Absolutely entangled sets of pure states for bipartitions and multipartitions

TL;DR

This paper investigates absolutely entangled sets of pure quantum states, providing conditions for their detection, constructing examples, and establishing bounds for bipartitions and multipartitions, with implications for quantum information robustness.

Contribution

It introduces new criteria and constructions for absolutely entangled sets across bipartitions and multipartitions, advancing understanding of entanglement invariance.

Findings

Sets of N > ((d1+1)(d2+1))/2 states are almost surely absolutely entangled for bipartitions.

Constructed explicit examples of absolutely entangled sets for various multipartitions.

Derived lower bounds on the size of absolutely entangled sets for multipartitions.

Abstract

A set of quantum states is said to be absolutely entangled, when at least one state in the set remains entangled for any definition of subsystems, i.e. for any choice of the global reference frame. In this work we investigate the properties of absolutey entangled sets (AES) of pure quantum states. For the case of a two-qubit system, we present a sufficient condition to detect an AES, and use it to construct families of $N$ states such that $N-3$ (the maximal possible number) remain entangled for any definition of subsystems. For a general bipartition $d=d_1d_2$, we prove that sets of $N>\left\lfloor{(d_{1}+1)(d_{2}+1)/2}\right \rfloor$ states are AES with Haar measure 1. Then, we define AES for multipartitions. We derive a general lower bound on the number of states in an AES for a given multipartition, and also construct explicit examples. In particular, we exhibit an AES with respect to any possible multi-partitioning of the total system.