Local indistinguishability and incompleteness of entangled orthogonal bases: Method to generate two-element locally indistinguishable ensembles

TL;DR

This paper explores the relationship between local indistinguishability and unextendibility of entangled bases, providing methods to generate two-element indistinguishable ensembles with potential applications in quantum information sharing.

Contribution

It introduces a systematic method to generate two-element locally indistinguishable ensembles based on unextendible entangled bases, applicable to bipartite and multipartite systems.

Findings

All two-qubit unextendible entangled bases are of size three.

Two-element orthogonal ensembles can be constructed to be locally indistinguishable.

Certain multipartite unextendible bases maintain unextendibility across all bipartitions.

Abstract

We relate the phenomenon of local indistinguishability of orthogonal states with the properties of unextendibility and uncompletability of entangled bases for bipartite and multipartite quantum systems. We prove that all two-qubit unextendible entangled bases are of size three and they cannot be perfectly distinguished by separable measurements. We identify a method of constructing two-element orthogonal ensembles, based on the concept of unextendible entangled bases, that can potentially lead to information sharing applications. Two-element ensembles form the fundamental unit of ensembles, and yet does not offer locally indistinguishable ensembles for pure state elements. Going over to mixed states does open this possibility, but can be difficult to identify. The method provided using unextendible entangled bases can be used for their systematic generation. In multipartite systems, we find a class of unextendible entangled bases for which the unextendibility property remains conserved across all bipartitions. We also identify nonlocal operations, local implementation of which require entangled resource states from a higher-dimensional quantum system.