We study steady-states of quantum Markovian processes whose evolution is described by local Lindbladians. We assume that the Lindbladian is gapped and satisfies quantum detailed balance with respect to a unique full-rank steady state σ. We show that under mild assumptions on the Lindbladian terms, which can be checked efficiently, the Lindbladian can be mapped to a local Hamiltonian on a doubled Hilbert space that has the same spectrum and a ground state that is the vectorization of σ1/2. Consequently, we can use Hamiltonian complexity tools to study the steady states of such open systems. In particular, we show an area-law in the mutual information for the steady state of such 1D systems, together with a tensor-network representation that can be found efficiently.

Understanding the structure and physical properties of open many-body quantum systems is a major problem in condensed matter physics. Over the years, these systems have been studied using a plethora of methods from statistical physics, many-body quantum theory, and functional analysis. More recently, with the advent of quantum computation and quantum information, these systems have also been studied using techniques coming from quantum information. Coming from these research paradigms, one would typically want to understand the computational complexity of these systems, the type of entanglement and correlations that they can create, and whether or not they can be represented efficiently on a classical computer.

This line of research has been well-established in the context of closed systems. In particular, there are many results characterizing the complexity of ground states of local Hamiltonians defined on a lattice. For example, it has been shown that for several families of Hamiltonians it is QMA-hard to approximate the ground state energy,1,2 implying that the ground state of such systems does not posses an efficient classical description unless QMA = NP. On the other hand, it is generally believed that gapped local Hamiltonians on a D-dimensional lattice satisfy an area-law of entanglement entropy3 and can be well approximated by efficient tensor network states. This has been rigorously proven in 1D,4 with some partial results in higher dimension.5–11 

A natural question to ask is whether, and to what extent, tools and techniques that are used to analyze the complexity of ground states of closed systems can be used to study the complexity of steady states of open systems. Specifically, in this paper we consider open systems that are described by Markovian dynamics that is generated by a local Lindbladian superoperator L=iLi (see Sec. II for a formal definition). The steady state of the system is then given by a density operator σ for which
L(σ)=iLi(σ)=0.
This is a homogeneous linear equation, which has a striking similarity to the corresponding problem in closed system
(Hϵ01)Ω=ihiϵ01Ω=0,
where H = ihi is a local Hamiltonian with ground state Ω and ground energy ϵ0. Moreover, as described in Sec. II, the spectrum of L is in the Re ≤ 0 part of the complex plane, and we can define the spectral gap of the Linbladian to be the largest non-zero real part of the spectrum. It is therefore tempting to try and use local Hamiltonian techniques to characterize σ. However, a quick inspection reveals the main obstacle for such a simple plan to work: Whereas the Hamiltonian is a self-adjoint (Hermitian) operator, which therefore has a real spectrum with a set of orthonormal eigenstates, the same is not true for the Lindbladian; it is generally not a self-adjoint operator, and consequently its spectrum might be complex with non-orthogonal eigenoperators. To make L self-adjoint, one might consider L*L, but this come at the price of losing locality. Alternatively, we might consider 12(L+L*), but it is not clear how the eigenstates of this operator are related to the eignstates of L.

A more sophisticated way of obtaining self-adjointness is by assuming that the Linbladian satisfies quantum detailed-balance.12,13 For such Linbladians there is a way of defining an inner product with respect to which L* is self-adjoint13–16 (see Subsection II C). This inner product, however, can be highly non-local (with respect to the underlying tensor-product structure) and might deform the natural geometry of the Hilbert space significantly. It is therefore not clear how to use it to bound correlations and other measures of locality in the steady state.

Nevertheless, in this paper we show that the quantum detailed-balance condition provides a surprisingly simple map that takes a local Lindbladian to a self-adjoint local super-operartor (i.e., a super Hamiltonian), which is then mapped to a local Hamiltonian using vectorization. All this is done while maintaining a direct relation between the steady state of the former and the ground state of the latter. Consequently, many of the bounds and properties that were proved for ground states, easily transform to the steady states of detailed-balance Lindbladians. In particular, we show that under mild conditions, which can be efficiently checked, steady states of gapped 1D Lindbladian with detailed balance satisfy an area-law for the mutual information. Moreover, just as in the 1D area-law case for local Hamiltonians, these steady states are well-approximated by an efficient tensor network.

We conclude this section by noting that our mapping is not new; it has already been used before, e.g., in Refs. 15 and 1720. Our main contribution is showing that this mapping results in a local Hamiltonian, rather than a general Hermitian operator, which enables us to take advantage of the local Hamiltonian machinery (see Subsection III A). It is important to note that the locality of Hamiltonian is non-trivial for general detailed balance local Lindbladian. For example, in Refs. 17 and 19, locality was achieved due to the further assumptions made in each work.

Several works studied the entanglement structure of steady states of open systems, see for example Refs. 18 and 2123. Here we will concentrate on Refs. 18 and 21, which study problems that are very close to ours.

In Ref. 18 the authors considered steady states of local Lindbladians on a D-dimensional lattice with unique, full-rank steady state. Under the assumption of gapped, detailed-balanced Linbladian, they have shown exponential decay of correlations in the steady state. To prove an area law, the authors additionally assumed a very strong form of fast convergence to the steady state, known as system-size independent log-Sobolev constant (see Ref. 24 for a definition). Under this assumption, and using the Lieb-Robinson bounds for open systems,25–27 they showed clustering of correlations in terms of the mutual information, and as a consequence, an area law for the mutual information. Specifically, for every region A in the lattice, the mutual information is bounded by
I(A:Ac)cloglog(σ1)|A|.
(1)
Notice, however, that for full rank steady states, ‖σ−1‖ grows at least exponentially with system size, and might even grow doubly exponential with system size, in which case (1) is no longer an area-law.
In Ref. 21 the authors proved an area-law for the steady state of what they refer to as a uniform family of Linbladians. Essentially, this is a local Linbladian defined on an infinite lattice, together with a set of boundary conditions that allow one to restrict the dynamics to finite regions {Λ} in the infinite lattice. In this setup, the authors assumed another strong form of convergence, which is called rapid mixing. Roughly, it assumes that the restriction of the system to any region Λ on the lattice has a unique fixed point σΛ, to which it converges exponentially fast from any initial state,
ρ(t)σΛ1c|Λ|δeγt,
(2)
where c, δ, γ are some region-independent constants. They manged to prove an area-law by assuming either that the Lindbladian is frustration free, or that the steady state is a pure state. Notice that the latter assumption addresses setups relevant to dissipative quantum state engineering,28 although rapid mixing restricts it to low-entangled quantum states (see Ref. 28). Under these assumptions, using Lieb-Robinson bounds, they have managed to show the following area-law with logarithmic corrections
I(A:Ac)c|A|log(|A|).
(3)

It is interesting to contrast these two results with several area-law results for ground states of gapped local Hamiltonians.4,11,29–31 In both cases, the proof follows the intuitive logic in which a fast convergence to the steady state (or ground state) can yield a bound on its entanglement. Indeed, given a region A in the lattice, we can prepare the system in a product state of this region with the rest of the system and then drive the system toward its steady state. If the convergence is quick and the underlying dynamic is local, not too much entanglement is created, which bounds the entanglement in the steady state.

There is, however, a highly non-trivial caveat in this program. It is not a priori clear that there exists a product state with a large overlap with the fixed point. If the overlap is exponentially small, it might take a long (polynomial) time for the dynamics to converge, even if the convergence is exponentially fast. This is the hard step in proving an area-law for ground states of gapped local Hamiltonians. In Hastings’ 1D area-law Proof,4 the initial state is ρAρAc, where ρA, ρAc are the reduced density matrices of the ground state on the regions A, Ac. Then a non-trivial overlap with ρAρAc is shown using an ingenious argument about the saturation of mutual information. In Refs. 11 and 2931 it is done by constructing an approximate ground state projector (AGSP) using a low-degree polynomial of the Hamiltonian. If the Schmidt rank of this AGSP times its approximation error is smaller than unity, we are promised that there exists some product state AAc with a large overlap with the groundstate.

In the open system area-law proof of Refs. 18 and 21 there is no parallel argument to lower-bound the overlap between the initial product state and the steady state. In addition, the convergence to the fixed point is always via the natural etL map—which might not be the most efficient one (in terms of the amount of entanglement that is generated). In Ref. 18 a worse-case overlap is assumed, which leads to the log(‖σ−1‖) factor in (3). (‖σ−1‖ is the smallest possible overlap of full-supported σ with another state.) In Ref. 21, it is assumed that fast convergence is independent of the initial state. This assumption, together with additional local fixed-point uniqueness assumptions that are also made in that work, imply that local expectation values at the fixed point can be efficiently calculated by a classical computer, which might make it less interesting from a computational point of view. In that respect, we believe that our Lindbladian → Hamiltonian mapping paves the way for a more fine-grained analysis of the problem. Finally, we remark that the assumptions of a gap plus detailed-balance do not generally imply short mixing times. Indeed, in the East model32,33 there is a gap and yet the mixing time is linear. This separates our result from Refs. 18 and 21.

The structure of the paper is as follows. In Sec. II we introduce the notation and background of the paper. In Sec. III we state and prove our main results. In Sec. IV we describe two non-trivial models that satisfy the assumptions we make, and present explicit forms for their super-Hamiltonian. We explicitly derive resultant local Hamiltonians and show it is indeed a local (or exponentially local). We then use Hamiltonian complexity techniques from Refs. 34 and 35 to demonstrate a finite gap in those Hamiltonians (and correspondingly in the original L) for a specific example. In  Appendix A we prove a lemma used in Theorem 3.2 regarding square root of sparse matrices. In  Appendices B and  C we prove that our examples indeed satisfy all the requirements and explicitly derive the super-Hamiltonian.

Throughout of this manuscript we use the big-O notation of computer science. If n is the asymptotic variable, then X=OY means that there exists C > 0 such that for sufficiently large n, X(n) ≤ C · Y(n). On the other hand, X=ΩY means that there exists a constant c > 0 such that X(n) ≥ c · Y(n) for sufficiently large n. Finally, X=ΘY means that there exist constants C > 0, c > 0 such that c · Y(N) ≤ X(n) ≤ C · Y(n) for suffciently large n.

We consider many-body systems composed of sites with d-dimensional local Hilbert space (“qudits,” which could physically correspond to spins, fermions, or hard core bosons) that reside on a lattice Λ of n sites and fixed spatial dimension (1D, 2D or 3D). The d-dimensional local Hilbert space at site x ∈ Λ is denoted by Hx, so that the global Hilbert space is
H=xΛHx,N=defdim(H)=d|Λ|.
We denote the space of linear operators on H by L(H), which possess a tensor-product structure as well, i.e., L(H)=xΛL(Hx). L(H) is by itself a finite-dimensional Hilbert-space with respect to the Hilbert-Schmidt inner product
O1,O2=defTr(O1O2),O1,O2L(H).
(4)
We denote the pth Schatten norm of an operator X by Xp=defTr(|X|p)1/p. The p = 1 case is the trace norm, commonly used to measure distance between density matrices. The p = 2 case is the norm derived from the Hilbert-Schmidt inner-product. The p = ∞ norm, defined by the maximal singular value of X, is the operator norm, and is denoted in this paper by ‖X‖.

Given a subset of the lattice S ⊆ Λ, we denote its local Hilbert space by HS, i.e., HS=xSHx. An operator OL(H) is supported on a subset S ⊆ Λ if it can be written as O=OS1HSc where OSL(HS) and 1HSc is the identity operator on the complementary Hilbert space.

