Framework for distinguishability of orthogonal bipartite states by one-way local operations and classical communication

TL;DR

This paper introduces a framework to analyze the perfect distinguishability of orthogonal bipartite quantum states using one-way LOCC, based on the dimension of a specific subspace of hermitian matrices, applicable to systems of arbitrary dimensions.

Contribution

It proposes a novel method linking the existence of one-way LOCC protocols to the dimension of a hermitian matrix subspace, enabling broad results across various bipartite systems.

Findings

The dimension of the subspace $ ilde{T}^{(i)}$ determines the distinguishability.

The framework applies to bipartite systems of arbitrary dimensions.

Provides sweeping results for the (in)distinguishability based on subspace dimension.

Abstract

In the topic of perfect local distinguishability of orthogonal multipartite quantum states, most results obtained so far pertain to bipartite systems whose subsystems are of specific dimensions. In contrast very few results for bipartite systems whose subsystems are of arbitrary dimensions, are known. This is because a rich variety of (algebraic or geometric) structure is exhibited by different sets of orthogonal states owing to which it is difficult to associate some common property underlying them all, i.e., a common property that would play a crucial role in the local distinguishability of these states. In this paper, I propose a framework for the distinguishability by one-way LOCC ($1$-LOCC) of sets of orthogonal bipartite states in a $d_A \otimes d_B$ bipartite system, where $d_A, d_B$ are the dimensions of both subsytems, labelled as $A$ and $B$. I show that if the $i$-th party (where $i=A,B$) can initiate a $1$-LOCC protocol to perfectly distinguish among a set of orthogonal bipartite states, then the information of the existence of such a $1$-LOCC protocol lies in a subspace of $d_i \times d_i$ hermitian matrices, denoted by $\Tb^{(i)}$, and that the method to extract this information (of the existence of this $1$-LOCC protocol) from $\Tb^{(i)}$ depends on the value of $dim \Tb^{(i)}$. In this way one can give sweeping results for the $1$-LOCC (in)distinguishability of all sets of orthogonal bipartite states corresponding to certain values of $dim \Tb^{(i)}$. Thus I propose that the value of $dim \Tb^{(i)}$ gives the common underlying property based on which sweeping results for the $1$-LOCC (in)distinguishability of orthogonal bipartite quantum states can be made.