Enhancement of entanglement concentration using catalysts

Entanglement transformations are a fundamental quantum information primitive, which convert a given initial state (or copies of the state) to a final desired one. Often, the allowed framework for such transformations falls within the “distant labs” paradigm where two experimenters, Alice and Bob, each possess and operate on part of a quantum system while coordinating their actions through classical communications. This paradigm of local operations and classical communication (LOCC) is relevant in many different settings; for example, in quantum communication, Alice and Bob may represent the nodes of a quantum key distribution network,¹ or they may represent sub-processors in distributed quantum computation² or spatially separated sensors in distributed quantum sensing.³ Since entanglement concentrated in the form of maximally entangled pure states is a crucial resource for such applications,^4–6 an important class of entanglement transformations is entanglement concentration (EC).⁷ In EC, one obtains maximally entangled pure states starting from some initial number of copies, N, of partially entangled pure states using LOCC⁸ between two parties. A variety of LOCC protocols for entanglement concentration are known,^7,9–11 which are, in general, non-deterministic with different transformation success probabilities.

Aided by a shared catalyst state, that is, an auxiliary entangled pure quantum state, Alice and Bob can enhance the transformation success probability of EC protocols^12,13 and thereby obtain a higher yield of maximally entangled pure states. In this process, known as an entanglement-assisted LOCC (ELOCC) process, the auxiliary state is utilized as part of an overall LOCC process to enhance the transformation success probability and is recovered intact at the end if the transformation is successful—much like the catalyst in certain chemical reactions. The possible catalyst states for a particular entanglement transformation can be determined by the non-increasing condition on the entanglement monotones^8,14,15 of the initial and final states. While catalyst states of higher dimensions are more efficient in enhancing the transformation success probability, for practical applications, optimal catalysts of smaller dimensions may be more suitable.

The use of catalysts for EC can be beneficial in the case when the desired transformation involves converting a finite number of copies, N, of a two-qubit pure state, $|α〉=α|00〉+1−α|11〉$⁠, to a single-copy of a maximally entangled two-qubit state (Bell state), $|ϕ〉=(|00〉+|11〉)/2$⁠, while the other (N − 1) states have zero entanglement. In the case of an asymptotic number of copies,⁷ N → ∞, of |α⟩, the number of Bell states that can be deterministically and reversibly obtained using LOCC is given by M = S_VN(α)N, where S_VN(α) = −α log₂(α) − (1 − α)log₂(1 − α) is the Von Neumann entropy of the reduced initial state. With just one copy, i.e., N = 1, a Bell state cannot be obtained deterministically or reversibly. The maximum probability of obtaining a single Bell state in this case is P = 2(1 − α) < 1,^9,10 where we have assumed without loss of generality that α ≥ 0.5. In both these limits, i.e., N → 1 and N → ∞, the use of a catalyst does not help, that is, it cannot increase the number of Bell states, M, obtained asymptotically nor can it increase the success probability, P, for a single copy transformation, |α⟩ → |ϕ⟩. Away from these limits on the number of copies, i.e., when the number of copies is 1 < N < ∞, catalyst-aided LOCC transformations can increase the success probability of concentrating the entanglement content of the copies of the initial state into a single copy of a Bell state.^12,16,17

In this paper, we show the enhancement of EC utilizing suitable catalysts for the LOCC transformation, |α⟩^⊗N → |ϕ⟩. Obtaining catalysts for arbitrary entanglement transformations is a difficult problem analytically. However, in catalyst-aided EC, it is possible to utilize the structure of the ordered Schmidt coefficients (OSCs) of the state |α⟩^⊗N to analytically obtain the optimal two-qubit catalyst and the maximum transformation success probability. For catalysts of higher dimensions, a numerical search in the space of Schmidt coefficients provides the same information. We compare the enhancement of the transformation success probability using catalysts to the case when no catalyst-aid is utilized. To close, we comment on ELOCC strategies to obtain multiple copies of Bell states.