Linear operators acting on the operators space L(H) are called superoperators, and will usually be denoted by curly letters (e.g., L,P). As in the case of operators, we say a superoperator B is supported on a subset S ⊆ Λ of the sites in the lattice if it can be written as BSISc, where BS is a superoperator on L(HS), and ISc is the identity operation on the complementary operators space.

Given a superoperator B, its dual map B* is the unique map that satisfies
O1,B(O2)=B*(O1),O2O1,O2L(H),
(5)
i.e., the adjoint under the Hilbert-Schmidt inner-product. It is easy to check that the adjoint of the superoperator B(ρ)=i,jcijFiρFj is given by B*(ρ)=i,jcij*FiρFj. We call B self-adjoint if B=B*.

In this section we provide some basic definitions and results about Markovian open quantum systems. For a detailed introduction to this subject we refer the reader to Refs. 36 and 37.

We study Markovian open quantum systems governed by a time-independent Lindbladian (also known as a Liouvillian) L in the Schrödinger picture. Formally, this means that the quantum state describing the system evolves continuously by a family of completely positive trace preserving (CPTP) maps {Tt}t0, which are given by Tt=defeLt so that
ρ(t)=Ttρ0tρ(t)=Lρ(t),ρ(0)=ρ0.
(6)
{Tt} is known in the literature as a dynamical semigroup or Quantum Markovian Semigroup due to the identity TtTs=Tt+s. A necessary and sufficient condition for L to be the generator of a semigroup is given by the following theorem (see, for example, Theorem 7.1 in Ref. 37):

Theorem 2.1.
Tt=eLt is a dynamical semigroup iff it can be written as
L(A)=i[H,A]+jLjALj12{LjLj,A},
(7)
where H is a Hermitian operator and Lj are operators.

The operator H is known as the Hamiltonian of the system, and it governs the coherent part of the evolution. The operators Lj are often called jump operators, and are responsible for the dissipative part of the evolution.

Given a Lindbladian L, the representation in (7) is not unique; for example, we can always change HH+c1 for c1 for cR without changing L. In addition, L will remain the same under the transformation LjLj+cj1 and HH+i2j(cj*LjcjLj). Therefore, we can assume without loss of generality that there is a representation of L in which both H and Lj are traceless.

This condition, however, does not fully fix the jump operators, and there can be several traceless jump operators representations of the same Lindbladian. The following theorem, which is an adaptation of Proposition 7.4 in Ref. 37, shows how these representations are related.

Theorem 2.2

(Freedom in the representation of L, Proposition 7.4 in Ref. 37). Let L be given by Eq. (7) with traceless {Lj} and H. If it can also be written using traceless {Li} and H, then H = Hand there exists a unitary matrix U such that Li=jUijLj, where the smaller set of jump operators is padded with zero operators.

The following are two important properties of Lindbladians that will be used extensively in this work. First, we define the notion of locality of a Lindbladian (which can be generalized to a general super-operator).

Definition 2.3

(k-body Lindbladians). We say that L is a k-body Lindbladian if it can be written as in (7) with jump operators Lj that are supported on at most k sites, and in addition the Hamiltonian H can be written as H = ihi, with every hi also supported on at most k sites. We say that L is a geometrically local k-body Lindbladian if, in addition to being k-body, every Lj and hi are supported on neighboring lattice sites.

Note that we are using the name “k-body” Lindbladian instead of “k-local:” as it is often done in the Hamiltonian complexity literature,1 the term k-local is reserved to local terms supported on at most k qubits. Here, we are allowing any constant local dimension d.

Next, we define the spectral gap of a Lindbladian.

Definition 2.4
(Spectral gap of a Lindbladian). The spectral gap of the Lindbladian is defined as the minimal real part of its non-zero eigenvalues
gap(L)=defmin0λSpec(L)|Re(λ)|.
(8)

The spectral gap of a Lindbladian controls the asymptotic convergence rate of the dynamics to a steady state,38 meaning that at long enough times, the distance to the steady state decays at rate lower bounded by the gap. However, it is important to notice that a finite gap by itself does not guarantee short-time convergence39 (see for example Refs. 32 and 33).

We conclude this section by listing few well-known facts about Lindbladians. We refer the reader to chapters 6 and 7 of Ref. 37 for proofs and details:

  • Fact 1: L is an hermicity-preserving superoperator: L(A)=L(A), and the same holds for L*.

  • Fact 2: There is always at least one quantum state σ that satisfies L(σ)=0, namely σ is a fixed point of the time-evolution. We refer to it as a steady state.

  • Fact 3: Tt is a contractive map and consequently, L has only non-positive real parts in its spectrum (Proposition 6.1 in Ref. 37).

In this work, we follow Refs. 13, 14, and 16 in defining the detailed-balance condition for the Lindbladian system. We begin by describing classical detailed-balance, and then use it to define the corresponding quantum condition.

Classically, let PMn be the transition matrix of a Markov chain over the discrete state of states {1, 2, …, n}, i.e., Pij=defProb(ji|j), and let π denote a probability distribution on these states. Then P is said to satisfy the detailed-balance condition with respect to a fully supported π (i.e., πi > 0 for all i ∈ [n]) if the probability of observing a ij transition is identical to the probability of observing a ji transition, when the system state is described by π. Mathematically, this means Pijπj = Pjiπi. This condition implies that π is a steady state of the Markov chain, however, the converse is not always true.40 

We can also write this condition in terms of matrices. Defining Γπ to be the diagonal matrix (Γπ)ij=πiδij, the detailed-balance condition can also be written as the matrix equality:
PΓπ=ΓπPT.
(9)
Alternatively, noting that Γπ is a positive definite matrix, the above condition is equivalent to the condition of Γπ1/2PΓπ1/2 being symmetric:
Γπ1/2PΓπ1/2=Γπ1/2PTΓπ1/2=(Γπ1/2PΓπ1/2)T.
(10)
These two conditions can be generalized to the quantum setting by changing PL and πσ for some reference quantum state σ. Just as in the classical case, we demand that σ is invertible, i.e., it has no vanishing eigenvalues. The quantum analog of Eq. (9) should be an equation over superoperators that act on quantum density operators. While P is replaced by L, Γπ should be replaced by a superoperator that multiplies an input state by the quantum state σ. But as σ does not commute with all quantum states (unless it is the completely mixed state), there are several ways to define this multiplication. In particular, for every s ∈ [0, 1], we might define the “multiplication by σ” superoperator (Here we do not use a curly letter to denote the superoperator Γs in order to be consistent with previous works.)
Γs(A)=defσ1sAσs.
(11)
It is easy to verify that, just as in the classical case, the superoperator Γs is self adjoint: Γs=Γs*. Moreover, Γs is invertible, and for every xR, we have Γsx(A)=σx(1s)Aσxs.

With this notation, the quantum detailed balance (QDB) condition is defined as the following generalization of the classical condition (9):

Definition 2.5
(Quantum detailed balance). A Lindbladian satisfies quantum detailed-balance with respect to some invertible (full-rank) state σL(H) and s ∈ [0, 1] if
LΓs=ΓsL*,
(12)
where Γs is the superoperator defined in (11).

We note that not every steady-state of a Lindbladian defines a superoperator Γs with respect to which the Lindbladian obeys detailed-balance.41 

As in the classical case, the quantum detailed-balance condition can be formulated in a few equivalent ways:

Claim 2.6.

Given a Lindbladian L, an invertible state σL(H), and s ∈ [0, 1], the following conditions are equivalent:

  1. (QDB1) L satisfies quantum detailed balance with respect to σ for some s.

  2. (QDB2) The superoperator Γs1/2LΓs1/2 is self-adjoint, namely Γs1/2LΓs1/2=Γs1/2L*Γs1/2.

  3. (QDB3) L* is self-adjoint with respect to the inner-product defined by
    A,Bs=defTr(AΓs(B))=Tr(Aσ1sBσs).
    (13)

Proof.
(QDB2) is equivalent to (QDB1) by a simple conjugation with Γ1/2, and (QDB3) is equivalent to (QDB1) by
L*(A),Bs=TrL*(A)Γs(B)=TrALΓs(B)=TrAΓsL*(B)=A,L*(B)s.

The s = 1 is commonly known as the Gelfand-Naimark-Segal (GNS) case, and its inner product A,B1=Tr(σAB) is often called the GNS inner-product. Similarly, the s = 1/2 case is called the Kubo-Martin-Schwinger (KMS) case, with A,B1/2 known as the KMS inner-product.

It was shown in Ref. 16 (see Lemmas 2.5 and 2.8 therein) that any superoperator that is self adjoint with respect to the inner product defined by some s ∈ [0, 1/2) ∪ (1/2, 1] is self adjoint with respect to the inner product defined by all s′ ∈ [0, 1], including s′ = 1/2. Therefore by QDB3, if L satisfies the detailed-balance condition for some s ∈ [0, 1/2) ∪ (1/2, 1], then it satisfies it for all other s ∈ [0, 1].

Finally, note that if L satisfies detailed balance with respect to σ, then σ is automatically a steady state, since Γs1(σ)=1 and therefore
L(σ)=ΓsL*Γs1(σ)=ΓsL*(1)=0.

We refer the reader to Refs. 15, 18, 42, and 43 for other definitions and generalization to quantum detailed-balance.

The QDB condition has well-known implications to the structure of the Lindbladian L. In this section we describe some of the central consequences of this condition, and in particular the so-called canonical form, which will be used later. None of the results in this section are new, as they already appeared in several works (see, for example, Refs. 13, 16, and 44). Nevertheless, we repeat some of the easy proofs for sake of completeness.

Our starting point is the modular superoperator, which is central for constructing a canonical representation of detail-balanced Lindbladians.

Definition 2.7
(The modular superoperator). Given an invertible state σ, its associated modular superoperator is defined by
Δσ(A)=defσAσ1.
(14)
We note that we are using a greek letter Δ to denote the modular superoperator instead of curly letter. This is done for being consistent with the notations of Refs. 16 and 20.

Below are few properties of Δσ, which follow almost directly from its definition.

Claim 2.8.

The modular superoperator Δσ has the following properties:

  1. Δσ1(A)=σ1Aσ and Δσ(A)=Δσ1(A).

  2. Self-adjointness: Δσ*=Δσ.

  3. Positivity: A,Δσ(A)>0 for all non-zero operators A.

Proof.
Properties 1 and 2 follow from definition. For 3, note that
A,Δσ(A)=Tr(AσAσ1)=Tr(σ1/2Aσ1/2)(σ1/2Aσ1/2)
if A ≠ 0, then by the invertability of σ it follows that also σ1/2−1/2 ≠ 0, hence the RHS above is positive.■

Properties 2 and 3 imply that Δσ is fully diagonalizable by an orthonormal eigenbasis of operators {Sα} and positive eigenvalues {eωα}, where α = 0, 1, 2, …, N2 − 1 is a running index. As Δσ(1)=1, we can fix S0=1N1, ω0 = 0, and conclude that Tr(Sα)=NS0,Sα=0 for every α > 0. Finally, from property (1) we find that
Δσ(Sα)=Δσ1(Sα)=eωαSα=eωαSα.
Therefore, Sα is also an eigenoperator of Δα with eigenvalue eωα. All of these properties are summarized in the following corollary, which defines the notion of a modular basis.

