Distinguishability of Quantum States by Positive Operator-Valued Measures With Positive Partial Transpose

We study the distinguishability of bipartite quantum states by positive operator-valued measures with positive partial transpose (PPT POVMs). The contributions of this paper include: 1) we give a negative answer to an open problem of showing a limitation of a previous known method for detecting nondistinguishability; 2) we show that a maximally entangled state and its orthogonal complement, no matter how many copies are supplied, cannot be distinguished by the PPT POVMs, even unambiguously. This result is much stronger than the previous known ones; and 3) we study the entanglement cost of distinguishing quantum states. It is proved that root 2/3 vertical bar 00 > + root 1/3 vertical bar 11 > is sufficient and necessary for distinguishing three Bell states by the PPT POVMs. An upper bound of entanglement cost of distinguishing a d circle times d pure state and its orthogonal complement is obtained for separable operations. Based on this bound, we are able to construct two orthogonal quantum states, which cannot be distinguished unambiguously by separable POVMs, but finite copies would make them perfectly distinguishable by local operations and classical communication. We further observe that a two-qubit maximally entangled state is always enough for distinguishing a d circle times d pure state and its orthogonal complement by the PPT POVMs, no matter the value of d. In sharp contrast, an entangled state with Schmidt number at least d is always needed for distinguishing such two states by separable POVMs. As an application, we show that the entanglement cost of distinguishing a d circle times d maximally entangled state and its orthogonal complement must be a maximally entangled state for d = 2, which implies that teleportation is optimal, and in general, it could be chosen as O(log d/d).