Entanglement as a resource for local state discrimination in multipartite systems

Abstract

We explore the question of using an entangled state as a universal resource for implementing quantum measurements by local operations and classical communication (LOCC). We show that for most systems consisting of three or more subsystems, there is no entangled state from the same space that can enable all measurements by LOCC. This is in direct contrast to the bipartite case, where a maximally entangled state is an universal resource. Our results are obtained showing an equivalence between the problem of local state transformation and that of entanglement-assisted local unambiguous state discrimination.

Optimal resource states for local state discrimination

“…Since there are 4 N possible outcomes, this implies that the orbit of P N in fact reaches all 2 N elements of S and that the set |Ψ * G ⊗ S can be perfectly distinguished with LOCC. On the other hand, the result in [16] asserts that for any optimal resource |Ψ , |Ψ * must be locally transformable into |Ψ G , which implies that for every entanglement measure, E (Ψ) ≥ E (Ψ G ) = E (Ψ * G ) and every local space must have dimension at least two. Hence, |Ψ * G is an optimal resource state for S. Note that the (m, m)-GHZ states are locally equivalent to the graph states corresponding to the complete graph K m on m vertices; hence Theorem 2 is a direct generalization of Theorem 1.…”

“…In multipartite systems, questions related to optimal resources may pose different kinds of challenges, especially because of the complex structure of the states, computability of entanglement measures and existence of multiple SLOCC equivalence classes [54,55]. In fact, the existence of multiple SLOCC classes led to a recent no-go result [16] which states that for a given multipartite system, a universal resource (a state which can optimally distinguish any set of locally indistinguishable states) almost always does not exist in the same state space. For example, one cannot find a three-qubit pure entangled state that can perfectly distinguish any three-qubit orthonormal basis by LOCC.…”

“…On the other hand, with sufficiently many entangled states any set of LI states can be optimally distinguished. For example, the teleportation protocol [8,12,16] can optimally distinguish any set of LI states in C d ⊗N while consuming (N − 1) log d ebits. However, from a resource perspective the teleportation protocol in general is not optimal.…”

Distinguishing multipartite orthogonal product states by LOCC with entanglement as a resource

“…We believe that any class of locally indistinguishable orthogonal product states can be perfectly distinguished by LOCC with enough entanglements as resources. On the other hand, Bandyopadhyay et al have proved that there is no entangled state as a universal resource for local state discrimination in multipartite systems [31]. This result says that distinguishability of multipartite orthogonal product states has to be dealt individually according to the system.…”

Alternative method for deriving nonlocal multipartite product states

“…More recently, S. Halder presented several sets that no states can be eliminated from the basis by performing orthogonality preserving measurements [33]. Another direction of related research is to study entanglement as a resource to distinguish quantum states of locally indistinguishable states [34][35][36].…”