we used in turn Eq. (), Eq. () and the last but one equation. This shows that Eq. () holds for j ∈ {1, …, m}.

The proof of Eq. () proceeds in the same way: For k = n − 1, we have V_>k = V_k+1 and

{\tilde{V}}_{k} = {(G_{k + 1}^{R})}^{⊺} = G_{> k}

⁠, i.e., () implies () for k = n − 1. Suppose that () holds for some k + 1 ∈ {m + 1, …, n}, i.e.,

\begin{array}{l} r k ({\tilde{V}}_{k}) & = r k ([1_{d_{k + 1}} \otimes (V_{> k + 1} G_{> k + 1})] {(G_{k + 1}^{R})}^{⊺}) = r k ([1_{d_{k + 1}} \otimes (G_{> k + 1})] {(G_{k + 1}^{R})}^{⊺}) \\ = r k (G_{> k}) . \end{array}

This implies

r k (V_{> k} G_{> k}) = r k (V_{k} {\tilde{V}}_{k}) = r k ({\tilde{V}}_{k}) = r k (G_{> k}) .

(A36)

We have used, in turn, (), () and the last but one equation. This implies that () holds for k ∈ {m, …, n − 1}.

The unfolding can be written as

t_{m} = G_{\leq m} {(G_{> m})}^{⊺}

[Eq. ()]. Applying Eqs. () and () and Proposition 3 provides

\begin{align} r k (t_{m}) & = r k (G_{\leq m} {(G_{> m})}^{⊺}) = r k (U_{\leq m} G_{\leq m} {(G_{> m})}^{⊺}) = r k (U_{\leq m} G_{\leq m} {(G_{> m})}^{⊺} {(V_{> m})}^{⊺}) \\ = r k (U_{\leq m} t_{m} {(V_{> m})}^{⊺}) . \end{align}

(A37)

This shows that Eq. () holds and finishes the proof.

In the last Lemma, it was possible to choose U_j as submatrices of permutation matrices. The following lemma shows that this implies that the U_≤j are submatrices of permutation matrices as well and that the position of the non-zero entries of U_≤j can be computed efficiently.

Lemma 37.

In the following, let j ∈ {1, …, m}. Choose matrices

U_{j} \in C^{D_{j} \times D_{j - 1} d_{j}}

which are submatrices of permutation matrices. Set U_≤0 = 1 and set

U_{\leq j} = U_{j} (U_{\leq j - 1} \otimes 1_{d_{j}}) \in C^{D_{j} \times d_{1} \dots d_{j}} .

(A38)

In the following, we also assume i_j ∈ {1, …, d_j} and b_j ∈ {1, …, D_j}. Denote the set of unit entries in a given row of U_j and U_≤j by

f_{j} (b_{j}) = {(b_{j - 1}, i_{j}) : {[U_{j}]}_{b_{j}, (b_{j - 1}, i_{j})} = 1},

(A39a)

f_{\leq j} (b_{j}) = {(i_{1}, \dots, i_{j}) : {[U_{\leq j}]}_{b_{j}, (i_{1} \dots i_{j})} = 1} .

(A39b)

The latter set is given by

f_{\leq j} (b_{j}) = \{(i_{1}, \dots, i_{j}) : (b_{j - 1}, i_{j}) \in f_{j} (b_{j}) a n d {(i}_{1, \dots, i_{j - 1)}} \in f_{\leq j - 1} (b_{j - 1})\},

(A40)

and U_≤j is a submatrix of a permutation matrix, i.e., we have |f_≤j(b_j)| ≤ 1 and the entries of U_≤j are given by

{[U_{\leq j}]}_{b_{j}, (i_{1} \dots i_{j})} = \{\begin{matrix} 1, & i f (i_{1}, \dots, i_{j}) \in f_{\leq j} (b_{j}), \\ 0, & o t h e r w i s e . \end{matrix})

(A41)

Proof.

