PPT-indistinguishable states via semidefinite programming

TL;DR

This paper introduces a semidefinite programming approach to determine the maximum probability of distinguishing orthogonal maximally entangled states using PPT measurements, providing tight bounds and generalizations for specific quantum states.

Contribution

It presents a simple semidefinite program for PPT state distinguishability, derives tight bounds for specific cases, and generalizes results to sets of states in tensor product spaces of dimension powers of two.

Findings

Semidefinite program computes maximum PPT distinguishability probability.

Tight bound of 7/8 for certain 4-dimensional maximally entangled states.

Construction of k-state sets in C^k ⊗ C^k with similar properties.

Abstract

We show a simple semidefinite program whose optimal value is equal to the maximum probability of perfectly distinguishing orthogonal maximally entangled states using any PPT measurement (a measurement whose operators are positive under partial transpose). When the states to be distinguished are given by the tensor product of Bell states, the semidefinite program simplifies to a linear program. In [Phys. Rev. Lett. 109, 020506 (2012) -- arXiv:1107.3224v1], Yu, Duan and Ying exhibit a set of 4 maximally entangled states in $C^4 \otimes C^4$, which is distinguishable by any PPT measurement only with probability strictly less than 1. Using semidefinite programming, we show a tight bound of 7/8 on this probability (3/4 for the case of unambiguous PPT measurements). We generalize this result by demonstrating a simple construction of a set of k states in $C^k \otimes C^k$ with the same property, for any k that is a power of 2. Finally, by running numerical experiments, we obtain some non-trivial results about the PPT-distinguishability of certain interesting sets of generalized Bell states in $C^5 \otimes C^5$ and $C^6 \otimes C^6$.