Corollary 2.9

(Modular basis). Given an invertible state σ, the modular superoperator Δσ has an orthonormal diagonalizing basis {Sα} with α = 0, 1, …, N2 − 1 and the following properties:

  1. Sα,Sβ=δαβ

  2. S0=1N1 and Tr(Sα) = 0 for α > 0.

  3. Δσ(Sα)=eωαSα and Δσ(Sα)=eωαSα.

  4. For every α there exists αsuch that Sα=Sα.

The basis {Sα} is called a modular basis.

Being a diagonalizing basis, the modular basis is unique up to unitary transformations within every eigenspace. The numbers ωα, which determine the eigenvalues of the modular superoperator, are called Bohr frequencies. We say that an operator O has well-defined Bohr frequency ω if Δσ(O) = eωO, i.e., it belongs to the eigenspace of Δσ with eigenvalue eω. For example, it is evident that σ itself (or any other state that commutes with it) has a well-defined Bohr frequency ω = 0 since Δσ(σ) = σ.

The Bohr frequencies are related to the eigenvalues of σ. To see this, we use an explicit construction of a modular basis. Given the spectral decomposition σ=ieEiii, we define the operators Oij=defij with i, j ∈ 1, …, N, and note that they form an orthonormal diagonalizing basis of Δσ:
Δσ(Oij)=σijσ1=eωijOij,ωij=defEiEj.

Finally, Lindbladians that satisfy the QDB condition for s ∈ [0, 1/2) ∪ (1/2, 1] with respect to an invertible state σ can be written in a particular canonical way. This was first proved in Ref. 13 under slightly different conditions. Here we will follow Ref. 16, and use an adapted form of Theorem 3.1 from that reference:

Theorem 2.10
(Canonical form of QDB Lindbladians, adapted from Theorem 3.1 in Ref. 16). A Lindbladian L satisfies quantum detailed-balance with respect to an invertiable state σ and s ∈ [0, 1/2) ∪ (1/2, 1] if and only if it can be written as
L(A)=αIγαeωα/2SαASα12SαSα,A
(15)
where ωα are Bohr frequencies, the jump operators {Sα} are taken from a modular basis, and {γα} are positive weights that satisfy (Recall that in accordance with the modular basis definition (see Corollary 2.9), for every index α, there exists an index α′ such that Sα=Sα and ωα = −ωα.) γα = γα, and in particular the set of indices I contains αfor every αI.

We end this section with a few remarks.

  1. In the original text of Ref. 16, the formula for L is given in the Heisenberg picture, which is the formula for L* in our notation. Additionally, the γα weights are missing, as they are instead absorbed into the Sα. The normalization condition Sα,Sβ=δαβ, however, remained unchanged, which we believe is a mistake.

  2. In Eq. (15) we sum over strictly positive γα and ignore the vanishing weights. In Ref. 16 [Eq. (3.3)] the authors considered the vanishing coefficients as well in the summation.

  3. It follows from the canonical representation that if L satisfies the QDB condition, it is purely dissipative, i.e., its Hamiltonian is vanishing. In the literature, QDB condition is often referred to the dissipator part of the Lindbladian.

  4. Theorem 2.10 applies to any Lindbladian defined on a finite dimensional Hilbert-space regardless of the many-body structure of the underlying Hilbert space, which can describe spins, fermionic, or hard-core bosonic systems.

  5. A physical example of a Lindbladian that satisfies the QDB condition (for s = 1) is the Davies generator of some Hamiltonian H.45 It is the principle example of a semigroup whose unique fixed point is a Gibbs state (i.e., thermal state), and is often referred to as a thermal semigroup.19,24,46

In this section we give precise statements of our main results about the mapping of detailed-balanced local Lindbladians to local Hamiltonians (Subsection III A), and discuss their application to the complexity of the steady states of these systems (Subsection III B). The proofs of the main results are given in Subsection III C.

We consider a geometrically-local k-body Lindbladian L defined on a finite D dimensional lattice Λ of qudits with local dimension d. We assume that L satisfies the QDB condition in Definition 2.5 with respect to some s ∈ [0, 1/2) ∪ (1/2, 1] and a unique, full-rank steady state σ. As L is detailed-balanced, it does not have a Hamiltonian part, and therefore it can be written as
L(ρ)=jLjρLj12{LjLj,ρ},
(16)
with k-body traceless jump operators Lj. We also assume that L has a spectral gap γ > 0. Note that since L obeys detailed-balance, it has a real spectrum, and gap(L)=γ.
Our goal is to map L and its steady state σ to a local Hamiltonian problem, where they can be analyzed using plethora of well-established tools.4,30,34,47,48 This is achieved by mapping the Lindbladian to the super-operator
H=defΓs1/2LΓs1/2,
(17)
which is self-adjoint according to Claim 2.6 (QDB2). We shall refer to H as the super-Hamiltonian, and study its vectorization as a local Hamiltonian. (To obtain a physical interpretation for the super Hamiltonian, one can relate its imaginary time evolution to the Lindblad evolution by similarity transformation, namely, eHt=Γs1/2eLtΓs1/2.)

Since H and L are related by a similarity transformation, they share the same spectrum (up to an overall global minus sign), and therefore also H has a spectral gap γ > 0. Moreover, it is easy to see that an eigenoperator A of L maps to an eigenoperator Γs1/2(A)=σ(1s)/2Aσs/2 of H, and in particular the steady state σ maps to σ which is in the kernel of H. We remark that the superoperator Γs1/2LΓs1/2 and its steady state σ were already studied in Refs. 15, 17, and 18 as tools for relating the χ2 decay constant to the spectral gap of a QDB Lindbladian, and in Ref. 19 for demonstrating strong clustering of information using the detectability lemma. While the locality of the super-Hamiltonian was not addressed in Refs. 15 and 18, it was used in Refs. 17 and 19. However, in these papers, locality was automatically achieved due to the steady state being a Gibbs state of a commuting local Hamiltonian, or the canonical (Davies) form having local jump operators. Our work is devoted to proving this in the general case.

We are left with the task of showing that H is geometrically local, or at least local with decaying interactions. We will prove this under two possible assumptions. In Theorem 3.1 we assume a certain linear-independence conditions on the jump operators, and consequently find that H is a k-body geometrically local. In Theorem 3.2 we expand the jump operators in terms of a local orthonormal basis, and show that if the coefficient matrix in that basis is gapped, then H is 2k-body local with coefficients that decay exponentially with the lattice distance.

Theorem 3.1.
Let {Lj} be a set of linearly independent and normalized jump operators such that for every index j, there exists an index π(j) such that Lj=Lπ(j). Assume that L satisfies the QDB condition (Definition 2.5) for s ∈ [0, 1/2) ∪ (1/2, 1] and is given by
L(ρ)=jcjLjρLj12{LjLj,ρ}
(18)
for some positive coefficients cj > 0. Then the super-Hamiltonian (17) is given by
H(ρ)=jcjcπ(j)LjρLjcj2{LjLj,ρ}.
(19)
Consequently, if L is k-body and geometrically local, then so is H.

To state the second theorem, we first expand our jump operators in Eq. (16) in terms of an orthonormal operators basis {Pa}, which we assume to be Hermitian and k-body geometrically local:
Lj=defaRajPa.
(20)
For example, if the local Hilbert dimension is d = 2, we can take {Pa} to be products of k Paulis on neighboring sites. Substituting Eq. (20) in Eq. (16) yields
L(ρ)=a,bCabPaρPb12{PbPa,ρ},
(21)
where Cab=jRajRbj* is a positive semi-definite (and hence Hermitian) matrix. With this notation, we have

Theorem 3.2.
Let L be a Lindbladinan given by Eq. (21) with {Pa} being a set of orthonormal hermitian operators, and assume that L satisfies the QDB condition (Definition 2.5) with respect to s ∈ [0, 1/2) ∪ (1/2, 1]. Then the super-Hamiltonian (17) is given by
H(ρ)=a,b(CC*)ab1/2PaρPb12Cab{PbPa,ρ}
(22)
where C* is the complex conjugate of C.

We note as a side-product of the proof, C, C* commute, and as they are both non-negative matrices, the square root (CC*)1/2 is well-defined. As a corollary, we achieve a sufficient condition for quantum detailed balance, which can be easily checked with a polynomial computation without the knowledge of the steady state σ. Let us formally state this result:

Corollary 3.3

(A necessary condition for quantum detailed-balance). Let L be a Lindbladinan given by Eq. (21) with {Pa} being a set of orthonormal hermitian operators, and assume that L satisfies the QDB condition (Definition 2.5) with respect to s ∈ [0, 1/2) ∪ (1/2, 1]. Then the matrix C from Eq. (21) commutes with its complex conjugate C*.

Assuming L is k-body and geometrically local, the matrix C is sparse: Cab ≠ 0 only for a, b for which Pa and Pb appear in the expansion of the same jump operator Lj, and therefore their joint support is at most gemotrically k-local. This means that the a,bCab{PbPa, ρ} term in Eq. (22) is geometrically-local k-body. However, the geometrical locality of the first term a,b(CC*)ab1/2PaρPb is not so clear, as it might mix Pa and Pb with distant supports (which might lead to a long-range H). In the following lemma, we show that as long as the smallest non-zero eigenvalue of C · C* is Ω(1), the coefficient (CC*)ab1/2 decays exponentially with the distance between the support of Pa, Pb, making the terms of H decay exponentially with distance.

Lemma 3.4.
Let λmin > 0 be the smallest non-zero eigenvalue of C · C*, and assume that |Cab| ≤ J for every a, b. Finally, let |ab| denote the lattice distance between the supports of Pa, Pb. Then
|(CC*)ab1/2|c1Jec2|ab|λmin/J2,
where c1, c2 are constants that depend only on the geometry of the lattice and k.

The proof of this lemma is given in  Appendix A.

Corollary 3.5.
Let L be a geometrically local k-body Lindbladian given by Eq. (21) that satisfies the QDB condition with respect to a full rank state and s ∈ [0, 1/2) ∪ (1/2, 1]. Let λmin be the smallest non-zero eigenvalue of C · C*, and assume that |Cab| ≤ J for all (a, b) pairs. Then the super-Hamiltonian in Theorem 3.2 is of the form
H=a,bHab,
where Hab is a 2k-local super operator that acts non-trivially on Supp(Pa) ∪ Supp(Pb), with an exponentially decaying interaction strength Hab=JeO(|ab|λmin/J2).

We conclude this section with two remarks:

  1. Related to Corollary 3.3, also the linear independence of {Lj} from Theorem 3.1, as well as the minimal non-vanishing eigenvalue of C · C* from Theorem 3.2, can be checked and calculated efficiently without any prior knowledge of the steady state or the gap. This might be beneficial in numerical applications of these mappings, as the formula for H is explicitly given in terms of the jump operators and C.

  2. Theorems 3.1 and 3.2 are valid for any Lindbladians defined on a finite-dimensional algebra of operators, as they do not use any commutativity or locality properties of the jump operators or basis. Therefore, these theorems apply to fermionic/hard-core bosonic systems as well.

Before providing proofs of Theorems 3.1 and 3.2 in Subsection III C, we will use Subsection III B to discuss their main consequences.

The mapping LH allows us to easily import results from the Hamiltonian realm into the Lindbladian realm. Here, we describe some of the main results that can be imported and discuss the underlying techniques. We begin by describing the vectorization mapping that allows us to map super-Hamiltonians and density operators to Hamiltonians and vectors, respectively.

