Unextendibility, uncompletability, and many-copy indistinguishable ensembles

TL;DR

This paper investigates unextendible and uncompletability concepts for operators positive under partial transpose, linking them to many-copy indistinguishable ensembles under LOCC, and introduces a unifying framework for their analysis.

Contribution

It introduces the notion of positive partial transpose uncompletability and connects it to many-copy indistinguishability, providing explicit constructions and simplifying proofs.

Findings

Maximum cardinality of nonorthogonal unextendible product bases (nUPBs) identified.

Many-copy indistinguishable ensembles can be characterized using unextendibility concepts.

Local indistinguishability increases as the number of mixed states decreases in certain ensembles.

Abstract

In this work, we explore the notions unextendible product basis and uncompletability for operators which remain positive under partial transpose. Then, we analyze their connections to the ensembles which are many-copy indistinguishable under local operations and classical communication (LOCC). We show that the orthogonal complement of any bipartite pure entangled state is spanned by product states which form a nonorthogonal unextendible product basis (nUPB) of maximum cardinality. This subspace has one to one correspondence with the maximum dimensional subspace where there is no orthonormal product basis. Due to these, the proof of indistinguishability of a class of ensembles under LOCC in many-copy scenario becomes simpler. Furthermore, it is now clear that there are several many-copy indistinguishable ensembles which are different construction-wise. But if we consider the technique of proving their indistinguishability property under LOCC, then, for many of them it can be done using the general notion of unextendible product basis. Explicit construction of the product states, forming nUPBs is shown. Thereafter, we introduce the notion of positive partial transpose uncompletability to unify different many-copy indistinguishable ensembles. We also report a class of multipartite many-copy indistinguishable ensembles for which local indistinguishability property increases with decreasing number of mixed states.