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arXiv:0912.5163 · Physical Review A

Local unambiguous discrimination with remaining entanglement

Yangjia Li, Runyao Duan, Mingsheng Ying

Published Sep 30, 2010• 3 citations
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Abstract

A bipartite state, which is secretly chosen from a finite set of known entangled pure states, cannot immediately be useful in standard quantum information processing tasks. To effectively make use of the entanglement contained in this unknown state, we introduce a way to locally manipulate the original quantum system: either identify the state successfully or distill some pure entanglement. Remarkably, if many copies are available, we show that any finite set of entangled pure states, whether orthogonal or not, can be locally distinguished in this way, which further implies that pure entanglement can be deterministically extracted from unknown entanglement. These results make it clear why a large class of entangled bipartite quantum operations including unitary operations and measurements that are globally distinguishable can also be locally distinguishable: They can generate pure entanglement consistently.

DOI
10.1103/physreva.82.032339

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A bipartite state, which is secretly chosen from a finite set of known entangled pure states, cannot immediately be useful in standard quantum information processing tasks. To effectively make use of the entanglement contained in this unknown state, we introduce a way to locally manipulate the original quantum system: either identify the state successfully or distill some pure entanglement. Remarkably, if many copies are available, we show that any finite set of entangled pure states, whether orthogonal or not, can be locally distinguished in this way, which further implies that pure entanglement can be deterministically extracted from unknown entanglement. These results make it clear why a large class of entangled bipartite quantum operations including unitary operations and measurements that are globally distinguishable can also be locally distinguishable: They can generate pure entanglement consistently.
In this work, we construct small sets of bipartite orthogonal pure states that cannot be perfectly distinguished by local operations and classical communication (LOCC). We mention that not all the states within the constructed sets are necessarily entangled. However, such a set contains at least one entangled state which cannot be conclusively identified by LOCC (with nonzero probability). Then, we show that the states of any such set can be perfectly distinguished by LOCC using a minimal-dimensional entangled resource state. Clearly, here the entangled resource state provides an advantage irrespective of the dimension of the given set. Using this result, we also prove that any pure entangled state is useful as a resource to distinguish the states of any present set unambiguously with nonzero probability under LOCC. These sets exist in all two-qudit Hilbert spaces. Furthermore, it is possible to decrease the average entanglement content of such a set or to increase the cardinality of the set without changing certain properties of the same.
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Suppose we want to identify an input state with one of two unknown reference states, where the input state is guaranteed to be equal to one of the reference states. We assume that no classical knowledge of the reference states is given, but a certain number of copies of them are available instead. Two reference states are independently and randomly chosen from the state space in a unitary invariant way. This is called the quantum state identification problem, and the task is to optimize the mean identification success probability. In this paper, we consider the case where each reference state is pure and bipartite, and generally entangled. The question is whether the maximum mean identification success probability can be attained by means of a local operations and classical communication (LOCC) measurement scheme. Two types of identification problem are considered when a single copy of each reference state is available. We show that a LOCC scheme attains the globally achievable identification probability in the minimum-error identification problem. In the unambiguous identification problem, however, the maximal success probability by means of LOCC is shown to be less than the globally achievable identification probability.
Entanglement is a useful resource because some global operations cannot be locally implemented using classical communication. We prove a number of results about what is and what is not locally possible. We focus on orthogonal states, which can always be globally distinguished. We establish the necessary and sufficient conditions for a general set of 2 x 2 quantum states to be locally distinguishable, and for a general set of 2 x n quantum states to be distinguished given an initial measurement of the qubit. These results reveal a fundamental asymmetry to nonlocality, which is the origin of "nonlocality without entanglement," and we present a very simple proof of this phenomenon.
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I consider the deterministic distinguishability of a set of orthogonal, bipartite states when only a single copy is available and the parties are restricted to local operations and classical communication, but with the additional requirement that entanglement must be preserved in the process. Several general theorems aimed at characterizing sets of states with which the parties can succeed in such a task are proven. These include (i) a maximum for the number of states when the Schmidt rank of every outcome must be at least a given minimum, (ii) an upper bound (equal to the dimension of Hilbert space if entanglement need not be preserved) for the sum over Schmidt ranks of the initial states when only one-way classical communication is allowed, and (iii) separately, a necessary and a sufficient condition on the states such that their original Schmidt ranks can always be preserved. Two additional theorems explicitly demonstrate a trade-off between the extent to which the set of states fill Hilbert space, as measured by their Schmidt ranks, and how refined the parties must make their measurements, an important factor in determining the Schmidt rank the state can retain after it has been identified. It is shown that our bound on the sum of Schmidt ranks can be exceeded if two-way communication is permitted, and this includes the case that entanglement need not be preserved, so that this sum can exceed the dimension of Hilbert space. Such questions, concerning how the various results are affected by the resources used by the parties (amount of classical communication and types of local operations), are addressed for each theorem. This subject is closely related to the problem of locally purifying an entangled state from a mixed state, which is of direct relevance to teleportation and dense coding using a mixed-state resource. In an appendix, I give an extremely simple and transparent proof of ``nonlocality without entanglement,'' a phenomenon originally discussed by Bennett and co-workers several years ago.