Nonlocal Entanglement Transformations Achievable by Separable Operations
For manipulations of multipartite quantum systems, it was well known that all local operations assisted by classical communication (LOCC) constitute a proper subset of the class of separable operations. Recently, Gheorghiu and Griffiths found that LOCC and general separable operations are equally powerful for transformations between bipartite pure states. In this letter we extend this comparison to mixed states and show that in general separable operations are strictly stronger than LOCC when transforming a mixed state to a pure entangled state. A remarkable consequence of our finding is the existence of entanglement monotone which may increase under separable operations.
The ability for quantum systems to exist in non-classical entangled states presents new possibilities for computational tasks and information processing Mermin (2007). Since entanglement is easily lost through interactions with the environment, any device that hopes to utilize the properties of entanglement must ultimately find a way to protect its machinery from excessive environmental noise. As the physical separation between different parts of a quantum system increases, any sort of manipulation on the system as a whole without introducing environmental interactions becomes even more challenging, a consideration obviously relevant to any long-distance quantum communication.
One solution to this problem is to concede the ability of global operations on the system and require that the only quantum operations performed are done so locally by different individuals having access to just a single part of the system. While still allowing parties to share classical information with one another, manipulation protocols of this type are called LOCC, an acronym for local operations and classical communication. All LOCC protocols belong to the class of separable operations Bennett et al. (1999) which has a very compact mathematical characterization. This fact turns out to be useful since LOCC impossibility results can be obtained by studying the limitations of separable operations, which Rains has done for the task of pure state distillation Rains (1997). However, it turns out that not all separable operations can be implemented in the form of an LOCC protocol. Specifically, as first demonstrated by the authors of Bennett et al. (1999), there exist sets of states that can be distinguished by separable operations but not by LOCC. Beyond the fact that all LOCC operations are separable, any further mathematical description of LOCC protocols is lacking. Furthermore, until now, the focus on tasks demonstrating the difference of separable operations and LOCC has been limited to state distinguishability which is unfortunate because it is precisely these tasks that expose the complex nature of LOCC protocols and illustrate their operational limitations.
In this Letter, we compare the transforming abilities of separable operations versus LOCC when acting on general quantum states and find the latter to be strictly more powerful. Formally, the most general state of any quantum system can be described by a density operator belonging to , the space of linear operators acting on some Hilbert space . Any change to the system’s state is referred to as a quantum operation and is mathematically expressed by a superoperator that acts on . It turns out that every physically realizable superoperator can be represented by a complete set of “Kraus” operators so that density operators are mapped according to where Schumacher (1996). For operations on -partite systems, if each can be factored in the form , the physical process described by is called a separable operation. As already noted, the Kraus operators representing every LOCC protocol can be factored as such and thus all LOCC operations are separable; it is the converse that fails to be true.
Given two bipartite quantum states and , we ask whether the existence of a separable operation such that necessarily implies the existence of an LOCC operation such that . Gheorghiu and Griffiths have recently shown that indeed, when both and are pure states, existence of one implies the existence of the other Gheorghiu and Griffiths (2008). Specifically, the authors prove that any pure state transformation by separable operations obeys the same necessary and sufficient conditions of an LOCC transformation, namely those given by Nielsen Nielsen (1999) and more generally by Vidal Vidal (1999) and Jonathan and Plenio Jonathan and Plenio (1999). The first step in learning whether such a result holds for all bipartite states is to consider arbitrary and examine the question of pure state distillation from a single mixed state. In other words, given some mixed state , is it possible to transform into a pure state under a given class of operations?
It has previously been shown that for mixed states in systems, one cannot obtain a pure entangled state via separable operations with any positive probability of success Kent (1998). Below, we prove that the same result holds for systems when the probability of success is one. Thus, for the task of deterministic pure state distillation on and systems, separable operations and LOCC can be considered as equally powerful. In sharp contrast to that, we are able to show that for systems, mixed states exist in which pure entanglement can be distilled by separable operations but not by LOCC. This naturally leads to the existence of entanglement monotones that increase under separable operations thus resolving a conjecture intimated in Ref. Horodecki et al. (2003).
