Entanglement for any definition of two subsystems

TL;DR

This paper introduces the concept of absolutely entangled sets of quantum states that remain entangled under any basis change, providing a new perspective on entanglement invariance across all bipartitions.

Contribution

It defines the notion of absolutely entangled sets, presents a minimal example, and develops a polynomial optimization method to quantify their entanglement.

Findings

Minimum example of an absolutely entangled set in $\

A quantitative measure for absolute set entanglement is proposed.

A convex optimization method over unitaries is developed to bound this measure.

Abstract

The notion of entanglement of quantum states is usually defined with respect to a fixed bipartition. Indeed, a global basis change can always map an entangled state to a separable one. The situation is however different when considering a set of states. In this work we define the notion of an "absolutely entangled set" of quantum states: for any possible choice of global basis, at least one of the states in the set is entangled. Hence, for all bipartitions, i.e. any possible definition of the subsystems, the set features entanglement. We present a minimum example of this phenomenon, with a set of four states in $\mathbb{C}^4 = \mathbb{C}^2 \otimes \mathbb{C}^2$. Moreover, we propose a quantitative measure for absolute set entanglement. To lower-bound this quantity, we develop a method based on polynomial optimization to perform convex optimization over unitaries, which is of independent interest.