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Quantum Reverse Shannon Theorem Revisited

Zahra Baghali Khanian, Debbie Leung

TL;DR

This work broadens the quantum reverse Shannon landscape by unifying classical and quantum channel simulations into a single framework that accommodates general mixed-state inputs and encoder-side side information. It introduces two rate functionals, $a(\rho^{AR},\gamma)$ and $u(\rho^{AR},\gamma)$, to characterize entanglement-assisted and unassisted simulation rates, proving the optimal entanglement-assisted rate equals $a(\rho^{AR},0)$ and providing a regularized unassisted rate via $u(\rho^{AR},\gamma)$. The results recover known special cases, such as identity channels and classical inputs, and connect to KI decomposition and quantum state redistribution to establish achievability and converse bounds. The study highlights the role of decoupling in channel simulation with mixed states and points to open questions on computability, single-letter characterizations, and resource trade-offs between entanglement, randomness, and quantum communication.

Abstract

Reverse Shannon theorems concern the use of noiseless channels to simulate noisy ones. This is dual to the usual noisy channel coding problem, where a noisy (classical or quantum) channel is used to simulate a noiseless one. The Quantum Reverse Shannon Theorem is extensively studied by Bennett and co-authors in [IEEE Trans. Inf. Theory, 2014]. They present two distinct theorems, each tailored to classical and quantum channel simulations respectively, explaining the fact that these theorems remain incomparable due to the fundamentally different nature of correlations they address. The authors leave as an open question the challenge of formulating a unified theorem that could encompass the principles of both and unify them. We unify these two theorems into a single, comprehensive theorem, extending it to the most general case by considering correlations with a general mixed-state reference system. Furthermore, we unify feedback and non-feedback theorems by simulating a general side information system at the encoder side.

Quantum Reverse Shannon Theorem Revisited

TL;DR

This work broadens the quantum reverse Shannon landscape by unifying classical and quantum channel simulations into a single framework that accommodates general mixed-state inputs and encoder-side side information. It introduces two rate functionals, and , to characterize entanglement-assisted and unassisted simulation rates, proving the optimal entanglement-assisted rate equals and providing a regularized unassisted rate via . The results recover known special cases, such as identity channels and classical inputs, and connect to KI decomposition and quantum state redistribution to establish achievability and converse bounds. The study highlights the role of decoupling in channel simulation with mixed states and points to open questions on computability, single-letter characterizations, and resource trade-offs between entanglement, randomness, and quantum communication.

Abstract

Reverse Shannon theorems concern the use of noiseless channels to simulate noisy ones. This is dual to the usual noisy channel coding problem, where a noisy (classical or quantum) channel is used to simulate a noiseless one. The Quantum Reverse Shannon Theorem is extensively studied by Bennett and co-authors in [IEEE Trans. Inf. Theory, 2014]. They present two distinct theorems, each tailored to classical and quantum channel simulations respectively, explaining the fact that these theorems remain incomparable due to the fundamentally different nature of correlations they address. The authors leave as an open question the challenge of formulating a unified theorem that could encompass the principles of both and unify them. We unify these two theorems into a single, comprehensive theorem, extending it to the most general case by considering correlations with a general mixed-state reference system. Furthermore, we unify feedback and non-feedback theorems by simulating a general side information system at the encoder side.

Paper Structure

This paper contains 9 sections, 12 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Setup
  3. Decoupling Condition and Rate Functionals
  4. Entanglement-assisted simulation
  5. Unassisted Simulation
  6. Discussion
  7. Proof of Lemma \ref{['lemma: decoupling']}
  8. Proof of Lemma \ref{['lemma: f properties']}
  9. Miscellaneous Lemmas and Facts

Key Result

Lemma 2

The fidelity criterion of Eq. (eq: F) implies that the decoded state on systems $A_1B_1$ is decoupled from the rest of the systems in the following sense where $\delta(n,\epsilon) = 4\sqrt{6\epsilon} \log(d_1) + \frac{2}{n}h(\sqrt{6\epsilon})$, with $d_1=\frac{\log|A_1|}{n}$ and the binary entropy $h(\epsilon)=-\epsilon\log\epsilon - (1-\epsilon)\log(1-\epsilon)$. The mutual information is with r

Theorems & Definitions (23)

  • Remark 1
  • Lemma 2
  • Definition 3
  • Remark 4
  • Lemma 5
  • Theorem 6
  • proof
  • Proposition 7
  • proof
  • Theorem 8
  • ...and 13 more