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

TL;DR

This paper investigates the limitations and requirements of PPT POVMs in distinguishing bipartite quantum states, revealing fundamental bounds, impossibility results, and entanglement cost estimates for various quantum state discrimination tasks.

Contribution

It provides new limitations of PPT POVMs, establishes entanglement cost bounds, and demonstrates the necessity of certain entangled states for state discrimination.

Findings

Maximally entangled states cannot be distinguished by PPT POVMs, even with multiple copies.

A specific entanglement resource is necessary for distinguishing Bell states by PPT POVMs.

Finite copies enable perfect discrimination of certain states by LOCC, but not by separable POVMs.

Abstract

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 [M. Horodecki $et. al$, Phys. Rev. Lett. 90, 047902(2003)] showing a limitation of their method for detecting nondistinguishability. (2). We show that a maximally entangled state and its orthogonal complement, no matter how many copies are supplied, can not be distinguished by PPT POVMs, even unambiguously. This result is much stronger than the previous known ones \cite{DUAN06,BAN11}. (3). We study the entanglement cost of distinguishing quantum states. It is proved that $\sqrt{2/3}\ket{00}+\sqrt{1/3}\ket{11}$ is sufficient and necessary for distinguishing three Bell states by PPT POVMs. An upper bound of entanglement cost of distinguishing a $d\otimes d$ 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 LOCC. We further observe that a two-qubit maximally entangled state is always enough for distinguishing a $d\otimes d$ pure state and its orthogonal complement by 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 $d\otimes d$ 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 $\mathcal{O}(\frac{\log d}{d})$.