Positive-partial-transpose-indistinguishable states via semidefinite programming

Abstract

We present 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), Yu, Duan, and Ying exhibit a set of four maximally entangled states in ${\mathbb{C}}^{4}\ensuremath{\bigotimes}{\mathbb{C}}^{4}$, 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 ${\mathbb{C}}^{k}\ensuremath{\bigotimes}{\mathbb{C}}^{k}$ with the same property, for any $k$ that is a power of 2. By running numerical experiments, we show the local indistinguishability of certain sets of generalized Bell states in ${\mathbb{C}}^{5}\ensuremath{\bigotimes}{\mathbb{C}}^{5}$ and ${\mathbb{C}}^{6}\ensuremath{\bigotimes}{\mathbb{C}}^{6}$ previously considered in the literature.