1. The vectorization map

We map the super Hamiltonian (17) to a local Hamiltonian using standard vectorization, which is the well-known isomorphism L(H)HH (see, for example, Ref. 49). We will denote this mapping by XX, where XL(H) is an operator and XHH is a vector. For the sake of completeness, we explicitly define it here and list some of its main properties. The map is defined first on the standard basis elements by
ijij,
and then linearly extended to all operators such that
c1ψ1ϕ1+c2ψ2ϕ2c1ψ1ϕ1*+c2ψ2ϕ2*.
Above, ϕ* is the state one obtains by complex conjugating the coefficients of ϕ in the standard basis. We note that the vectorization map preserves inner products, (A, B) = Tr(AB) = ⟨⟨A|B⟩⟩. It can also be extended naturally to many-body setup where we have a lattice Λ and the global space is H=xΛHx. In such case, we view HH as composite system in which at every site x ∈ Λ, there are two copies of the Hilbert space Hx: the original Hx and a fictitious Hx, so that HH=xΛ(HxHx) as described in Fig. 1. With this definition, the notion of locality in H can be naturally mapped to locality in HH. This point of view is very intuitive when considering the vectorization |σ1/2 of a quantum state σ. First note that |σ1/2 is a normalized vector since by the preservation of inner product,
σ1/2|σ1/2=σ1/2,σ1/2=Tr(σ1/2σ1/2)=Tr(σ)=1.
Moreover, it is easy to see that for every observable A on H,
σ1/2A1σ1/2=σ1/2,Aσ1/2=Tr(σA),
(23)
i.e., the expectation value of every observable A with respect to σ is equal to the expectation value of A1 with respect to |σ1/2. This implies that |σ1/2 is a purification of σ
σ=Trfict|σ1/2σ1/2|,
where Trfict denotes the tracing over the fictitious Hx Hilbert spaces. The above observations can be summarized in the following easy lemma, which relates the entanglement structure of |σ1/2 to that of σ.
FIG. 1.

The composite system in the vectorization of a many-body system on a lattice. Starting from lattice on which σ is defined [the black dots in (a)], we introduce a “fictitious” site next to every site in the lattice [the red dots in (b)]. The composite system can then be viewed as a new many-body lattice system in which the local dimension is d2.

FIG. 1.

The composite system in the vectorization of a many-body system on a lattice. Starting from lattice on which σ is defined [the black dots in (a)], we introduce a “fictitious” site next to every site in the lattice [the red dots in (b)]. The composite system can then be viewed as a new many-body lattice system in which the local dimension is d2.

Close modal

Lemma 3.6.

Let σ be a many-body quantum state on a D-dimensional lattice Λ, let |σ1/2 be the vectorization of σ1/2, and define ρ=def|σ1/2σ1/2|. Then:

  1. For any bi-partition Λ = AB,
    I(A:B)σ2S(A)ρ,
    (24)
    where I(A:B)σ is the mutual information between A, B in the state σ, and S(A)ρ is the entanglement entropy of region A in the composite system HH (where for every xA we include both the original system and its fictitious partner) with respect to the state |σ1/2. S(A)ρ is also known as the operator-space entanglement entropy of σ (see Refs. 50 and 51).
  2. If Λ is 1D and there exists an MPS ψD on the composite system with bond dimension D such that |σ1/2ψDϵ, then ΨD2=defTrfictψDψD can be described by an MPO with bond dimension D2 and σΨD212ϵ. A similar relation exists also for higher dimensions (replacing MPS with, say, PEPS).

Proof.

  1. By definition, I(A:B)ρ = S(A)ρ + S(B)ρS(AB)ρ, and as ρ is a pure state and A, B a bi-partition of the system, S(A)ρ = S(B)ρ and S(AB)ρ = 0. Therefore, I(A:B)ρ = 2S(A)ρ. As σ = Trfict ρ, it follows from the monotonicity of the relative entropy that
    I(A:B)σ=S(σσAσB)S(ρρAρB)=I(A:B)ρ=2S(A)ρ.
  2. First note that
    ϵσ1/2ψD=2(1ReψDσ1/2)2(1|ψDσ1/2|)
    and therefore ψDσ1/21ϵ2/2 and
    ρψDψD1=σ1/2σ1/2ψDψD1=21|σ1/2ψD|22ϵ.
    Therefore, by the monotonicity of the trace distance,
    σΨD21=TrfictρTrfictψDψD12ϵ.
It remains to show that ΨD2 can be written as an MPO with bond dimension D2. This can be understood from inspecting Fig. 2, which shows how ΨD2 is obtained by contracting the fictitious legs in the TN that describes ρ=σ1/2σ1/2.■

FIG. 2.

Obtaining an MPO description of TrfictψDψD from the MPS ψD. (a) The MPS ψD. Each leg is represented by two legs, where the solid leg corresponds to the original local Hilbert space and the dashed leg to the fictitious Hilbert space. (b) The TN representation of ψDψD. (c) To calculate TrfictψDψD we contract the fictitious legs, leading to an MPO with bond dimension D2.

FIG. 2.

Obtaining an MPO description of TrfictψDψD from the MPS ψD. (a) The MPS ψD. Each leg is represented by two legs, where the solid leg corresponds to the original local Hilbert space and the dashed leg to the fictitious Hilbert space. (b) The TN representation of ψDψD. (c) To calculate TrfictψDψD we contract the fictitious legs, leading to an MPO with bond dimension D2.

Close modal
We conclude the discussion by noting that vectorization also maps super-operators to operators. Indeed, for every super operator F there is a unique FL(HH) that satisfies
F(X)=defFX.
This map respects the notion of adjoints in the sense that FF iff F*F, and so a self-adjoint super Hamiltonian H is mapped to an Hermitian Hamiltonian H. It is easy to verify that if F is the superoperator F(X)=defAXB, then its vectorization is the operator F = ABT. Using this formula it is clear that a k-body super-operator maps to a 2k-body operator. For example, the super Hamiltonian (19) from Theorem 3.1 is mapped to the Hamiltonian
H=jcjcπ(j)LjLj*cj2LjLj1+1(LjLj)T,
and a similar expression arises for the super Hamiltonian from Theorem 3.2.

2. Area laws for steady states

In this subsection we use the notation Õ(X) for OXpoly(logX). Our first result will be an area law for 1D Lindbladians that satisfy the requirements of Theorem 3.1. These map to geometrically local 1D super Hamiltonians, for which we will use the following result:

Theorem 3.7

(Taken from Refs. 30 and 47). Let Λ be a 1D lattice of sites with local dimension d. Let H = ihi be a nearest neighbors Hamiltonian with a spectral gap γ > 0 and a unique ground state Ω. Assume, in addition, thathi‖ ≤ J for every i, where J is some energy scale. Then the entanglement entropy of Ω with respect to any bi-partition satisfies SE=Õlog3(d)γ/J. Moreover, there is a matrix product state (MPS) ψD of sublinear bond dimension D=eÕ(log3/4n/(γ/J)1/4) such that ΩψD1poly(n), and such MPS can be found efficiently on a classical computer.

Using the above theorem together with Lemma 3.6 and Theorem 3.1, we immediately obtain the following corollary.

Corollary 3.8.

Let L be a geometrically local two-body Lindbladian defined on a 1D lattice of local dimension d that satisfies the requirements of Theorem 3.1. Assume in addition that L has a gap γ > 0, a unique steady state, and that |cj| ≤ J for all j. Then:

  1. σ satisfies an area law for the operator-space entanglement entropy,50 that is, for any cut in the 1D lattice into Λ = LR the entanglement entropy of σ between the two parts satisfies
    S(L)ρ=Õlog3(d)γ/J.
    (25)
  2. σ satisfies an area-law for the mutual information. Specifically, for any cut in the 1D lattice, the mutual information between the two sides satisfies
    I(L:R)σ=Õlog3(d)γ/J.
    (26)
  3. There is a matrix product operator (MPO) ΨD with bond dimension D=eÕ(log3/4n/(γ/J)1/4) such that σΨD11poly(n). Moreover, a similar approximating MPO for σ can be found efficiently on a classical computer.

For the next result, we will use the main result of Ref. 31 adapted to the two-local settings with exponentially decaying interactions.

Theorem 3.9
(Taken from Ref. 31). Let Λ a 1D lattice of sites with local dimension d. Let H = i,jhij + ihi be a two-body Hamiltonian defined on Λ, and suppose that there exist a constant J such that hijJrijα,hiJ, and that H has a spectral gap γ > 0 and a unique ground state Ω. Then the entanglement entropy of Ω with respect to any cut in the 1D grid satisfies
SE=Õlog2dlogdγ/J1+2α2.
Moreover, there is a MPS ψD of bond dimension D=eOlog5/2(n) such that ΩψD1poly(n).

It should be noted that Theorem 3.9 can be restated for 1D k-local Hamiltonians and still give the same entropy bound — see Appendix B in Ref. 31. Using the above theorem, together with Lemma 3.6 and Corollary 3.5, we can choose an appropriate B for which HabB/|ab|α for any Pa, Pb for some α > 4. This implies the following corollary:

Corollary 3.10.

Let L be a geometrically local two-body Lindbladian defined on a 1D lattice that satisfies the requirements of Theorem 3.2, with a spectral gap γ > 0 and a steady state σ. Assume also that the smallest non-vanishing eigenvalue of C · C* is λmin=Ω1. Then

  1. σ satisfies an area law for the operator-space entanglement entropy.50 That is, for any cut in the 1D lattice into Λ = LR the entanglement entropy of σ between the two parts satisfies
    S(L)ρ=Õlog2dlogdγ/J2.
    (27)
  2. σ satisfies an area-law for the mutual information. Specifically, for any cut in the 1D lattice, the mutual information between the two sides satisfies
    I(L:R)σ=Õlog2dlogdγ/J2.
    (28)
  3. There is a matrix product operator (MPO) ΨD with bond dimension D=eOlog5/2(n) such that σΨD11poly(n).

We finish this part by noting that there are many more applications of the theorems in Subsection III A which we did not address. For example:

  1. One can import other ground state area-law results to the Linbladian settings, including results for 2D and higher dimensions such as those given in Refs. 5, 79, and 11. For that purpose, one should check that the conditions required in the references above are fulfilled for the super-Hamiltonian. For instance, suppose that L is a 2D Lindbladian given in a canonical form with local jump operators, and further assume it is locally gapped (in the sense discussed in Ref. 11). Then the super-Hamiltonian inherits the same locality structure, frustration-freeness and local gap of the original Lindbladian. The area law for the steady state is automatically achieved using the result of Ref. 11. An example to such model would be the 2D generalization of the toy-model described in Subsection IV A.

  2. One can use the results in Refs. 48 and 52 to show exponential decay of correlations for σ, which implies the same for σ using Eq. (23). This was already shown in Ref. 18 (see Theorem 9 in the paper), so we omit the details. Notice that our result are implied for any such Lindbladian without assuming local uniqueness (see regular Lindbladians in Ref. 18).

  3. One can demonstrate a gap in the Lindbladian by demonstrating a gap in the super-Hamiltonian instead. As the former task is somewhat challenging, the latter has been discussed more frequently in the literature. Below we will exemplify this by using the finite-size criteria from Ref. 34 to demonstrate a gap for a family of Lindbladians (see the proof of Claim 4.1, bullet 4 for details). We remark that this type of work has been also done in Ref. 17.