The rest of this paper is organized as follows: In Sec. II, we review the theory of deterministic LOCC transformations in Subsection II A and probabilistic LOCC entanglement transformations in Subsection II B including the criteria for such transformations based on entanglement monotones. In Sec. III, we first sketch how to obtain the relevant entanglement monotones for entanglement concentration using two-qubit catalysts in Subsection III A, derive the optimal probability of entanglement concentration using a two-qubit catalyst in Subsection III B, show the enhancement of EC using catalysts of higher dimensions in Subsection III C, compare the catalyst-aided LOCC success probability to that without the use of catalysts in Subsection III D, and finally provide strategies to obtain multiple Bell states in Subsection III E. Finally, we conclude with a discussion and open questions in Sec. IV.

A pure state LOCC entanglement transformation involves Alice and Bob performing local quantum operations on respective parts of a state $|I〉∈HA⊗HB$ in their possession where $HA$ and $HB$ are the Hilbert spaces of their systems. The local operations may involve unitary operations or measurements, and the two parties can coordinate their actions using classical communication. The aim of these transformations is to obtain a final state |F⟩ with a different final value of entanglement than the initial state. The entanglement value of quantum states can be measured in terms of monotones that are unitarily invariant, concave functions of the Schmidt coefficients of the state. Since entanglement is a non-local property of quantum states, LOCC operations cannot increase the expected value of any entanglement monotones in the final obtained state. As there exist an infinite number of such functions, therefore, in principle, the number of entanglement monotones is infinite.

For pure quantum states, there exist a minimal set of entanglement monotones, however, which can be used to determine whether the LOCC transformation can be performed irreversibly but deterministically, i.e., with unit probability. For a pure state with Schmidt decomposition $|I〉=∑j=1dλjI|jAjB〉$⁠, where |j_A⟩ and |j_B⟩ form orthonormal bases for $HA$ and $HB$⁠, respectively, Nielsen’s criterion¹⁴ states that the transformation from the initial state |I⟩ to a final state |F⟩ is possible with certainty, i.e., P(I → F) = 1, iff the entanglement monotones defined by (with $λ0I=λ0F=0$⁠)

which are functions of the squares of the non-increasingly ordered Schmidt coefficients (OSCs), $λ¯I=(λ1I≥λ2I≥⋯≥λdI)$ and $λ¯F=(λ1F≥λ2F≥⋯≥λdF)$⁠, obey the relation

Entangled quantum states whose OSCs obey relation (2) are called commensurate states with respect to LOCC entanglement transformations.

Initial and final quantum entangled states in a transformation whose OSCs do not obey Eq. (2) are said to be incommensurate. The transformation from |I⟩ → |F⟩ in that case is possible only probabilistically and irreversibly. The maximum probability¹⁵ of such a non-deterministic transformation is given by

That is, the maximum probability is obtained by finding the minimum of the ratios of corresponding entanglement monotones of the initial and final states.

For probabilistic transformations between incommensurate states, taking the help of a catalyst state can increase the transformation success probability. For a desired transformation |ψ⟩ → |ξ⟩, if one utilizes a catalyst state |C⟩, then the entanglement monotones $E(|〉I)$ and $E(|〉F)$ for the initial, |I⟩ = |ψ⟩|C⟩, and final states, |F⟩ = |ξ⟩|C⟩, can be such that

In the above equation, we have defined P_C(I → F) to be the catalyst-aided entanglement transformation success probability. For certain transformations, utilizing a catalyst state, |C⟩, can even enable a deterministic transformation that may not be possible otherwise,¹² i.e.,

where P_C(|ψ⟩|C⟩ → |ξ⟩|C⟩) = 1 but P(|ψ⟩ → |ξ⟩) < 1. For the case of entanglement concentration of a finite number of copies of two-qubit pure states, no catalyst can make the transformation deterministic but can provide a significant boost to the success probability as we discuss in Sec. III. First, however, we provide specific examples of catalyst-aided entanglement transformations in the following.