As U_j is a submatrix of a permutation matrix, we have |f_j(b_j)| ≤ 1 and the entries of U_j are given by

{[U_{j}]}_{b_{j}, (b_{j - 1}, i_{j})} = \{\begin{matrix} 1, & if (b_{j - 1}, i_{j}) \in f_{j} (b_{j}), \\ 0, & otherwise. \end{matrix})

(A42)

The entries of U_≤j are given by [Eqs. () and ()]

\begin{array}{l} {[U_{\leq j}]}_{b_{j}, (i_{1} \dots i_{j})} & = \sum_{b_{j - 1}} {[U_{j}]}_{b_{j}, (b_{j - 1}, i_{j})} {[U_{\leq j - 1}]}_{b_{j - 1}, (i_{1} \dots i_{j - 1})} = \sum_{\begin{array}{c} b_{j - 1} : \\ (b_{j - 1}, i_{j}) \in f_{j} (b_{j}) \end{array}} {[U_{\leq j - 1}]}_{b_{j - 1}, (i_{1} \dots i_{j - 1})} \\ = \{\begin{matrix} {[U_{\leq j - 1}]}_{b_{j - 1}, (i_{1} \dots i_{j - 1})} where (b_{j - 1}, i_{j}) \in f_{j} (b_{j}) & if | f_{j} (b_{j}) | = 1, \\ 0 & otherwise. \end{matrix}) \end{array}

Here, we used that there is at most one element (b_j−1, i_j) ∈ f_j(b_j). Note that |f_≤j(b_j)| ≤ 1 and Eq. () hold for j = 0 (as f_≤0(1) = {1}); assume that the two conditions hold for some j − 1 ∈ {0, …, m − 1}. In the following, we show that they also hold for j. If we apply the assumption to the last equation, we obtain

{[U_{\leq j}]}_{b_{j}, (i_{1} \dots i_{j})} = \{\begin{matrix} 1, & if (b_{j - 1}, i_{j}) \in f_{j} (b_{j}) and (i_{1}, \dots, i_{j - 1}) \in f_{\leq j - 1} (b_{j - 1}), \\ 0, & otherwise. \end{matrix})

(A43)

The set f_≤j(b_j) has at most one element because f_j(b_j) has at most one element and because we assumed that f_≤j−1(b_j−1) (for the single possible value of b_j−1) has at most one element as well. This finishes the proof.

5. Sequence of local quantum operations as PMPS representation

This section introduces the locally purified matrix product state (PMPS) representation and discusses the known fact that a sequentially prepared mixed quantum state can be represented as a PMPS or as an MPO (Lemma 38). The PMPS representation^12,26 provides an alternative to the MPO representation for positive semidefinite operators such as mixed quantum states. The purification is given in terms of n ancilla systems of dimensions $d_{k}^{'}$ with bases ${| φ_{i_{k}^{'}}^{(k)} ⟩)}_{k = 1}^{d_{k}^{'}}$ ⁠. A PMPS representation of ρ is given by

\begin{matrix} ρ & = & {T r}_{1^{'} \dots n^{'}} (|Ψ⟩ ⟨Ψ|), D_{0} = D_{n} = 1, \\ ⟨ ϕ_{i_{1}}^{(1)} φ_{i_{1}^{'}}^{(1)} \dots ϕ_{i_{n}}^{(n)} φ_{i_{n}^{'}}^{(n)} |) Ψ ⟩ & = & G_{1} (i_{1}, i_{1}^{'}) G_{2} (i_{2}, i_{2}^{'}) \dots G_{n} (i_{n}, i_{n}^{'}), i_{k} \in {1, \dots, d_{k}}, \\ G_{k} (i_{k}, i_{k}^{'}) & \in & C^{D_{k - 1} \times D_{k}}, i_{k}^{'} \in {1, \dots, d_{k}^{'}} . \end{matrix}

(A44)

Given the tensors G_k of a PMPS representation, the tensors ${\tilde{G}}_{k}$ of an MPO representation are given by

{\tilde{G}}_{k} (i_{k}, j_{k}) = \sum_{i_{k}^{'} = 1}^{d_{k}^{'}} G_{k} (i_{k}, i_{k}^{'}) \otimes \bar{G_{k} (i_{k}^{'}, j_{k})},

