Semidefinite hierarchies for diagonal unitary invariant bipartite quantum states

TL;DR

This paper studies semidefinite hierarchies for detecting entanglement in bipartite quantum states with specific symmetries, improving computational efficiency and extending duality characterizations.

Contribution

It introduces block diagonalization of the DPS hierarchy for diagonal unitary invariant states and characterizes the dual hierarchy for Bose symmetric states, expanding the theoretical framework.

Findings

Block diagonalization enhances computational efficiency.

Dual hierarchy characterized via sums of squares of polynomials.

Tested relaxations on various example classes.

Abstract

We investigate questions about the cone $\mathrm{SEP}_n$ of separable bipartite states, consisting of the Hermitian matrices acting on $\mathbb{C}^n\otimes\mathbb{C}^n$ that can be written as conic combinations of rank one matrices of the form $xx^*\otimes yy^*$ with $x,y\in\mathbb{C}^n$. Bipartite states that are not separable are said to be entangled. Detecting quantum entanglement is a fundamental task in quantum information and a hard computational problem. We explore the Doherty-Parrilo-Spedaglieri (DPS) hierarchy of semidefinite conic approximations for $\mathrm{SEP}_n$ when the bipartite states have some additional structural properties: first, (i) for states with diagonal unitary invariance, and second (ii) for states with Bose symmetry. In case (i) we show that the DPS hierarchy can be block diagonalized, which, combining with its moment reformulation, leads to a substantially more efficient implementation. In case (ii), we give a characterization of the dual hierarchy, in terms of sums of squares of Hermitian complex polynomials, extending a known result in the generic case. It turns out that the completely positive cone $\mathrm{CP}_n$, its dual cone $\mathrm{COP}_n$, and their sums-of-squares based conic approximations $\mathcal{K}^{(t)}_n$, play a central role in these two settings (i),(ii). We clarify these connections and test the block diagonal relaxations on classes of examples.