Consider a pair of initial and final four-qubit states written in the Schmidt basis,

It can be verified that the Schmidt coefficients of these states do not satisfy the inequality in Eq. (2) for all l = 1, 2, 3, 4. Therefore, a deterministic transformation, |I⟩ → |F⟩, using a LOCC procedure is not possible. The maximum success probability of such a transformation evaluates to P(I → F) = 0.8 using Eq. (3). However, when supplied with a two-qubit catalyst state $|C〉=.7|00〉+.3|11〉$⁠, the transformation |I⟩|C⟩ → |F⟩|C⟩ can be achieved with certainty since the Schmidt coefficients of the product initial and final states satisfy Eq. (2).

Consider a different pair of initial and final four-qubit states written in the Schmidt basis,

In this case, again, it can be verified that the Schmidt coefficients of these states do not satisfy the inequality in Eq. (2) for all l = 1, 2, 3, 4. Therefore, here too, a deterministic transformation, |I⟩ → |F⟩, using a LOCC procedure is not possible with the maximum success probability being P(I → F) = 0.72. Moreover, no catalyst state can make this transformation deterministic. The optimal two-qubit catalyst in this case is $|C〉=.591967|00〉+.408033|11〉$⁠, using which the transformation |I⟩|C⟩ → |F⟩|C⟩ can be achieved with a probability of 0.88.

In this subsection, we briefly sketch how the presence of a catalyst can enhance the process of entanglement concentration that we have fully described elsewhere.¹⁶ In this problem, a finite number of copies of two-qubit pure states, $|α〉=α|00〉+1−α|11〉$⁠, are available, and the goal is to finally obtain one maximally entangled two-qubit state as output. Thus, the initial and final states are of the form

which will be provided entanglement assistance via the catalyst state |C⟩. For a fixed number of copies, N, in the above, if the states |ψ⟩ → |ϕ⟩ are incommensurate, no catalyst can make the transformation deterministic.¹⁶ Therefore, for a fixed N ≥ 1, we focus on transformations |ψ⟩^⊗N → |ϕ⟩ whose success probability can benefit from the presence of a catalyst even though they remain non-deterministic under LOCC.

The OSCs of the two states in (8) form probability vectors of length 2^N and are given by

where the Schmidt coefficients α^N−p(1 − α)^p of |ψ⟩ have multiplicities of $Np$ and 0.5 ≤ α ≤ 1. The maximum success probability for such a transformation as given by Eq. (3) is

For non-deterministic LOCC transformations, the minimum in the RHS above is less than unity. Therefore, we have that 2(1 − α^N) < 1 $⇒$α > (1/2)^1/N. For such states, different two-qubit catalysts, $|C〉=c|00〉+1−c|11〉$⁠, provide varying gains to the success probability, P_C(I → F), of the transformation,

where |ψ⟩ and |ϕ⟩ are as in Eq. (8). The largest transformation success probability obtainable using a two-qubit catalyst will be denoted by $PCmax(I→F)$⁠, that is, $PCmax(I→F)=max|C〉PC(I→F)$ where the maximization is over all two-qubit catalyst states.

We obtain P_C(I → F) by evaluating the relevant entanglement monotones of initial and final states in transformation (11). For this, we utilize OSCs of the initial and final states. The OSCs of the final state |F⟩ are

where the first and second entries in the list each have a multiplicity of 2 and 0.5 ≤ c < 1. The zeros following the non-zero entries make the length of $λ¯F$ match the dimension of the initial state dim(|I⟩) = 2^N × 2. Note that the minimization problem in Eq. (3) is restricted to the first four values of l since E_l(|F⟩) = 0∀l ≥ 5, and thus, the ratios, r_l(α, c) ≔ E_l(|I⟩)/E_l(|F⟩) = ∞, for l ≥ 5 do not contribute to the complexity of the minimization in our case. Therefore, only the first four monotones, E_l, of the initial and final states are required. These can be obtained if the first three entries of the OSCs of the initial and final states are known. For the final state [in the entire domain c ∈ (0.5, 1)], we have that

For the initial state |I⟩, we again need the first three entries of the OSCs for the case when 0.5 < c ≤ α and when α < c < 1. These entries are

which holds for 0.5 < c ≤ α, whereas they are

for α < c < 1—note the repeated second largest OSC for $λI2$ above due to its multiplicity (the second largest eigenvalue has a multiplicity of N). Thus, for the two parts of the domain for c, the monotones E_l(|I⟩) of the initial state evaluate to

From Eqs. (13) and (16), we have the four ratios of the entanglement monotones as functions of α, c, and N,

The minimum among the ratios in Eq. (17) for a given value of α and N determines the transformation success probability. Therefore, by a suitable choice of the catalyst, i.e., the value of c, one may be able to achieve P_C(I → F) > P(ψ → ϕ). In Subsection III B, we sketch the arguments using which the minimum among the monotone ratios can be determined.