(A45)

where the overline denotes the complex conjugate. Equation (A45) shows that given a PMPS representation of bond dimension D, we can directly construct an MPO representation with bond dimension D². However, an MPO representation with bond dimensions smaller than D² can exist. It has been shown that there is a family of quantum states on n systems which can be represented as an MPO with bond dimension independent of n but the bond dimension of any PMPS representation of those states increases with n.¹² This is an advantage of the MPO representation, but on the other hand, deciding whether a given MPO representation represents a positive semidefinite operator is an NP-hard problem, i.e., a solution in polynomial (in n) time is unlikely.¹⁸ The PMPS representation has the advantage that it always represents a positive semidefinite operator by definition. The relative merits of the MPO and PMPS representations of a mixed quantum state depend on the application.

Suppose that a quantum state $ρ \in D (H_{1, \dots, n})$ was prepared via quantum operations $W_{k} : B (H_{k - 1}) \to B (H_{k - 1, k})$ ⁠, i.e.,

ρ = W_{n} W_{n - 1} \dots W_{3} W_{2} (σ), σ \in D (H_{1}) .

(A46)

Clearly, this is an efficient representation of the quantum state ρ as it is described by at most nd⁶ parameters. It is known that such a representation can be efficiently—i.e., with at most poly(n) computational time—converted into an MPO representation or a PMPS representation.^17,18 The following lemma provides the technical details of the conversion.

Lemma 38.

Let

Y_{k}^{'}

(k ∈ {1, …, n}) be systems with

\dim (Y_{n}^{'}) = 1

⁠. Let

ρ_{1} \in B (1, Y_{1}^{'})

and

W_{k} \in B (B (Y_{k - 1}^{'}); B (k, Y_{k}^{'}))

⁠. A linear operator

ρ \in B (H_{1 \dots n})

is given by

ρ = ρ_{n}, ρ_{k} = (i d \otimes W_{k}) (ρ_{k - 1}) (k \in {2, \dots, n}) .

(A47)

Let

F_{i_{k}}^{(k)}

and

F_{b_{k}}^{(Y_{k}^{'})}

be orthonormal operator bases of systems k and

Y_{k}^{'}

⁠. An MPO representation of ρ is given by

{[G_{1 (i_{1})}]}_{b_{0}, b_{1}} = ⟨ F_{i_{1}}^{(1)} \otimes F_{b_{1}}^{(Y_{1}^{'})}, ρ_{1} ⟩,

(A48a)

{[G_{k} (i_{k})]}_{b_{k - 1}, k_{k}} = ⟨ F_{i_{k}}^{(k)} \otimes F_{b_{k}}^{(Y_{k}^{'})}, W_{k} (F_{b_{k - 1}}^{(Y_{k - 1}^{'})}) ⟩, (k \in {2, \dots, n}) .

(A48b)

The bond dimensions of the representation are given by

D_{k} = {(d_{Y_{k}^{'}})}^{2}

(k ∈ {1, …, n − 1}, D₀ = D_n = 1). Now, let ρ₁ be a quantum state and

W_{k}

be CPTP linear maps. In addition, let

ρ_{1} = \sum_{i = 1}^{r_{1}} |ψ_{i}⟩ ⟨ψ_{i}|, W_{k} (\cdot) = \sum_{i = 1}^{r_{k}} E_{i}^{(k)} \cdot {(E_{i}^{(k)})}^{*} (k \in {2, \dots, n})

(A49)

be a decomposition of ρ₁ into r₁ = rk(ρ₁) orthogonal vectors and a Kraus decomposition of

