In order to better understand the class of quantum operations that preserve the positivity of partial transpose (PPT operations) and its relation to the widely used class of local operations and classical communication (LOCC), we study the problem of distinguishing orthogonal maximally entangled states (MES) by PPT operations. Firstly, we outline a rather simple proof to show that the number of
d⊗d PPT distinguishable MES is at most
d, which slightly generalizes existing results on this problem. Secondly, we construct 4 MES in
4⊗4 state space that cannot be distinguished by PPT operations. Before our work, it was unknown whether there exists
d MES in
d⊗d state space that are locally indistinguishable. This example leads us to a novel phenomenon of Entanglement Catalysis Discrimination. Moreover, we find there exists a set of locally indistinguishable states
K such that
K⊗m is locally distinguishable for some finite
m. As an interesting application, we exhibit a quantum channel with one sender and two receivers, whose one-shot zero-error local capacity is not optimal, but multi-use would enhance the capacity to achieve the full output dimension even without entangled input. Finally, we consider the entanglement cost of distinguishing three Bell states and a
2⊗2 entangled basis. In the former case a bipartite pure entangled state with the largest Schmidt coefficient at most 2/3 is necessary and sufficient, while in the latter case an additional Bell state, or one ebit, should be supplied.
Abstract In this paper, we use graph theory to study the distinguishability of lattice states under local operations and classical communication (LOCC) in C p 2 ⨂ C p 2 . Firstly, we present that for the basis of lattice unitary matrices, there are (p 2 + 1)(p + 1) distinct maximal commuting sets in p 2-dimensional system. Secondly, for any set of lattice states, we can obtain a graph G which consists of some k-complete subgraphs. Using the graph G, we show that the set can be distinguished by one-way LOCC if n ( p + 1 ) − ∑ k = 1 p + 1 s k ( k − 1 ) < ( p 2 + 1 ) ( p + 1 ) , where n is the number of vertices of a graph G and s k is the number of k-complete subgraphs. Moreover, we also proved that this conclusion is necessary and sufficient in C 2 2 ⨂ C 2 2 . For C p 2 ⨂ C p 2 system, our result can cover the results in (2015, Phys. Rev. A 92, 042 320); (2020, Sci. China-Phys. Mech. Astron. 63, 280 312).
We show that there exist sets of three mutually orthogonal
d-dimensional maximally entangled states which cannot be perfectly distinguished using one-way local operations and classical communication (LOCC) for arbitrarily large values of
d. This contrasts with several well-known families of maximally entangled states, for which any three states can be perfectly distinguished. We then show that two-way LOCC is sufficient to distinguish these examples. We also show that any three mutually orthogonal
d-dimensional maximally entangled states can be perfectly distinguished using measurements with a positive partial transpose (PPT) and can be distinguished with one-way LOCC with high probability. These results circle around the question of whether there exist three maximally entangled states which cannot be distinguished using the full power of LOCC; we discuss possible approaches to answer this question.