When a single copy of |α⟩ is available, i.e., N = 1, one can verify that the minimum of the ratios in the set of equations (17) is given by r₄(α, c, N) = 2(1 − α), which is equal to the LOCC probability without a catalyst for all values of 0.5 < c < 1. Therefore, a catalyst cannot help increase the success probability of a LOCC transformation of a single copy of |α⟩ to |ϕ⟩.

For more than one copy of |α⟩, i.e., N ≥ 2, the minimum of the ratios r_l(α, c, N) for l = 2, 3, 4 determines the probability of a successful catalyzed conversion from |ψ⟩^⊗N → |ϕ⟩ [since r₁(α, c, N) = 1]. This can be obtained using the derivatives and continuity properties of r₂, r₃, r₄ to determine the minimum among the three.¹⁶ It turns out that the maximum probability of a LOCC conversion, |I⟩ → |F⟩, is obtained where the curves for r₂(α, c, N) and r₃(α, c, N) with respect to c intersect for a fixed α and N; see Fig. 1. The intersection point, c^opt(α, N), is obtained from the solution of one of the quadratic equations, $r2(α,c,N)=r3c≤α(α,c,N)$ or $r2(α,c,N)=r3c>α(α,c,N)$⁠, as given by Eq. (17). We find that the latter has solutions c = 0 or c > 1, which are unacceptable for a physically meaningful catalyst state, whereas the former equation provides an acceptable solution,

The Schmidt coefficient, c^opt(α, N), provides a two-qubit catalyst pure state, $|Copt(α,N)〉=copt(α,N)|00〉+1−copt(α,N)|11〉$⁠, that provides the maximum success probability in a catalyst-aided procedure for obtaining a maximally entangled two-qubit state from N-copies of partially entangled pure states. This probability is given by the value of r₂(α, c^opt, N) or r₃(α, c^opt, N),

The analytical form of the ratios of the entanglement monotones [Eq. (17)] and that of the optimal catalyst [Eq. (18)] shed light on some interesting properties. They show that an optimal two-qubit catalyst state always exists for N ≥ 2-copies of every state |α⟩ with α ∈ ((1/2)^1/N, 1) and that the optimal catalyst state is always more entangled than |α⟩ since c^opt(α, N) < α. Interestingly, any pure entangled two-qubit state can act as a catalyst, that is, provide a positive boost to the success probability of the |ψ⟩^⊗N → |ϕ⟩, N ≥ 2 transformation in a catalyst-aided procedure. However, optimal self-catalysis is not possible, that is, c^opt(α, N) ≠ α for any N and α < 1. The optimal catalyst state |C^opt(α, N)⟩ becomes less entangled as the state |α⟩ becomes less entangled (α → 1) since the derivative dc^opt(α, N)/dα > 0 in the region α ∈ (0.5, 1)∀N ≥ 2.

We remark that the transformation |I⟩ → |F⟩ in Eq. (11) can be achieved via LOCC operations jointly on the N-copies of the initial state and one-copy of the catalyst state in a two step procedure,^9,14,15 which we briefly outline. In the first step, a temporary state |Γ⟩ that majorizes the initial state is obtained with certainty, i.e., |I⟩ ≺ |Γ⟩, via a sequence of LOCC operations on corresponding two-dimensional subspaces of Alice’s and Bob’s systems (of Hilbert space dimension 2^N+1 each). That is, a single LOCC operation involves two-levels |i⟩_A, |j⟩_A on Alice’s systems and the corresponding two levels |i⟩_B, |j⟩_B on Bob’s systems with i, j ∈ [1, 2^N+1]. Note that the operations on states, ${|i〉A,B}i$⁠, involve the collective manipulation of N-qubits of the shared initial state and one-qubit of the shared catalyst state. The number of such (α, c)-dependent two-level operations is upper bounded by (2^N+1 − 1). In the second step, Bob performs a two-outcome generalized measurement on his portion of the shared state |Γ⟩. For one of the outcomes, which occurs with probability given by Eq. (19), the post-measurement state obtained is |F⟩; therefore, in this case, the catalyst state is recovered along with a Bell state, whereas the other outcome signals the failure of the catalytic process and the post-measurement state may be discarded.

