Pairwise Completely Positive Matrices and Conjugate Local Diagonal Unitary Invariant Quantum States

TL;DR

This paper introduces pairwise completely positive matrices, explores their properties, and links them to quantum entanglement, providing new criteria for state separability and applications to important quantum states.

Contribution

It generalizes completely positive matrices to pairs, establishes conditions for their identification, and connects this to quantum state separability, advancing entanglement theory.

Findings

Pairwise completely positive matrices are characterized by testable conditions.

Determining PCP status is equivalent to assessing separability of certain quantum states.

Many states with positive partial transpose are shown to be separable.

Abstract

We introduce a generalization of the set of completely positive matrices that we call "pairwise completely positive" (PCP) matrices. These are pairs of matrices that share a joint decomposition so that one of them is necessarily positive semidefinite while the other one is necessarily entrywise non-negative. We explore basic properties of these matrix pairs and develop several testable necessary and sufficient conditions that help determine whether or not a pair is PCP. We then establish a connection with quantum entanglement by showing that determining whether or not a pair of matrices is pairwise completely positive is equivalent to determining whether or not a certain type of quantum state, called a conjugate local diagonal unitary invariant state, is separable. Many of the most important quantum states in entanglement theory are of this type, including isotropic states, mixed Dicke states (up to partial transposition), and maximally correlated states. As a specific application of our results, we show that a wide family of states that have absolutely positive partial transpose are in fact separable.