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arXiv:quant-ph/0302066 · Physical Review A

Condition for unambiguous state discrimination using local operations and classical communication

Anthony Chefles

Published May 28, 2004• 86 citations
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Abstract

We obtain a necessary and sufficient condition for a finite set of states of a finite-dimensional multiparticle quantum system to be amenable to unambiguous discrimination using local operations and classical communication. This condition is valid for states which may be mixed, entangled, or both. When the support of the set of states is the entire multiparticle Hilbert space, this condition is found to have an intriguing connection with the theory of entanglement witnesses.

DOI
10.1103/physreva.69.050307

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We obtain a necessary and sufficient condition for a finite set of states of a finite-dimensional multiparticle quantum system to be amenable to unambiguous discrimination using local operations and classical communication. This condition is valid for states which may be mixed, entangled, or both. When the support of the set of states is the entire multiparticle Hilbert space, this condition is found to have an intriguing connection with the theory of entanglement witnesses.
Any separable measurement with a finite number of outcomes on a finite dimensional multiparticle quantum system can, with non-zero probability, be implemented with local operations and classical communication (LOCC). I use this fact to obtain a necessary and sufficient condition for a finite set of states of a finite dimensional multiparticle quantum system to be amenable to LOCC unambiguous state discrimination. This condition is valid for states which may be be mixed, entangled or both. It is also found to have an intriguing connection with the theory of entanglement witnesses.
We consider an infinite class of unambiguous quantum state discrimination problems on multipartite systems, described by Hilbert space H\mathcal{H}H, of any number of parties. Restricting consideration to measurements that act only on H\mathcal{H}H, we find the optimal global measurement for each element of this class, achieving the maximum possible success probability of 1/21/21/2 in all cases. This measurement turns out to be both separable and unique, and by our recently discovered necessary condition for local quantum operations and classical communication (LOCC) it is easily shown to be impossible by any finite-round LOCC protocol. We also show that, quite generally, if the input state is restricted to lie in H\mathcal{H}H, then any LOCC measurement on an enlarged Hilbert space is effectively identical to an LOCC measurement on H\mathcal{H}H. Therefore, our necessary condition for LOCC demonstrates directly that a higher success probability is attainable for each of these problems using general separable measurements as compared to that which is possible with any finite-round LOCC protocol.
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.
Entangled states can help in quantum-state discrimination by local operations and classical communication (LOCC). For example, a Bell state is necessary (and sufficient) to perfectly discriminate a set of either three or four Bell states by LOCC. In this paper, we consider the task of LOCC discrimination of the states of noisy Bell ensembles, where a given ensemble consists of the states obtained by mixing the Bell states with an arbitrary two-qubit state with nonzero probabilities. It is proved that a Bell state is required for optimal discrimination by LOCC, even though the ensembles do not contain, in general, any maximally entangled state and, in specific instances, any entangled state.
The paper studies unambiguous discrimination of fermionic states through local operations and classical communication (LOCC). In the task of unambiguous discrimination, no error is tolerated but an inconclusive result is allowed. We show that contrary to the quantum case, it is not always possible to distinguish two fermionic states through LOCC unambiguously with the same success probability as if global measurements were allowed. Furthermore, we prove that we can overcome such a limit through an ancillary system made of two fermionic modes, independently of the dimension of the system, prepared in a maximally entangled state: in this case, LOCC protocols achieve the optimal success probability.