The fact that separable transformations on mixed states are more powerful than LOCC should not be overly surprising. This is because mixed states can be regarded as an ensemble of pure states such that mixed state distillation is equivalent to the bulk transformation between two sets of pure states where, up to complex coefficients, the target set consists of only one pure state. We already know that when the final set contains more than just a single pure state, there exist source and target sets transformable by separable operations but not LOCC. This is indirectly the main result of Ref. Bennett et al. (1999) and more explicitly implied in Ref. Duan et al. (). We state this observation and (thanks to the findings in Ref. Bennett et al. (1999)) its simple proof here for definitiveness.
Let be any set of states distinguishable by separable operations but not by LOCC. Then there exists sets such that is achievable by separable operations but not by LOCC.
Proof. Although a non-trivial fact, such sets do exist and as pointed out in Ref. Duan et al. (), any set of distinguishable states can always be converted to a set of LOCC distinguishable states through some separable operation. Hence, conversion is impossible through LOCC since by locally distinguishing the elements of , the separated parties would have a way to distinguish the elements of .
When contains just a single pure state, this distinguishability argument no longer applies and we return to the central investigation of this Letter. It should be noted that previous research has been done concerning bulk transformations between sets of states using unrestricted quantum operations Chefles et al. (2003). In what follows, we will make frequent use of a pure state’s Schmidt rank which is the minimum number of product states needed to express the given state. Thus, every entangled state has a Schmidt rank of at least two. We begin by examining the differences in separable and LOCC pure state distillation on lower dimensional systems and find the following:
Let be any or mixed state. Then for any entangled state , no separable operation can transform into with probability one. Hence, deterministic pure state distillations on and systems cannot be achieved by either separable operations or LOCC.
Proof. As Kent has already proven this to be true for systems Kent (1998), it is enough to examine systems. Suppose there exists some separable operation with Kraus operators that can perform the indicated transformation. Without loss of generality, we may assume the rank of is two and let and denote its orthogonal eigenstates. Note that the separable operation must also transform any superposition of and into which implies that and each have Schmidt rank at least two. Then there exists some operator in the set of Kraus operators such that and where both and is nonzero, or either only one of or is nonzero. Consider case . Since must be full rank, we have that , or equivalently . But then is a product vector, which is impossible since any state in the linear span of and should be entangled. Consider now case and without loss of generality, take to be nonzero. Then must be full rank which means for to be zero, must be rank one. But this is impossible since has Schmidt rank two.
We now move onto the larger dimensional state space of and show that separable and LOCC pure state distillation abilities are no longer the same. The following theorem is the main result of this Letter.
(a) Let be any mixed state and let be any entangled pure state. Then it is impossible to deterministically convert to by LOCC. (b) There are mixed state and pure entangled state such that the transformation of to can be achieved with certainty by some separable operation. Thus, separable operations are strictly stronger than LOCC for entanglement transformations between mixed states and pure states.
Proof. Without loss of generality, we may assume that is of rank two and let and be its orthogonal eigenstates. Again note that the stated conversion is possible if and only if there exists an LOCC protocol such that . In other words, transforms and into simultaneously. Now, every bipartite LOCC protocol consists of alternating rounds where one party, say Alice, makes a measurement on her subsystem and then reports the result to Bob. In the next round, Bob selects a particular measurement to make on his subsystem based on the information given by Alice. The outcome of this measurement is reported back to Alice and the cycle repeats. Hence a tree of all possible outcomes emerges in which each branch represents a series of operators alternately applied by Alice and Bob. Specifically, let index all the branches after the round of measurement so that describes the net operations throughout all rounds along branch . We first prove the following claim: for any branch , if and only if . By induction, assume it to be true. Then suppose Alice (or Bob) makes the next measurement. For any , let correspond to the branch . Since this is a deterministic transformation, implies and hence (by assumption) must both be states with Schmidt rank at least two. Furthermore, if is applied with a nonzero probability on either state, it must also be at least rank two, and thus cannot eliminate any state with Schmidt rank at least two. As a result, we have if and only if . Clearly this argument also applies when , so we have proven the claim.