W_{k}

where r_k equals the Kraus rank of

W_{k}

(i.e.,

r_{1} \leq d_{1} d_{Y_{1}^{'}}

and

r_{k} \leq d_{k} d_{Y_{k}^{'}} d_{Y_{k - 1}^{'}}

⁠). Let

{| i_{k} {⟩)}_{k}}_{i_{k}}

and

{| b_{k} {⟩)}_{Y_{k}^{'}}}_{b_{k}}

be orthonormal bases of

H_{k}

and

Y_{k}^{'}

⁠, respectively. A PMPS representation of ρ is given by

{[G_{1} (i_{1}, i_{1}^{'})]}_{b_{0}, b_{1}} =_{1} ⟨) i_{1} |_{Y_{1}^{'}} ⟨ b_{1} | ψ_{i_{1}^{'}} ⟩,

(A50a)

{[G_{k} (i_{k}, i_{k}^{'})]}_{b_{k - 1}, b_{k}} =_{k} ⟨) i_{k} |_{Y_{k}^{'}} {⟨ b_{k} | E_{i_{k}^{'}}^{(k)} | b_{k - 1} ⟩}_{Y_{k - 1}^{'}} (k \in {2, \dots, n}) .

(A50b)

The bond dimensions of the representation are given by

D_{k} = d_{Y_{k}^{'}}

(k ∈ {1, …, n − 1}, D₀ = D_n = 1) and the ancilla dimensions are given by r_k.

Proof.

For the MPO representation, evaluate the operator basis elements of ρ from Eq. (A47) and compare with the operator basis elements of the representation [Eq. (15)]. For the PMPS representation, evaluate the matrix entries of ρ from Eq. (A47) inserting Eq. (A49) and compare with the matrix entries of the representation [Eq. (A44)].

6. General linear maps as measurements

Consider a quantum state $ρ \in D (Y)$ and an arbitrary linear map $N \in B (B (Y); B (X))$ ⁠. Above, we work with linear maps $N$ which are not necessarily CPTP and therefore do not represent a physical operation on the quantum state. Such a map $N$ is of relevance only if $N (ρ)$ can be obtained from the outcomes of physical measurements on ρ. In this section, we show how this can be achieved, allowing the reconstruction scheme from Sec. III C to be used for quantum state tomography.

First, we construct a set of observables $G_{i} \in B (Y)$ (⁠ $i \in {1, \dots, 2 d_{X}^{2}}$ ⁠) whose expectation values Tr(G_iρ) in the state ρ can be used to compute $N (ρ)$ ⁠. Second, we construct a positive-operator-valued measure (POVM) with $2 d_{x}^{2} + 1$ elements $E_{i} \in B (Y)$ such that the outcome probabilities Tr(E_iρ) in the state ρ can also be used to compute $N (ρ)$ ⁠.

We denote the components of $N (ρ)$ in an operator basis $F_{i}^{(X)}$ of X by s_i,

s_{i} = ⟨ F_{i}^{(X)}, N (ρ) ⟩ i \in {1, \dots, d_{X}^{2}} d_{X} = \dim (X) .

(A51)

The key tool is the following property of the map $N^{*}$ ⁠:

s_{i} = ⟨ N^{*} (F_{i}^{(X)}), ρ ⟩ = T r (H_{i} ρ), H_{i} = {(N^{*} (F_{i}^{(X)}))}^{*} .

(A52)

Since H_i may not be Hermitian, we use its Hermitian and skew-Hermitian components

G_{i} = \frac{1}{2} (H_{i} + H_{i}^{*}) G_{i + d_{X}^{2}} = \frac{1}{2 i} (H_{i} - H_{i}^{*}) H_{i} = G_{i} + i G_{i + d_{X}^{2}} .

(A53)

Using the observables G_i, the components s_i can be expressed as follows:

s_{i} = ⟨ N^{*} (F_{i}^{(X)}), ρ ⟩ = T r (G_{i} ρ) + i T r (G_{i + d_{X}^{2}} ρ) .

(A54)

In other words, the expectation values of the G_i provide the real and imaginary parts of s_i,

R e (s_{i}) = T r (G_{i} ρ) I m (s_{i}) = T r (G_{i + d_{X}^{2}} ρ) .

