Separability of Bosonic Systems

TL;DR

This paper investigates the separability of bosonic quantum states, establishing criteria based on matrix positivity, confirming a conjecture, and demonstrating NP-hardness for general cases.

Contribution

It introduces a novel connection between separability witnesses and Power Positive Matrices, and provides a separability criterion for multi-qubit states with Dicke states.

Findings

Separable states correspond to positive semidefinite Hankel matrices.

Confirmed the conjecture that such states are separable iff their partial transpose is non-negative.

Determined that separability testing is NP-hard for general $d$ in $d\otimes d$ systems.

Abstract

In this paper, we study the separability of quantum states in bosonic system. Our main tool here is the "separability witnesses", and a connection between "separability witnesses" and a new kind of positivity of matrices--- "Power Positive Matrices" is drawn. Such connection is employed to demonstrate that multi-qubit quantum states with Dicke states being its eigenvectors is separable if and only if two related Hankel matrices are positive semidefinite. By employing this criterion, we are able to show that such state is separable if and only if it's partial transpose is non-negative, which confirms the conjecture in [Wolfe, Yelin, Phys. Rev. Lett. (2014)]. Then, we present a class of bosonic states in $d\otimes d$ system such that for general $d$, determine its separability is NP-hard although verifiable conditions for separability is easily derived in case $d=3,4$.