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The existence of distinguishable bases in three-dimensional subspaces of qutrit-qudit systems under one-way local operations and classical communication

Zhiwei Song, Lin Chen, Dragomir Z. Djokovic

Abstract

We show that every three-dimensional subspace of qutrit-qudit complex or real systems has a distinguishable basis under one-way local operations and classical communication (LOCC). In particular this solves an open problem proposed in [J. Phys. A, 40, 7937, 2007]. We construct a three-dimensional space whose locally distinguishable basis is unique and apply the uniqueness property to the task of state transformation. We also construct a three-dimensional locally distinguishable multipartite space assisted with entanglement. On the other hand, we show that four-dimensional indistinguishable bipartite subspaces under one-way LOCC exist. Further, we show that the environment-assisted classical capacity of every channel with a three-dimensional environment is at least $\log_2 3$, and the environment-assisting classical capacity of any qutrit channel is $\log_2 3$. We also show that every two-qutrit state can be converted into a generalized classical state near the quantum-classical boundary by an entanglement-breaking channel.

The existence of distinguishable bases in three-dimensional subspaces of qutrit-qudit systems under one-way local operations and classical communication

Abstract

We show that every three-dimensional subspace of qutrit-qudit complex or real systems has a distinguishable basis under one-way local operations and classical communication (LOCC). In particular this solves an open problem proposed in [J. Phys. A, 40, 7937, 2007]. We construct a three-dimensional space whose locally distinguishable basis is unique and apply the uniqueness property to the task of state transformation. We also construct a three-dimensional locally distinguishable multipartite space assisted with entanglement. On the other hand, we show that four-dimensional indistinguishable bipartite subspaces under one-way LOCC exist. Further, we show that the environment-assisted classical capacity of every channel with a three-dimensional environment is at least , and the environment-assisting classical capacity of any qutrit channel is . We also show that every two-qutrit state can be converted into a generalized classical state near the quantum-classical boundary by an entanglement-breaking channel.
Paper Structure (3 sections, 9 theorems, 31 equations)

This paper contains 3 sections, 9 theorems, 31 equations.

Table of Contents

  1. APPENDIX A: The proof of K-M conjecture
  2. APPENDIX B: The proof of a necessary lemma supporting Theorem \ref{['3dim']}
  3. APPENDIX C: The real K-M conjecture and its proof

Key Result

Theorem 1

The K-M conjecture is true. Equivalently, any three-dimensional subspace of $\mathbb{C}^3\otimes \mathbb{C}^n$ has an orthonormal basis which is distinguishable under one-way LOCC.

Theorems & Definitions (10)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Corollary 5
  • Lemma 6
  • Lemma 7
  • Conjecture 8
  • Lemma 9
  • Theorem 10