Any $2\otimes n$ subspace is locally distinguishable

TL;DR

This paper proves that all subspaces in a bipartite system with a qubit and an n-dimensional subsystem are locally distinguishable, resolving a long-standing open problem and linking it to quantum channel capacities.

Contribution

It demonstrates that any $2 ext{-}n$ bipartite subspace is locally distinguishable, solving an open problem since 2005 and extending understanding of local distinguishability.

Findings

Any $2\otimes n$ subspace contains a basis distinguishable by LOCC.

All such subspaces are locally distinguishable, unlike higher-dimensional cases.

Quantum channels with two Kraus operators have optimal environment-assisted classical capacity.

Abstract

A subspace of a multipartite Hilbert space is called \textit{locally indistinguishable} if any orthogonal basis of this subspace cannot be perfectly distinguished by local operations and classical communication. Previously it was shown that any $m\otimes n$ bipartite system such that $m>2$ and $n>2$ has a locally indistinguishable subspace. However, it has been an open problem since 2005 whether there is a locally indistinguishable bipartite subspace with a qubit subsystem. We settle this problem by showing that any $2\otimes n$ bipartite subspace is locally distinguishable in the sense it contains a basis perfectly distinguishable by LOCC. As an interesting application, we show that any quantum channel with two Kraus operations has optimal environment-assisted classical capacity.