In this section we give the proofs of our main results, Theorems 3.1 and 3.2. The two proofs follow the same outline: Examining the continuous family of superoperators Lx=defΓsxLΓsx, we would like to show that it obeys some notion of locality at x = 1/2. In fact, we will show that it is local for all xR. By definition, L0=L, and by the QDB condition (12), L1=Γs1LΓs=L*, both of which are local. To prove locality for all other xR we use the modular basis (see Corollary 2.9). We will show that Lx remains diagonal in the modular basis for every x. Then, we will use Theorem 2.2 to establish a connection between the local representation of L and the modular basis. Finally, we will employ the fact that L1 is local to argue that it must remain local for all xR.

Let us then begin by proving that Lx remains diagonal in the modular basis.

Lemma 3.11
(Adaptation of Lemma 7 from Ref. 20). Let L satisfy the quantum detailed-balance condition with respect to s ∈ [0, 1/2) ∪ (1/2, 1]. Then for every xR, the superoperator ΓsxLΓsx is diagonal in the modular basis, and is given by
(ΓsxLΓsx)(A)=αIγαeωα/2exωαSαASα12SαSα,A.
Note: the above formula is identical to the formula for L in Theorem 2.10, except for the exωα factor in front of the SαASα term.

Proof.
Recall that Γsx(A)=σx(1s)Aσxs. We consider the terms SαASα and {SαSα,A} separately. The conjugation of SαASα gives
Γsx(ASαASα)Γsx(A)=ΓsxSαΓsx(A)Sα=σx(1s)Sασx(1s)AσxsSασxs.
(29)
Recalling that
σx(1s)Sασx(1s)=Δσx(1s)(Sα)=ex(1s)ωαSα,σxsSασxs=Δσxs(Sα)=exsωαSα,
we find that, overall, Γsx(ASαASα)Γsx(A)=exωαSαASα.
For the anti-commutator conjugation, we have:
Γsx(A{SαSα,A})Γsx(A)=Γsx{SαSα,Γsx(A)}=σx(1s){SαSα,σx(1s)Aσxs}σxs=σx(1s)SαSασx(1s)A+AσxsSαSασxs={SαSα,A},
where in the last equality we used the fact that for every t,
σtSαSασt=σtSασtσtSασt=Δσt(Sα)Δσt(Sα)=etωαSαetωαSα,=SαSα.

With Lemma 3.11 at hand, we turn to the Proof of Theorem 3.1.

1. Proof of Theorem 3.1

By assumption, L is given by
L(A)=jcjLjALj12{LjLj,A},
but in addition, by the QDB condition and Theorem 2.10, it is given by a canonical form
L(A)=αIγαeωα/2SαASα12SαSα,A.
Therefore, by Theorem 2.2, there must be a unitary Uα,j that connects these two bases, i.e.,
γαeωα/4Sα=jUα,jcjLj.
(30)
Let us now consider the superoperator Lx=defΓsxLΓsx. By Lemma 3.11, it is given by
Lx(A)=αIγαeωα/2exωαSαASα12SαSα,A,
and therefore by Eq. (30), it can also be written in terms of the Lj basis as
Lx(A)=α,j,jexωαUα,jUα,j*cjcjLjALj12jcjLjLj,A.
Note that the anti-commutator term remained unchanged. To analyze the first term, we define the matrix B
Bjj=defαeωαUα,jUα,j*.
Note that B=Udiag({eωα})U, and therefore Bx=Udiag({exωα})U, hence
Lx(A)=j,jBjjxcjcjLjALj12jcjLjLj,A.
(31)
To prove the locality of Lx, we would like to show that Bjjxδjj, i.e., that Bx is diagonal, for at least one x̃0, which will then imply that it also diagonal for all xR. This is done using the QDB condition (12), which will let show diagonality for x̃=1. The QDB condition is equivalent to
L1(A)=L*(A)=jcjLjALj12{LjLj,A}.
Comparing to Eq. (31) gives the condition
jjBjjcjcjLjALj=jcjLjALj=jcπ(j)LjALj,
where in the last equality we used the definition of π(j), Lπ(j)=Lj. Let us now use the assumption that the jump operators {Lj} are linearly independent. This implies that the coefficients of LjALj on both sides of the equation should be identical, and therefore,
Bjjcjcj=cπ(j)δjj.
Using the assumption that cj > 0, we conclude that
Bjj=cπ(j)cjδjj,
and therefore,
Lx(A)=jcπ(j)xcj1xLjALjcj2LjLj,A.
(32)
Substituting x = 1/2 concludes the proof.■

2. Proof of Theorem 3.2

As in the Proof of Theorem 3.1, we show the locality of H using its modular basis representation. We start by noting that all the modular basis elements Sα that appear in L in (15) (those with γα > 0) can be unitarily expressed by the orthonormal basis {Pa}. Indeed, starting from the jump operators representation in Eq. (16), we conclude from Theorem 2.2 that whenever γα > 0, Sα can be written in terms of the jump operators Lj. But as Lj can be expanded in terms of the local {Pa} operators, it follows that this also holds for Sα as well. Finally, by the orthonormality of both sets of operators, we conclude that they are unitarily related: There exists a unitary U, where α runs over all the indices α for which γα > 0, such that
Sα=aUaαPa.
(33)
Plugging the expansion (33) into the canonical form (15) gives
L(ρ)=a,b,αUaαγαeωα/2Ubα*PaρPb12{PbPa,ρ}.
Comparing with Eq. (21), we use the fact that once an orthonormal basis is fixed, the coefficients of the Lindbladians are also fixed (see, e.g., Theorem 2.2 in Ref. 44). Therefore,
C=UDU,D=defdiag{γαeωα/2},
where C is the coefficient matrix of the expansion of L in terms of the {Pa} operators [Eq. (21)].
We now examine the superoperator Lx=defΓsxLΓsx, which by Lemma 3.11 is given by
Lx(ρ)=αIγαeωα(x1/2)SαρSα12αIγαeωα/2SαSα,ρ.
Using Eq. (33), we can rewrite it in terms of the {Pa} operators as
Lx(ρ)=a,bCab(x)PaρPb12a,bCab{PbPa,ρ},
(34)
where we defined
C(x)=defUdiag{γαeωα(x1/2)}U.
(35)
Note that C(x) commutes with C(x′) for any x, x′, as they are both diagonalized by U. As in the proof of the previous theorem, we know that C(x) is local for x = 0 and we would like to prove locality for every xR. We do this by using the QDB condition to show that also the x̃=1 is local, and by showing that any other xR, C(x) is a simple function of C(0) and C(1).
Consider then the x̃=1 point. By the QDB condition (12), it follows that
L1=Γs1LΓs=L*=a,bCab*PaρPb12{PaPb,ρ}.
Comparing it to Eq. (34) and using the fact that C is Hermitian, we conclude that C(1) = C*, and therefore, C* commutes with C = C(0). Finally, a simple algebra shows that
C1x(C*)x=Udiag{γα1xeωα(1x)/2}diag{γαxexωα/2}U=Udiag{γαeωα(x1/2)}U=C(x).
which can also be written as C(x) = C1−x(0) · Cx(1). Substituting x = 1/2, and using Eq. (34) proves the theorem.

In this section, we describe two exactly solvable models whose dynamics is governed by a Lindbladian satisfying, respectively, the requirements of Theorems 3.1 and 3.2 from Sec. III. In Subsection IV A we describe classical Metropolis-based dynamics that satisfies the conditions of Theorem 3.1, and in Subsection IV B we describe a family of quadratic fermion models that satisfy the conditions of Theorem 3.2. For both models, we prove detailed balance, unique steady-state and constant spectral gap. As the models are exactly solvable, we give explicit expressions for their super Hamiltonians, which are derived independently of the results of Subsection III A.

Here we describe a system that obeys the requirements of Theorem 3.1. First, we specify the steady state, which is the Gibbs state of a classical Hamiltonian, and then write the corresponding Lindbladian that annihilates it; this Lindbladian gives rise to classical thermalization dynamics of the diagonal elements of the density matrix, while dephasing away the off-diagonal elements. The full proofs are given in  Appendix B.

We consider a 1D lattice with periodic boundary conditions, where a single qubit occupies each site. Intuitively, we think of each site i as being able to hold a particle (e.g., a fermion or a hard core boson) with some energy ϵi, or being empty. The state of the system is then described by a binary string x̲=(x1,,xn){0,1}n, where xi determines the occupation of the ith site (either 0 or 1). To achieve a non-trivial Gibbs state, we define a constant interaction u acting between nearest neighbors, which is non-zero if an only if both neighboring sites are simultaneously occupied by a particle. To write down the Hamiltonian of the system, we let ni=def12(1Zi)=11i denote occupation number operator at site i, and let μ > 0 denote a chemical potential. Our Hamiltonian is then given by
H=defi=1n(ϵiμ)ni+ui=1nnini+1.
This is a classical Hamiltonian that is diagonal in the computational basis x̲=x1xn, and its Gibbs state is
σ=1Zx̲{0,1}neβExx̲x̲,
(36)
Ex̲=defi=1n(ϵiμ)xi+ui=1nxixi+1,
(37)
Z=defx̲{0,1}neβEx.
(38)
We now introduce a Lindbladian that drives the system into σ. It is a geometrically three-local Lindbladian that is given by
L=defk,bγk,bLk,b,
(39)
where k ∈ {1, …, n}, b ∈ {0, 1, 2} and {γk,b} are some positive (possibly random) O(1) constants. For each k, b, the superoperator Lk,b act locally on qubits k − 1, k, k + 1 and is defined by:
Lk,b(ρ)=defeβωk,b/2Lk,bρLk,b12{Lk,bLk,b,ρ}+eβωk,b/2Lk,bρLk,b12Lk,bLk,b,ρ,
(40)
where ωk,b=def(ϵkμ)ub and Lk,b are three-local jump operators given by
Lk,b=defσkΠk1,k+1b1rest.
(41)
Above, σk=def01k is the “annihilation operator” on site k, and Πk1,k+1b is the projector onto the subspace in which the sum of the occupation numbers of sites k − 1 and k + 1 is equal to b (for b ∈ {0, 1, 2}).

We will now show that σ is the unique steady state of L, and that L is gapped and satisfies QDB. The key is to show explicitly that the representation of L in Eq. (40) is the canonical representation, from which it will follow by Theorem 2.10 that L satisfies QDB with respect to σ, and therefore it is a steady state of L. Formally, we claim:

Claim 4.1.

The following properties hold for the Lindbladian L defined in (39):

  1. The jump operators {Lk,b,Lk,b} are taken from a modular basis (up to normalization), and so L is given in the canoniconical representation.

  2. Lk,b satisfy local detailed-balance with respect to σ,
    Lk,bΓs=ΓsLk,b*s[0,1].
  3. If γk,b > 0, then L has a unique steady state σ defined in Eq. (36).

  4. If γk,bα > 0 and ϵkμ,u=O1, then L is gapped with gapL=Ωα.

Proof.
To show that L is given in the canonical form, we will show that {Lk,b} is a proper modular basis with well-defined Bohr frequencies given by βωkb. First, it is easy to see that {Lk,b} are orthogonal and traceless. To see that they have well-defined Bohr frequencies, we need to show that Δσ(Lk,b)=eωk,bLk,b, where Δσ(Lk,b) = σLk,bσ−1. By Eq. (36),
Δσ(Lk,b)=x̲,x̲eβ(Ex̲Ex̲)x̲x̲Lk,bx̲x̲.
(42)
However, by the definition of Lk,b,
x̲Lk,bx̲=1,xk=1,xk=0,xi=xiforallik,andxk1+xk+1=b0,otherwise.
(43)
Therefore, whenever x̲Lk,bx̲0,
ExEx=i=1n(ϵiμ)(xixi)+ui=1n(xixi+1xixi+1)=(ϵkμ)ub=ωk,b.
Plugging this into Eq. (42), we get
Δσ(Lk,b)=eβωk,bx̲,x̲x̲x̲Lk,bx̲x̲=eβωk,bLk,b.
A similar calculation shows that Δσ(Lk,b)=eβωk,bLk,b.

