Threshold entanglement sharing: quantum states with absolutely separable marginals

TL;DR

This paper introduces threshold entanglement states that distribute entanglement in quantum networks, demonstrating their existence for certain qubit numbers and establishing bounds on their properties.

Contribution

It defines TE states, proves their existence for specific qubit counts, and improves bounds on marginals' purity, advancing understanding of entanglement distribution.

Findings

TE states exist for four and seven qubits.

Bounds on purity of marginals are improved.

TE states of eight qubits cannot exist.

Abstract

Motivated to understand how entanglement resources can be distributed in quantum networks, we introduce threshold entanglement (TE) states. These are multipartite quantum states whose entanglement across bipartitions forces all marginals of half or less local systems to be (absolutely) separable. First, in contrast to states used for quantum secret sharing, we demonstrate that TE states exist for four and seven qubits. Second, between four and nine qubits, we delimit the average entanglement that TE states must have by combining two semidefinite programming relaxations: (i) lower bounds on the minimal purity of pure state marginals, and (ii) upper bounds on the maximal purity of mixed absolutely separable states. Besides delimiting the existence regions of TE states, our approach independently improves the best known bounds on both of the above problems. Moreover, these improved bounds show that TE states of eight qubits cannot exist. Numerical evidence suggests that TE states accommodate significant amounts of entanglement and magic, which are resources needed for quantum advantage in quantum computing.