Separability of diagonal symmetric states: a quadratic conic optimization problem

Abstract

We study the separability problem in mixtures of Dicke states i.e., the separability of the so-called Diagonal Symmetric (DS) states. First, we show that separability in the case of DS in $C^d\otimes C^d$ (symmetric qudits) can be reformulated as a quadratic conic optimization problem. This connection allows us to exchange concepts and ideas between quantum information and this field of mathematics. For instance, copositive matrices can be understood as indecomposable entanglement witnesses for DS states. As a consequence, we show that positivity of the partial transposition (PPT) is sufficient and necessary for separability of DS states for $d\le 4$. Furthermore, for $d\ge 5$, we provide analytic examples of PPT-entangled states. Second, we develop new sufficient separability conditions beyond the PPT criterion for bipartite DS states. Finally, we focus on $N$-partite DS qubits, where PPT is known to be necessary and sufficient for separability. In this case, we present a family of almost DS states that are PPT with respect to each partition but nevertheless entangled.

31 pages (17 + appendices), 4 figures. Version accepted for publication in Quantum

References in corpus (8)

• Device-independent security of quantum cryptography against collective attacks
• Nonlinear atom interferometer surpasses classical precision limit
• Experimental entanglement of six photons in graph states
• Experimental entanglement of a six-photon symmetric Dicke state
• Mesoscopic Entanglement Induced by Spontaneous Emission in Solid-State Quantum Optics
• Experimental bound entanglement in a four-photon state
• Bounding the set of classical correlations of a many-body system
• Detection and typicality of bound entangled states