Journal of Physics A Mathematical and Theoretical

On the locally distinguishable three-dimensional subspace of bipartite systems

Lin Chen

Published Feb 22, 2018• 3 citations

Abstract

Abstract It has been conjectured and numerically supported in King and Matysiak (2007 J. Phys. A: Math. Theor . 40 7939) that any three-dimensional subspace of C 3 ⊗ C n has an orthonormal basis that can be reliably distinguished using one-way local operations and classical communications (LOCC). Since then there has been little progress towards the conjecture. We propose and prove a few cases of the conjecture. We also present a few equivalent or sufficient conditions for the conjecture, and thus moderately simplify the conjecture.

DOI: 10.1088/1751-8121/aab199

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Journal of Physics A Mathematical and Theoretical

On the locally distinguishable three-dimensional subspace of bipartite systems

Lin Chen

Published Feb 22, 2018• 3 citations

Publisher

Abstract

DOI: 10.1088/1751-8121/aab199

Related papers

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{d}_{A},{d}_{B}

are the dimensions of both subsytems, labeled as

A

and

B

. I show that if the

i\mathrm{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 ${d}_{i}\ifmmode\times\else\texttimes\fi{}{d}_{i}$ Hermitian matrices, denoted by ${\mathcal{T}}_{\ensuremath{\perp}}^{(i)}$, and that the method to extract this information (of the existence of this 1-LOCC protocol) from ${\mathcal{T}}_{\ensuremath{\perp}}^{(i)}$ depends on the value of $\mathrm{dim}{\mathcal{T}}_{\ensuremath{\perp}}^{(i)}$. 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 $\mathrm{dim}{\mathcal{T}}_{\ensuremath{\perp}}^{(i)}$. Thus I propose that the value of $\mathrm{dim}{\mathcal{T}}_{\ensuremath{\perp}}^{(i)}$ gives the common underlying property based on which sweeping results for the 1-LOCC (in)distinguishability of orthogonal bipartite quantum states can be made.

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 in negative by showing that any $2\ensuremath{\bigotimes}n$ bipartite subspace contains a basis that is locally distinguishable. As an interesting application, we show that any quantum channel with two Kraus operators has optimal environment-assisted classical capacity.

\mathbb{C}^{d_A \times d_A} \otimes \mathbb{C}^{d_B \times d_B}

, we strengthen this existence result by constructing an

O(d_A^2 d_B^2)

-time algorithm that specifies such a perfect one-way LOCC protocol. Finally, we establish the equivalence between Walgate et al.'s result and the fact that the one-shot environment-assisted classical capacity of every quantum channel is at least 1 bit per channel use, thereby clarifying the literature on these notions. At the core of all of these results is the fact that every operator with vanishing trace admits a basis where its diagonal entries are all zero.