Catalysts of higher dimensions, $|C〉=∑j=1dccj|jA〉|jB〉$⁠, with d_c ≥ 3 being its Schmidt dimension,⁶ can also be used to aid an EC transformation such as in Eq. (11). However, in that case, there are multiple Schmidt coefficients, c_j, j = 1, …, d_c, which need to be determined to obtain the optimal catalyst. Even with the symmetric structure of the Schmidt coefficients of |α⟩^⊗N as in Eq. (9), the optimal catalyst is difficult to obtain analytically. However, we use a numerical search in the space of Schmidt coefficients, c_j ∈ [0, 1], for the set ${cj}j$ that maximizes P_C(I → F) given by the left hand side of Eq. (4) with the constraint that $∑j=1dccj=1$ using a mesh of suitable spacing. This can identify the optimal catalysts for the EC procedure for given choices of the initial and final states |ψ⟩, |ϕ⟩ in Eq. (11). We show the result of such a search in Fig. 2. We find that the incremental advantage of using catalysts of higher dimensions reduces as one uses catalysts of larger dimensions. This can be seen in Fig. 2 by noticing that the increase in the optimal success probability at any fixed value of the parameter α in going from a catalyst of dimension 2 × 2 to a catalyst of dimension 3 × 3 is larger than the gain in going from a catalyst of dimension 3 × 3 to a catalyst of dimension 4 × 4. This implies that catalysts of very high dimensions would be required for states with a low value of initial entanglement, i.e., as α → 1.

The enhancement of EC using catalysts can be observed by comparing the success probability for a transformation aided by catalysts to that for other LOCC procedures that do not use any catalysts. We show the results of such a comparison in Fig. 3. The least efficient of these methods is filtration,^18,19 which is, however, a simple LOCC procedure that works with single copies of |α⟩ and requires no communication between the parties, whose success probability is obtained to be P_filtration(ψ → ϕ) = 2(1 − α). One can do better by performing purification via entanglement swapping, which requires two-copies of the state. Here, an identical copy of the state |α⟩ is utilized by one of the parties for entanglement swapping^10,20 with a successful swap heralding the desired Bell state. In this case, the success probability is P_pur-swap(ψ → ϕ) = 4α(1 − α). Finally, Fig. 3 shows that even the optimal LOCC process¹⁵ that involves collective local operations on multiple copies (N) of |α⟩ and conditional operations based on communication does not perform as well as the equally involved operations for catalyst-aided EC. The success probability for the optimal LOCC without the use of catalysts is 2(1 − α²) as obtained from Eq. (10), which can be seen to be lower than the probability of catalyst-aided LOCC for the optimal choice of catalyst in Fig. 3. Essentially, the entanglement present in the catalyst allows the collective manipulation of |α⟩^N to be more efficiently concentrated into a Bell state.

Higher dimensional catalysts are more efficient for obtaining multiple copies of Bell states. For example, the initial state |α⟩^⊗N (with even N) can be transformed to |ϕ⟩^⊗m with a catalyst of the form |C^opt(α,2)⟩^⊗N/2 in a pairwise ELOCC procedure where the number of obtained Bell states, m = 0, 1, 2, …, n = N/2, is binomially distributed. The probability of obtaining m Bell states is given by $pm=N/2mpm(1−p)N/2−m$ with $p=PCmax(I→F)$ as in Eq. (19), where |I⟩ = |α⟩^⊗2|C^opt(α, 2)⟩ and |F⟩ = |ϕ⟩|C^opt(α, 2)⟩. The expected entanglement, $⟨E⟩=∑mpm*m=(N/2)PCmax(I→F)$⁠, in this entanglement concentration procedure, which we will call strategy-1, is linear in the number of copies N of the initial state |α⟩.