Using Theorem 2.10 it follows that for every k, b, Lk,b satisfies the QDB condition for s ∈ [0, 1/2) ∪ (1/2, 1] (and therefore also for s = 1/2) with respect to σ, i.e., ΓsLk,b=Lk,b*Γs. As shown in the end of Subsection II C, this implies that σ is annihilated by Lk,b. Therefore, σ is a fixed point of L, and, moreover, it is a “frustration-free” Lindbladian.

Showing uniqueness and gap (bullets 3,4) are technically more involved, requiring the use of Knabe and Perron-Frobenious theorems, and are therefore deferred to  Appendix B.■

Using the claim above, specifically bullet 2, the superoperator Hk,b=defΓs1/2Lk,bΓs1/2 is self-adjoint and annahilates σ. Then, using bullet 1, we can apply Lemma 3.11 to Lk,b given in Eq. (40) to get
Hk,b=Lk,bρLk,b+Lk,bρLk,beβωk,b/22{Lk,bLk,b,ρ}eβωk,b/22Lk,bLk,b,ρ.
(44)
Consequently, H=k,bγk,bHk,b is a frustration-free local super-Hamiltonian. While the locality of H is guaranteed by Theorem 3.1, frustration-freeness is an extra feature that follows from that fact that every Lk,b is locally QDB. This allows us to use tools of frustration-free Hamiltonians to study H and L. For example, we prove bullet 4 of Claim 4.1 using techniques from Refs. 34, 35, and 53. We believe this method may be beneficial for physicists when investigating detailed-balance semigroups and specifically Davies generators.19,45,54 As a final remark, we mention that this could be extended to more complicated Lindblad operators, higher dimensional lattices, and other degrees of freedom (e.g., bosons with a larger finite set of allowed occupancies per site or qudits).

We now describe a family of models that satisfies the requirements of Theorem 3.2. The models consist of free spinless fermions on a lattice, coupled to two particle reservoirs, one with infinitely high chemical potential and one with an infinitely low chemical potential. The high chemical potential reservoir emits particles into the system, and the low chemical potential reservoir takes particles out of the system. The reservoirs are assumed to be large and Markovian, such that they can be integrated out and result in a dissipative Lindblad evolution of the lattice. A physical realization of such settings can be found in, e.g., Sec. 3 of Ref. 55; such Lindbladians can be used for the dissipative generation of topologically non-trivial states.56,57

The models are governed by quadratic Lindbladians of the form55,58,59
L(ρ)=i,jγijinaiρaj12{ajai,ρ}+γjioutaiρaj12{ajai,ρ},
(45)
ai, ai are, respectively fermionic annihilation and creation operators obeying the standard anticommutation relations, and where γin, γout are positive semi definite matrices with eigenvalues dkin,dkout, respectively. Physically, γin (γout) is responsible for the absorption (emission) of particles from (into) the environment. This model can be seen as a special case of a scenario which is treated by Theorem 3.2: This can be seen, for example, by expanding the fermionic operators in a Majorana basis,58 which is Hermitian, orthonormal and local (in the fermionic sense). Then, expressing the coefficients matrix C in terms of the elements of γin, γout it is easy to show that C · C* is gapped if and only if γin and γout are, and Theorem 3.2 will indicate that the super-Hamiltonian has exponentially decaying interactions. However, we find it more illuminating to not use it, but rather prove directly the properties of this specific model. Under the assumptions pronounced in the claim below, we show that the Lindbldian satisfies QDB with respect to a unique steady state which is Gaussian. This is proven similarly to 4.1, by employing the canonical representation of L.

Claim 4.2.

Let L be the Lindbladian in (45). Suppose that γin and γout are commuting, full-rank matrices. Then

  1. There is a unique steady state of a Gaussian form:
    σ=1Zexp(Hss),whereHss=i,jhijaiaj.
  2. L satisfies quantum detailed balance with respect to σ for any s ∈ [0, 1].

  3. Let α > 0 be a number such that γin+γoutα1, then gap(L)α2. In particular, L is gapped if the minimal eigenvalue of γin + γout is Ω1.

Proof.
We prove the last statement by transforming into a more convenient single-particle basis for which the Lindbladian decomposes to a sum of single mode sub-Lindbladians. To achieve this, we use the commutativity of γin and γout, which implies that they can be simultaneously diagonalized:
γijin=i,juikdkinujk*,γijout=i,juikdkoutujk*.
Plugging this expansion into L gives
L(ρ)=i,j,kuikdkinujk*aiρaj12{ajai,ρ}+ujkdkoutuik*aiρaj12{ajai,ρ}=kdkinckρck12{ckck,ρ}+dkoutckρck12{ckck,ρ}=defkLk(ρ),
(46)
where we defined ck=defiui,k*ai, the annihilation operators corresponding to the single-particle eigenmodes of γin and γout. Notice that in the new basis, L decomposes into a sum of single mode Lindbladians Lk, as promised. By calculating the spectrum of each Lk, we see that the
gap(L)=minkgap(Lk)=minkdkin+dkout2
which is bullet 3. Moreover, each Lk has a unique zero state σk. Solving for each Lk it is easy to see that it has the form σkexpϵkckck with ϵk=deflogdkindkout. The global steady state is then unique, being the product of all single mode zero states, σ=kσkexpkϵkckck. Going back to the original aj basis completes the proof of bullet 1. Finally, bullet 2 follows from the observation that {ck,ck} are canonical jump operators, namely, they have well-defined Bohr frequencies:
σckσ1=eϵkck,σckσ1=eϵkck.
The above identities can be easily checked in the eigenbasis of the occupation number operators, ckck.

We remark that: (1) The above can be also proven using correlation matrix formalism and the continuous Lyaponouv equation.59 (2) Starting from a Gaussian steady state and going to the corresponding single-particle eigenbasis one can show that γin, γout must be diagonal in the same basis. Hence, QDB is satisfied if and only if γin, γout commute.

As a corollary, the super-Hamiltonian can be derived, thus verifying the results of Theorem 3.2 for the current model:

Claim 4.3.
Let L be a Lindbladian as in (45) with commuting γin, γout. Then H=Γ1/2LΓ1/2 is given by
H(ρ)=i,j(γinγout)ijaiρaj+(γinγout)ij*aiρajγijin2{ajai,ρ}γjiout2{ajai,ρ}.
(47)

Proof.
We prove this in a similar fashion to Lemma 3.11. Recall that ck,ck are canonical jump operators, that is, σckσ1=eϵkck=dkoutdkinck and σckσ1=eϵkck=dkindkoutck. Using the Lindblad form in Eq. (46), we see that
H(ρ)=Γ1/2LΓ1/2(ρ)=kdkin(σ(1s)/2ckσ(1s)/2)ρ(σs/2ckσs/2)kdkout(σ(1s)/2ckσ(1s)/2)ρ(σs/2ckσs/2)12{}=kdkin(e(1s)ϵk/2ck)ρesϵk/2ckkdkout(e(1s)ϵk/2ck)ρ(esϵk/2ck)12{}=kdkineϵk/2ckρckkdkouteϵk/2ckρck12{}=kdkoutdkin(ckρck+ckρck)12{}
Similarly to Lemma 3.11, it can be checked that the anti-commutator terms {⋯} are unchanged by the transformation. Switching back to the original (local) creation and annihilation operators ak yields
H(ρ)=i,j,kuikdkoutdkinujk*(aiρaj)+uik*dkoutdkinujk(aiρaj)12{}=i,j(γinγout)ijaiρaj+(γinγout)ij*aiρaj12{}.

Since we are interested in local Lindbladians, we consider fermions that occupy the sites of a lattice. The locality properties of γinγout then from Lemma 3.4. For the purpose of illustration, let us treat explicitly the case of a 1D lattice with periodic boundary conditions. We focus on commuting γin, γout that connect nearest neighbors sites only, i.e., tridiagonal matrices (augmented by the upper-right and lower-left elemetns). Moreover, we consider translation invariant systems (Toeplitz matrices), such that
γin=γ0inγ1in00γ1inγ1inγ0inγ1in00γ1inγ0inγ0inγ1in00γ1inγ0inγ1inγ1in00γ1inγ0in,γout=γ0outγ1out00γ1outγ1outγ0outγ1out00γ1outγ0outγ0outγ1out00γ1outγ0outγ1outγ1out00γ1outγ0out.
(48)
The canonical jump operators ck of such a Lindbladian are achieved by diagonalizing γin and γout, which, due to translation invariance, are given by the a discrete Fourier transform of the original aj:
ck=1nje2πinkjaj,k=1,,n.
These are in-fact non-local operators, but linear combinations of such (as anticipated by the Proof of Theorem 3.2). By applying the discrete Fourier transform to 48, we see that dkin=γ0in+2γ1incos(2πkn) and similarly dkout=γ0out+2γ1outcos(2πkn). Therefore, γin, γout are gapped (in the sense that their smallest non-zero eigenvalue does not vanish as n → ∞) when γ0in2γ1in=Ω1>0 and γ0out2γ1out=Ω1>0, respectively. We also require that γ0in,γ0out=O1 for the Lindbladian to have bounded norm. As we now show, the gaps in γin and γout stated above are responsible for the decay of the matrix elements of γinγout, and therefore determine the locality of the super-Hamiltonian (47). This is due to the following lemma:

Lemma 4.4.
The matrices in Eq. (48) satisfy
|γinγoutij|2γ0in12γ1inγ0in2γ0out12γ1outγ0out22γ1inγ0ind(i,j)/2+2γ1outγ0outd(i,j)/2
where d(i,j)=defmin{|ij|,n|ij|} is the metric on the circle.

As a result, provided that γin and γout are gapped, H given in (47) is geometrically two-local (quadratic) with exponentially decaying interactions. The Proof of Lemma 4.4 is technical and thus left to  Appendix C.

We remark that the derivation in  Appendix C suggests that when the gap in γin (or γout) closes, that is, when γ0in=2γ1in, the super-Hamiltonian becomes long range with polynomially decaying interactions. The degree of the polynomial does not allow an area-law statement for the steady state, according to the results of Ref. 31. See also Fig. 3 for an illustration of the decay in γinγout.

FIG. 3.

Plot of the matrix elements, (γinγout)ij, for n = 100. The horizontal and vertical axis represent i and j, respectively. Left panel: gapped γin · γout, with γ0in,γ1in=(3,1), γ0out,γ1out=(2,0.5), Right panel: gapless γin · γout, with γ0in,γ1in=(3,1), γ0out,γ1out=(2,1).

FIG. 3.

Plot of the matrix elements, (γinγout)ij, for n = 100. The horizontal and vertical axis represent i and j, respectively. Left panel: gapped γin · γout, with γ0in,γ1in=(3,1), γ0out,γ1out=(2,0.5), Right panel: gapless γin · γout, with γ0in,γ1in=(3,1), γ0out,γ1out=(2,1).