Next consider any branch in the protocol that transforms to . As just proven, must also be transformed to along this branch. Since and are linearly independent but ultimately become the same state, there must be some round on this branch such that and are linearly independent, but where is some complex scalar. Here, denotes the resultant state of the system originally in state after round , and it is assumed without loss of generality that Alice makes the round measurement. Because both and can be converted into by the same LOCC protocol, it follows by linearity of the protocol that any superposition of and must be entangled. But then it is impossible that since is of rank at least two, and we arrive at a contradiction.
We give an explicit separable transformation. Consider the state where and . Work by Cohen reveals that these states cannot be distinguished by LOCC while preserving their Schmidt rank, but at the same time, such a discrimination is possible by separable operations Cohen (2007). By modifying a POVM given in Ref. Cohen (2007), we obtain a separable operation described by the following set of Kraus operators }:
where and . One can readily verify that deterministically transforms into with .
Theorem 2 leads directly to the existence of entanglement monotones that increase under separable operations. Recall that an entanglement monotone is some function mapping density operators to the real numbers that does not increase on average under LOCC transformations Vidal (2000). As demonstrated in Ref. Vidal (2000), any deterministic transformation not achievable by LOCC necessarily implies the increase in some entanglement monotone. Thus, we obtain the following consequence.
There exist entanglement monotones that can increase under separable operations.
The results of Ref. Gheorghiu and Griffiths (2008) show that any entanglement monotone defined by a convex roof extension of some pure state measure will necessarily be monotonic under separable operations. Consequently, the monotone that increases under separable operations must be defined in another manner such as the one provided in Theorem 1 of Ref. Vidal (2000). Specifically, for any states and , let be the maximum probability of obtaining from by LOCC. Then where the maximum is taken over all LOCC protocols transforming into an ensemble where output state occurs with probability . As a result, cannot increase on average under LOCC since the maximum average after any LOCC protocol is precisely . The source and target states of Theorem 2 provide an example of when increases under separable operations.
No matter how we define our entanglement monotone, product states will always have the lowest possible amount of entanglement since any product state can be obtained by LOCC from any original state. Because separable operations map product states to product states, they will never be able to increase the entanglement of systems originally unentangled, regardless of what monotone we use. Thus it is interesting to note that only if the original state is entangled can separable operations increase its entanglement according to some entanglement monotone, and as now evident, in some cases it does.
Many open questions exist concerning the items discussed in this Letter. It would be desirable to know if separable and LOCC transformations are equally strong on systems. For , it is not difficult to construct mixed states from which pure entanglement can be deterministically distilled. However in all such examples we found an LOCC protocol that could achieve the same transformation. Another relevant question is whether separable operations are more powerful than LOCC on and systems when the target state is mixed. With the class of separable operations being so large, this seems highly plausible.
When viewed in conjunction with Cohen’s work Cohen (2007), our results suggest that a critical difference between LOCC and separable operations is their ability to preserve entanglement when acting on multiple pure states. Specifically, since LOCC protocols consist of multi-round measurements, for the tasks we consider the Schmidt rank of a state must be preserved after each stage, whereas with separable operations, it only needs to be preserved after a “single shot”. Thus it is no surprise that the ability to perform certain quantum operations by LOCC is strictly less than separable operations. On the other hand, the phenomenon of nonlocality without entanglement Bennett et al. (1999), i.e., the existence of orthogonal product states indistinguishable by LOCC but not by separable operations, implies that preservation of entanglement cannot be the whole story to the difference between LOCC and separable operations. As further effort is put forth to understand the complicated nature of LOCC protocols, we hope that the results presented here generate useful tools and considerations to assist in this endeavor. At the very least, we have introduced a new practical scenario distinct from state distinguishability in which the capabilities of LOCC protocols cannot simply be obtained by studying separable operations.
We thank Yaoyun Shi and Andreas Winter for helpful discussions and comments. This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 60702080, 60736011, and 60621062), the FANEDD under Grant No. 200755, and the Hi-Tech Research and Development Program of China (863 project) (Grant No. 2006AA01Z102). Eric Chitambar was also partially supported by the National Science Foundation of the United States under Awards 0347078 and 0622033.
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