To obtain a target number, m_*, of Bell states, however, a different method, strategy-2, may be more beneficial. In such a strategy, the initial N-copies of |α⟩ may be grouped into m_* sets each of cardinality N_j such that $∑j=1j=m*Nj=N$⁠. The probability of obtaining m_* Bell states will then be the maximum of the product of probabilities maximized over the size of the sets, $pm*=Max{Nj}j∏j=1j=m*Pj$⁠, where P_j is the probability of the transformation, $|α〉⊗Nj→|ϕ〉$⁠. For sets with N_j ≥ 2, one can use an ELOCC transformation procedure so that for such sets, $Pj=PCmax(I→F)$ with $|I〉=|α〉⊗Nj|Copt(α,Nj)〉$ and |F⟩ = |ϕ⟩|C^opt(α, N_j)⟩. The different cardinalities, N_j, of the sets allow one to maximize the catalysis success probability using the appropriate catalyst |C^opt(α, N_j)⟩ for each set.

The choice of the advantageous strategy depends on the number of copies available N, the value of α, and the number of copies of the Bell state m_* desired as the output of the catalyzed entanglement concentration procedure. To compare, strategies-1 and 2, as described above, consider as an example the case when N = 6 and α = 0.99. If m_* = 2 copies of Bell states are desired as output, then strategy-1 yields a probability of 0.034, whereas strategy-2, utilizing 2 sets of 3-copies of |α⟩ each, yields a probability of 0.065. On the other hand, if only a single copy of a Bell state is the desired output, i.e., m_* = 1, then strategy-1 yields a probability of 0.391, whereas strategy-2, utilizing 1 set of 6-copies of |α⟩ each, yields a probability of 0.362.

In summary, we have shown the enhancement of EC using catalysts, which results in an increase in the LOCC transformation success probability. The simplest catalysts of dimension 2 × 2 can be obtained analytically, while higher dimensional catalysts are amenable to a numerical search. A higher dimensional catalyst cannot make EC deterministic; however, it makes the transformation more efficient by providing a larger boost to the success probability compared to transformations with no catalysts. We numerically demonstrated the increased success probability of EC using catalysts over LOCC methods that do not utilize catalysts. Catalyst-aided strategies for obtaining multiple Bell states were discussed.

The enhancement of EC using an entangled catalyst demonstrates the advantage of collective coherent manipulation of quantum states relative to transformations involving a smaller number of states. This is evidenced by the lower success probability of LOCC procedures that require single-copy manipulation as in filtration^18,19 or the use of two-copies as in purification by swapping.^10,20 Furthermore, for a fixed value of α, if a sufficient number of copies, N > log_1/α 2, of the state |α⟩ are available, then the transformation |α⟩^N → |ϕ⟩ can be performed deterministically without the need for any catalyst. Physically, this implies that the entanglement content of these N-copies is sufficient to obtain a single-copy of a Bell state even with the use of the less efficient LOCC procedure that does not utilize catalysts.

The advantage of many quantum protocols relies on the availability of high-fidelity Bell pairs. Therefore, the enhancement of EC using catalysts is relevant for implementations. Catalyst-aided EC requires prior knowledge of the state |α⟩ both for identifying suitable catalysts and for the LOCC operations that depend on the parameters of the state and the catalyst. This limitation needs to be overcome or mitigated to some extent if the enhancement of EC using catalysts can be utilized in practical settings. In such settings, the prior information about the states is not complete. Therefore, one may not be able to use or even determine the optimal catalyst. An open question therefore is on obtaining the optimal probability of EC or entanglement purification when we have information only about the distribution of pure states of a system, i.e., in the mixed state setting. Furthermore, relevant in practical settings is the question of the optimized sequence of quantum operations¹⁷ for catalyst-aided EC that can minimize the number of operations and the rounds of communication that each party performs. We believe that the results here can spur research in these directions.

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