Close modal

In this work we have shown how a detailed-balance Lindbladian L can be mapped to a local, self-adjoint superoperator H, which we call a super Hamiltonian. The mapping is via a similarity transformation, hence we are guaranteed that the Lindbladian and the super Hamiltonian share the same spectrum (up to an overall minus sign). Moreover, if σ is the steady state of the Lindbladian, σ is the steady state of H. By vectorizing the super Hamiltonian we get a local Hamiltonian whose ground state is σ1/2. As a side consequence of our mapping, we also found a necessary condition for a Lindbladian to satisfy detailed balance, which can be checked efficiently.

We observed that local expectation values in σ map to local expectation values in the ground state σ1/2, and that the mutual information in σ is bounded by the entanglement entropy in σ1/2. Consequently, several well-known results about the structure of gapped ground states of local Hamiltonians can be imported to the steady state of gapped, detailed-balanced Lindbladians. In particular, we have shown how under mild conditions that can be checked efficiently, the steady state of 1D, gapped, detailed-balanced Lindbladians satisfies an area-law in mutual information, and can be well approximated by an efficient MPO. These results cover many new systems for which the results of Refs. 18 and 21 are not known to apply.

The mapping applies for Lindbladians with traceless jump operators and vanishing Hamiltonian part (a consequence of detailed balance). However, it also applies to Lindbladians with a Hamiltonian that commutes with the steady state, since it leaves the steady state invariant. An example to such a Lindbldian is the Davies generator45 that describes a thermalization process. The addition of the corresponding Hamiltonian should not break primitivity,60 and we expect that in many cases it will not close the spectral gap.

Our work leaves several open questions and possible directions for future research. First, it would be interesting to see what other results/techniques can be imported from local Hamiltonians to Lindbladians using our mapping. It would also be interesting to see if this mapping can be used in numerical simulations. For example, one can apply DMRG to find the ground state of H, and then plug it back to the Lindbladian to see if this is indeed the fixed point. Since H can be easily obtained from L, this procedure can be used even if we do not know if the Lindbladian satisfies QDB.

It would also be interesting to further study the various necessary conditions that are needed to prove an area-law for steady states of local Lindbladians, and in particular, find whether the conditions under which our mapping applies can be relaxed. Is there a weaker condition that still ensure a local H?

Finally, it would also be interesting to understand if our mapping can be used in the opposite direction. Given a local Hamiltonian, one might ask if it is the vectorization of a super Hamiltonian that comes from some QDB Lindbladian. In such cases it might be possible to probe the ground state of the local Hamiltonian by simulating the time evolution of the Lindbladian on a quantum computer. This might show that the local Hamiltonian problem for this class of Hamiltonians in inside BQP. It would be interesting to characterize this class of Hamiltonians, and see if they can lead to interesting quantum algorithms.

We are thankful for Curt von Keyserlingk and Jens Eisert for insightful discussions. We are also grateful for an anonymous referee for pointing our attention to the results of Refs. 32 and 33 on the East model. M.G. was supported by the Israel Science Foundation (ISF) and the Directorate for Defense Research and Development (DDR&D) Grant No. 3427/21, and by the US-Israel Binational Science Foundation (BSF) Grant No. 2020072. I.A. acknowledges the support of the Israel Science Foundation (ISF) under the Individual Research Grant No. 1778/17 and joint Israel-Singapore NRF-ISF Research Grant No. 3528/20.

The authors have no conflicts to disclose.

Raz Firanko: Conceptualization (equal); Formal analysis (equal); Writing – original draft (equal). Moshe Goldstein: Conceptualization (equal); Methodology (equal); Visualization (equal). Itai Arad: Conceptualization (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

In this appendix, we prove Lemma 3.4. For convenience, we first restate it here.

Lemma A.1.
Let λmin > 0 be the smallest non-zero eigenvalue of C · C*, and assume that |Cab| ≤ J for every a, b. Let |ab| denote the lattice distance between the supports of Pa, Pb. Then
|(CC*)ab1/2|c1Jec2|ab|λmin/J2,
where c1, c2 are constants that depend only on the geometry of the lattice and on k.

Proof.

Set A=defCC*, and let λmax, λmin be the largest and smallest non-zero eigenvalues of A. Let |ab| denote the lattice distance between the support of Pa and Pb. Recall that Cab ≠ 0 only for Pa, Pb that intersect a geometrically local region of k sites, and therefore Cab ≠ 0 only for |ab| ≤ k. Similarly, for any integer > 0, Cab0 only when |ab| ≤ kℓ, and therefore, as also Cab*0 only for |ab| ≤ k, we conclude that Aab=(CC*)ab0 only when |ab| ≤ 2ℓk.

Following Ref. 61, we assume that there exists a family of polynomial approximations {Pm(z)} to the function z (indexed by their degree) with the following properties:

  1. |Pm(z)z|τ1eτ2m for every z ∈ [λmin, λmax] for some constants τ1, τ2 that depend on λmin, λmax, but not on m.

  2. Pm(0) = 0.

We will soon find such family, but for now let us discuss its consequences.

We first note that as the spectrum of A is in {0} ∪ [λmin, λmax], then for any m, APm(A)τ1eτ2m. Consider now a pair of indices (a, b), and let m be the largest integer for which 2km < |ab|. Then by the discussion above Pm(A)ab=0, and therefore by the triangle inequality,
|(A)ab||(A)abPm(A)ab|+|Pm(A)ab|=|APm(A)ab|APm(A)τ1emτ2.
However, as m is the largest integer for which 2km < |ab|, then 2k(m + 1) ≥ |ab| and therefore m ≥ |ab|/(2k) − 1, from which we deduce
|(A)ab|τ1eτ2e|ab|τ2/(2k).
(A1)
Our next step is to show that the family of polynomials exists and find the dependence of τ1, τ2 on λmin, λmax. We will find an m − 1 degree polynomial approximation to 1/z in [λmin, λmax], and then multiply it by z. Following Ref. 10, we use the expansion 1/1+z=j=014j2jjzj in the following manner:
1z=1λmax11+(z/λmax1)=1λmaxj=014j2jjzλmax1j.
(A2)
For z ∈ [λmin, λmax], |z/λmax − 1| < 1, and so the above series converges absolutely. Define Qm(z) to be the m − 1 degree polynomial that is the sum of the terms in (A2) with degree m1, and let Rm(z) be the sum of all the higher order terms. Then 1z=Qm(z)+Rm(z), and using the fact that 2jj4j, we get
|Rm(z)|1λmaxj=m|z/λmax1|j=1λmax|z/λmax1|m1|z/λmax1|=λmaxz1z/λmaxm.
Multiplying by z, and setting Pm(z)=defzQm(z), we find that
|zPm(z)|λmax1z/λmaxmλmaxemz/λmaxλmaxemλmin/λmax.
Therefore, τ1=λmax,τ2=λmin/λmax1. Plugging to (A1) yields
|(CC*)ab1/2|eλmaxe|ab|λmin/(2kλmax).

To conclude the proof, we need to show that λmaxηJ2, where η is a constant that is a function of the lattice geometry and k, independent of the system size. To do that, note that λmax = ‖C · C*‖ ≤ ‖C2. By definition, C is a sparse matrix, since at every row Pa there is only a constant number of Pb that overlap the same geometrically k-local region. Call this constant η, and note that it only depends on k and the geometry of the lattice (i.e., dimension, etc.). Therefore, assuming that |Cab| ≤ J for all a, b, we deduce that for any normalized vector v, CvJη, and so λmax = ‖C2ηJ2.■

This appendix is devoted to proving bullets 3 and 4 in Claim 4.1.

1. Proving uniqueness of the steady state (bullet 3)

We prove uniqueness of the steady state in two steps. First, we show that the steady state of the restriction of L to the diagonal elements is unique. Then we show that all off-diagonal elements decay due to the dissipative dynamics. We start by writing the action of the Lindbladian on a general diagonal element xx where x ∈ {0, 1}n:
L(xx)=k,bγk,bLk,b(xx)=xxfxxxxgxxx,
where the sum runs over all strings that can be achieved from x by flipping one spin. The corresponding weights are give by:
fxx=γk,bkeβωk,b/2,xk=1,eβωk,b/2,xk=0,gx=x;xxfxx,
where bk = bk(x) = xk−1 + xk+1, and k is defined by the spin that is flipped when xx′. Note that gx is responsible for ensuring that TrL(xx)=0.
It is known that a Lindbladian, being a generator of a CPTP semigroup, induces a (classical) continuous time Markov Process on the diagonal elements {xx} which is defined by62,
Pxy(t)=defTr(xxeLt(yy)).
To see this, Notice that TrxxeLt(yy)0 due to eLt being completely positive. One can also check the sum of each column to see that
xPxy(t)=xTrxxeLt(yy)=TreLt(yy)=1,
due to eLt being trace preserving. To deduce the uniqueness of a stationary distribution of Pxy, we use the Perron-Frobenious theorem and the connectivity of the Markov chain. Specifically, we show that Pxy(t) > 0 for any x, y, t, and uniqueness will follow as a consequence from Perron-Frobenious (see Theorem 1.1 in Ref. 40).

Claim B.1.

Pxy(t)=TrxxeLt(yy)>0 for any t > 0.

Proof.
Let G = (V, E) be a graph where V = {0,1}n is the set of bit-strings and (x, x′) ∈ E if and only if they have Hamming distance dH(x, x′) = 1 (x′ is obtained from x by flipping one bit). Notice that this is also the connectivity graph of L with self edges omitted. For x, yV, define = (0, …, ||+1) to be the shortest path in G from x to y (i.e., dH(x, y) = ||). Take c > 0 such that c > gz for any zV. As a result, (Lt+c1)xy0 for any x, y, and in particular (L+c1)xy>0 if (x, y) ∈ E. Let us expand eLt
eLt=eceLt+c1=eck=01k!(Lt+c1)k.
(B1)
Notice that (Lt+c1)kxy0 if and only if dH(x, y) ≤ k, therefore Pxy receives its first non-zero contributions from the k = ||th order in the expansion. Moreover, this first contribution is positive, since
x|(Lt+c1)k|y=r1,,rk1x|Lt+c1|r1r1|Lt+c1|r2rk1|Lt+c1|y=x|Lt+c1|11|Lt+c1|2k1|Lt+c1|y+{r1,,rk1}x|Lt+c1|r1r1|Lt+c1|r2rk1|Lt+c1|y>0,
where we abuse of notation by writing x|T|y instead of TrxxT(yy), exploiting the fact that the action of T=Lt+c1 is closed on the diagonal elements xx. Notice that the first term after the equality sign is greater than zero using the path and graph connectivity of the semigroup, and the rest of the sum is greater than or equal to zero, hence the whole term is greater than zero. The higher k terms are greater or equal to zero (since all matrix elements are), thus we conclude the claim.■

The only thing left to show now is that the off-diagonal elements decay. Let us write the action of the Lindbladian on the off diagonal terms explicitly:
L(xy)=x,yfxyxyxy12gx+12gyxy,
where fxyxy is non-zero only if there are k ∈ {1, …, n} and b ∈ {0, 1, 2} such that Lk,bxyLk,b0 or Lk,bxyLk,b0 (that is, if xk = yk, xk−1 + xk+1 = yk−1 + yk+1), and then it would give fxyxy = fxx = fyy. The observation is that
(x,y)fxyxy<12gx+12gy,
namely, looking at the representing matrix L|{mn}mn, the sum of the off-diagonal elements in each column is strictly smaller than the absolute value of the element on the diagonal. We claim that the eigenvalues of such matrix must be strictly negative.