(A55)

If these expectation values can be measured, we already obtain a way to obtain $N (ρ)$ from physical measurements on ρ even if $N$ is not CPTP. Furthermore, we construct a POVM whose measurement on ρ also allows to determine $N (ρ)$ ⁠. We choose coefficients $c_{i} \in R$ and c > 0 such that the following operators become positive semidefinite:

G_{i} + c_{i} 1 \geq 0 1 - c \sum_{i = 1}^{2 d_{X}^{2}} (G_{i} + c_{i} 1) \geq 0 .

(A56)

We define $2 d_{X}^{2} + 1$ POVM elements by

E_{0} = 1 - \sum_{i = 1}^{2 d_{X}^{2}} E_{i} E_{i} = c (G_{i} + c_{i} 1) .

(A57)

Clearly, the expectation values of the G_i are related to the POVM probabilities by

T r (G_{i} ρ) = \frac{1}{c} (T r (E_{i} ρ) - c_{i}) .

(A58)

The coefficients s_i of $N (ρ)$ can be obtained from these expectation values using Eq. (A55). As a consequence, the POVM probabilities of the given POVM allow us to determine $N (ρ)$ even if $N$ is not CPTP.

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2018

Author(s)

Petz recovery versus matrix reconstruction

I. INTRODUCTION

II. PRELIMINARIES

A. Notation and basic definitions

B. MPS, MPO, and PMPS representations

C. The Petz recovery map

III. RECONSTRUCTION OF BIPARTITE STATES SUBJECTED TO LOCAL OPERATIONS

A. Reconstruction of low-rank matrices

B. Stability of the reconstruction under perturbation

C. Reconstruction of bipartite states

IV. PETZ RECOVERY OF BIPARTITE STATES SUBJECTED TO LOCAL QUANTUM OPERATIONS

V. COMPARISON OF PETZ RECOVERY AND STATE RECONSTRUCTION

A. Comparison for four-partite systems

VI. EFFICIENT RECONSTRUCTION OF STATES ON SPIN CHAINS VIA RECURSIVELY DEFINED MEASUREMENTS

A. Reconstruction of states from local marginal states

B. Long-ranged measurements

1. Recovery from long-ranged measurements

2. Reconstruction from long-ranged measurements

3. Recovery vs. reconstruction for long-ranged measurements

ACKNOWLEDGMENTS

APPENDIX:ADDITIONAL RESULTS AND BACKGROUND INFORMATION

1. Optimality of the stability bound

2. The stability bound for matrices with non-unit operator norm

3. Alternative proof of the stability bound

4. Known results on matrix product representations

5. Sequence of local quantum operations as PMPS representation

6. General linear maps as measurements

REFERENCES

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Petz recovery versus matrix reconstruction

I. INTRODUCTION

II. PRELIMINARIES

A. Notation and basic definitions

B. MPS, MPO, and PMPS representations

C. The Petz recovery map

III. RECONSTRUCTION OF BIPARTITE STATES SUBJECTED TO LOCAL OPERATIONS

A. Reconstruction of low-rank matrices

B. Stability of the reconstruction under perturbation

C. Reconstruction of bipartite states

IV. PETZ RECOVERY OF BIPARTITE STATES SUBJECTED TO LOCAL QUANTUM OPERATIONS

V. COMPARISON OF PETZ RECOVERY AND STATE RECONSTRUCTION

A. Comparison for four-partite systems

VI. EFFICIENT RECONSTRUCTION OF STATES ON SPIN CHAINS VIA RECURSIVELY DEFINED MEASUREMENTS

A. Reconstruction of states from local marginal states

B. Long-ranged measurements

1. Recovery from long-ranged measurements

2. Reconstruction from long-ranged measurements

3. Recovery vs. reconstruction for long-ranged measurements

ACKNOWLEDGMENTS

APPENDIX:ADDITIONAL RESULTS AND BACKGROUND INFORMATION

1. Optimality of the stability bound

2. The stability bound for matrices with non-unit operator norm

3. Alternative proof of the stability bound

4. Known results on matrix product representations

5. Sequence of local quantum operations as PMPS representation

6. General linear maps as measurements

REFERENCES

Citing articles via

Submit your article

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Related Content

Resources

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