Claim B.2.

Let Aij be a matrix with non-negative off diagonal elements and strictly negative diagonal elements, such that j;jiAij < −Aii for any i. Then Spec(A) ⊂ (−∞, 0).

Proof.
Consider the exponentiation of A. This is a semi-stochastic matrix (its matrix-elements are non-negative), since
eA=eceA+c1=eck=01k!A+c1k
where c > 0 is chosen such that Aii + c > 0 for each i. Therefore, (A+c1)i,j, and correspondingly (eA)ij, are greater or equal to zero for any i, j. Using the Perron-Frobenious theorem, eA has a maximal eigenvalue r > 0 with an eigenvector v in which vi ≥ 0 for any i. We deduce that v is also an eigenvector of A with largest eigenvalue, due to the monotonicity of the exponent:
jAijvj=λvii.
(B2)
We finally show that λ is negative due to the assumption in the claim, and the remaining spectrum will be negative as well (since it must be below λ). Indeed, let i0 = arg max{vi}, and take
λvi0=jAi0jvj=Ai0i0vi0+ji0Ai0jvjAi0i0vi0+vi0ji0Ai0j<0,
where in the last inequality we used ji Aij < −Aii.■

2. Proving that L is gapped (bullet 4)

To show a gap in the system, we import the finite-size criteria for frustration-free Hamiltonians, originally introduced by Knabe34 and used, for example, in Refs. 35 and 53. First we introduce the method for generic systems defined on finite dimensional Hilbert spaces, and then use it to show a gap for the system under consideration. We change our notation correspondingly, e.g., we first discuss projectors on a generic Hilbert space and denote them by Pe, and in the next paragraph we refer to projectors on operators space (super-projectors) which we denote by Pe.

Claim B.3.
(Finite size criteria). Let G = (V, E) be a regular graph of degree δ. Assign a local Hilbert-space of dimension d to each vertex. Let H be a nearest-neighbors frustration free Hamiltonian defined on the joint Hilbert space of the vertices of the form:
H=eEPe,
(B3)
where each Pe is a projector defined on the bond e. We define the local gap to be γloc=defmineegapPe+Pe. Then
gap(H)2(δ1)γloc1(2(δ1))1.
(B4)

Proof.
Note that due to frustration-freeness and spectral decomposition, an inequality of the form
H2cH
(B5)
immediately implies that gap(H) ≥ c. This is the principle that lies at heart of the method. To achieve a bound of the form (B5), Knabe used the Hamiltonian’s local structure and the local spectral gap, as explained in the following. We start by squaring H and rearranging the double sum:
H2=ePe2=ePe2+ee(PePe+PePe)+2ee=PePe.
(B6)
Since we are looking for an inequality of the form H2cH, we can ignore the rightmost term in (B6), as it is positive semi-definite, being a sum of products of non-overlapping projectors. Using Pe2=Pe, one should notice that the first term in Eq. (B6) reduces to H itself. Hence we obtained H2H + Q, where Q=defeePePe+PePe. To achieve the desired bound, we write down a lower bound for Q using H. To do so, let us consider
eePe+Pe2.
Expanding the square, we see that the expression equals
eePe+Pe+PePe+PePe=2(δ1)H+Q,
where the 2(δ − 1) factor stems from the fact that for a simple regular graph, each edge intersects 2(δ − 1) distinct edges (δ − 1 on each vertex). On the other hand, due to frustration freeness, we have Pe+Pe2gapPe+PePe+Pe, and obtain
2(δ1)H+Q2(δ1)γlocH.
Plugging this into Eq. (B6) gives
H2H+Q2(δ1)Hγloc112(δ1).
(B7)
Using inequality (B5) (H2c · H) for c=2(δ1)γloc112(δ1), we see that
gap(H)2(δ1)γloc112(δ1),
meaning that if γloc is greater than 112(δ1), a global gap in H confirmed.■

We remark that the method can be generalized to larger sub-regions and to open boundary conditions.34,63

a. Application to H

Now we apply the method to the system under consideration to show a global gap in L, thus completing bullet 4 of Claim 4.1. First, we reduce the super-Hamiltonian to the sum-of-local-projectors Hamiltonian presented in Eq. (B3) (up to a multiplicative factor). Then, we numerically verify the statement that γloc>112(δ1)=12 for the 1D ring, which will assure the global gap by Eq. (B7).

Recall the parent Hamiltonian, H=k,bγk,bHk,b, where Hk,b=Γs1/2Lk,bΓs1/2 and γk,bα > 0. Also recall that H is frustration-free, where Hk,b0 has a ground state σ with energy 0. Then H satisfies
Hαk,bHk,b=αkHk,whereHk=defb=0,1,2Hk,b.
Let Pk denote the local projector onto the positive spectrum of Hk such that HkgkPk where gk=defgap(Hk). By simply calculating the spectrum of Hk we see that gk=minb{0,1,2}{eβωk,b}=eβ(ϵkμ+2u), which is of order 1 provided that ϵkμ,u=O1. We thus conclude that
HαgminkPk,
where we defined gmin=defminkgk. Moreover, since both sides have a common ground space which is spannded by σ with ground energy 0, using Claim B.3 we have
gap(H)αgmingapkPk2αgmin(γloc1/2),
where γloc=minkgap(Pk+Pk+1). To show a gap in H, we assure that gap(Pk+Pk+1)>1/2. Instead of giving a tedious derivation of the local gap, we numerically verify it in the following table for μ = 0 and different choices of {ϵk} and u (Table I).
TABLE I.

Numerical values of the local gap in L for a few choices of {ϵk} and u. The random energies (first line) were picked from the uniform distribution on [0,1], and the local gap was averaged over 100 instances.

Local Gap γloc
Modelu = −1u = 0.5u = 2
Random {ϵk0.756 0.887 0.678 
Const ϵk = 1 0.769 0.913 0.778 
Const ϵk = 0.5 0.755 0.891 0.698 
ϵ1 = 1, ϵ2 = 10 0.993 0.998 0.997 
Local Gap γloc
Modelu = −1u = 0.5u = 2
Random {ϵk0.756 0.887 0.678 
Const ϵk = 1 0.769 0.913 0.778 
Const ϵk = 0.5 0.755 0.891 0.698 
ϵ1 = 1, ϵ2 = 10 0.993 0.998 0.997 

We remark that the derivation of the finite size criteria considers two-local interaction (graph interactions), and the model under consideration here is three-local. However, the derivation and results are still valid. This is due to terms that overlap on a single qubit (e.g., Hk and Hk+2) commute with each other, as they act trivially on their shared qubits. Therefore, when squaring H in Eq. (B6), their product can be absorbed in the positive “leftover” [the last term in (B6)].

This appendix is devoted to proving Lemma 4.4, which demonstrates the exponential decay in the coefficients of the super-Hamiltonian with respect to the distance between the real space fermionic operators.

First, recall the settings of the lemma. To relate the model to the subject of our work, which discusses local Lindbladians that satisfy quantum detailed-balance, we focus on commuting γin, γout that connect nearest neighbors (i.e., tridiagonal matrices). For simplicity, we consider translation invariant systems, such that
γin=γ0inγ1in00γ1inγ1inγ0inγ1in00γ1inγ0inγ0inγ1in00γ1inγ0inγ1inγ1in00γ1inγ0in,γout=γ0outγ1out00γ1outγ1outγ0outγ1out00γ1outγ0outγ0outγ1out00γ1outγ0outγ1outγ1out00γ1outγ0out.
The canonical jump operators are given by a discrete Fourier transform of the original ak:
ck=1nje2πinkjaj,k=1n.
These are non-local operators, but a linear combination of such.
To prove the lemma, we want to show that the matrix γinγout has elements that decay exponentially with the distance from the main diagonal. First, we show that γin and γout individually satisfy
(γin)ij=Oγ0in12γ1inγ0in2γ1inγ0ind(i,j),(γout)ij=Oγ0out12γ1outγ0out2γ1outγ0outd(i,j).
Then, we use this to establish an upper-bound on γinγoutij.
To show a decay in γin (the treatment of γout is identical), we notice that
γin=γ0inγ0inγ0inγ0in+0γ1inγ1inγ1in0γ1inγ1in0γ1inγ1inγ1in0=defγ0in1+2γ1inW.
A key observation is that W is the (symmetric) random walk matrix Wij=12δj,i±1. Plugging the Taylor expansion 1+x=k0αkxk produces
γin=γ0in1+2γ1inW=γ0in1+2γ1inγ0inW=γ0ink0αk2γ1inγ0inkWk.
Note that (Wk)ij is the transition probability for ij after k steps of the walk, which is less than one and non-zero only when k ≥ min{|ij|, n − |ij|}; Assume without loss of generality that the minimum is attained at |ij|. Moreover, the sequence {αk} is smaller than one in absolute value (polynomially decaying to be precise). Then the matrix entry (γin)i,j satisfies
(γin)i,j=γ0ink0αk2γ1inγ0inn(Wk)ijγ0ink|ij|2γ1inγ0ink=2γ1inγ0in|ij|γ0in12γ1inγ0in.
(C1)
This shows that (γin)ij=Oe|ij| provided that γin is gapped and γin=O1. The analogous argument for γout is straightforward. Note that for γ0in=2γ1in, the gap in γin closes, and a polynomial decay in γinij appears instead (see Fig. 3). This is because for large k’s, the probability for the random walk yields (Wk)ij=12k+1k|ij|+k2=Ok1/2 (see p. 291 in Ref. 64), and also we get that αk=O1k. (This can be derived using the expression αn=(2n3)!22n2n!(n2)! and applying the Stirling’s formula for large n.)
It is now left to show that the product B=defγinγout has exponentially decaying matrix elements as well. We show this for first row elements B1k for convenience, as it automatically generalized to any Bij due to translation invariance. Let us write down B1k, explicitly and use the former bounds for γin and γout. Suppose that kn/2 as before; then
|B1k|m|γin1mγoutmk|mγ0in12γ1inγ0inγ0out12γ1outγ0out2γ1inγ0ind(m,1)2γ1outγ0outd(k,m)=γ0in12γ1inγ0inγ0out12γ1outγ0outm=k/2+1k+k/22γ1inγ0ind(m,1)2γ1outγ0outd(k,m)+m=3k/2+1k/22γ1inγ0ind(m,1)2γ1outγ0outd(k,m),
where the last sum is performed periodically, i.e., m = 3k/2 + 1, 3k/2 + 2, …, n − 1, n, 1, 2, …, k/2. All we did is to separate the sum to a k/2 length neighborhood of the kth site and the complement (see Fig. 4). At the first sum, the decay is dominated by 2γ1inγ0ind(m,1)=2γ1inγ0inm12γ1inγ0ink/2, and at the second sum the 2γ1outγ0outd(k,m)2γ1outγ0outk/2 term dominates. Therefore we write
|B1k|γ0in12γ1inγ0inγ0out12γ1outγ0outm=k/2+1k+k/22γ1inγ0inm1+m=3k/2+1k/22γ1outγ0outd(k,m),
where we used the fact that both 2γ1inγ0in and 2γ1outγ0out are smaller than 1. Since both sums are geometric, they can be bounded to achieve the desired bound
|B1k|2γ0in12γ1inγ0in2γ0out12γ1outγ0out22γ1inγ0ink/2+2γ1outγ0outk/2.
FIG. 4.

The neighborhoods for the partial sums in the calculation for B1k.

FIG. 4.

The neighborhoods for the partial sums in the calculation for B1k.

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