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Fundamental Quality Bound on Optical Quantum Communication

Tobias Rippchen, Ludovico Lami, Gerardo Adesso, Mario Berta

TL;DR

This work shifts quantum optical communication analysis from capacity to quality by bounding the two-way assisted error exponent with the single-letter reverse relative entropy of entanglement of the channel’s Choi state. It shows that for teleportation-simulable channels, the exponent is tightly controlled by D(\mathrm{SEP}\|\rho_{\mathcal{N}}(r)) and that, in the Gaussian regime, this bound is computable via a finite convex program over covariance matrices. The authors provide closed-form results for key one-mode Gaussian channels (thermal attenuator, amplifier, and additive noise) and give a rigorous operational interpretation of the reverse REE in entanglement distillation under non-entangling operations, extending to infinite dimensions. This framework sharpens theoretical benchmarks for quantum optical networks and offers practical computability for important channels, informing design of robust quantum communication systems. The results connect entanglement theory and quantum communication in a way that highlights quality as a resource and sets the stage for future exploration of resource theories in continuous-variable settings.

Abstract

Sending quantum information reliably over long distances is a central challenge in quantum technology in general, and in quantum optics in particular, since most quantum communication relies on optical fibres or free-space links. Here, we address this problem by shifting the focus from the quantity of information sent to the quality of the transmission, i.e. the rate of decay of the transmission error with respect to the number of channel uses. For the general class of teleportation-simulable channels, which includes all channels arising in quantum optical communication, we prove that the single-letter reverse relative entropy of entanglement of the Choi state upper bounds the error exponent of two-way assisted quantum communication - paralleling the celebrated capacity bound of [Pirandola et al., Nat. Comm. (2017)] in terms of the regularised relative entropy of entanglement. Remarkably, for Gaussian channels our bound can be computed efficiently through a convex program with simple constraints involving only finite-dimensional covariance matrices. As a prototypical application, we derive closed-form analytical expressions for several one-mode Gaussian channels that are fundamental to optical communication. Extending recent work [Lami et al., arXiv:2408.07067 (2024)] to infinite-dimensional systems, we further endow the reverse relative entropy of entanglement with an exact operational interpretation in entanglement testing, and show that it characterises the rate of entanglement distillation under non-entangling operations. These findings offer a new perspective on entanglement as a resource and sharpen the theoretical benchmarks for future quantum optical networks.

Fundamental Quality Bound on Optical Quantum Communication

TL;DR

This work shifts quantum optical communication analysis from capacity to quality by bounding the two-way assisted error exponent with the single-letter reverse relative entropy of entanglement of the channel’s Choi state. It shows that for teleportation-simulable channels, the exponent is tightly controlled by D(\mathrm{SEP}\|\rho_{\mathcal{N}}(r)) and that, in the Gaussian regime, this bound is computable via a finite convex program over covariance matrices. The authors provide closed-form results for key one-mode Gaussian channels (thermal attenuator, amplifier, and additive noise) and give a rigorous operational interpretation of the reverse REE in entanglement distillation under non-entangling operations, extending to infinite dimensions. This framework sharpens theoretical benchmarks for quantum optical networks and offers practical computability for important channels, informing design of robust quantum communication systems. The results connect entanglement theory and quantum communication in a way that highlights quality as a resource and sets the stage for future exploration of resource theories in continuous-variable settings.

Abstract

Sending quantum information reliably over long distances is a central challenge in quantum technology in general, and in quantum optics in particular, since most quantum communication relies on optical fibres or free-space links. Here, we address this problem by shifting the focus from the quantity of information sent to the quality of the transmission, i.e. the rate of decay of the transmission error with respect to the number of channel uses. For the general class of teleportation-simulable channels, which includes all channels arising in quantum optical communication, we prove that the single-letter reverse relative entropy of entanglement of the Choi state upper bounds the error exponent of two-way assisted quantum communication - paralleling the celebrated capacity bound of [Pirandola et al., Nat. Comm. (2017)] in terms of the regularised relative entropy of entanglement. Remarkably, for Gaussian channels our bound can be computed efficiently through a convex program with simple constraints involving only finite-dimensional covariance matrices. As a prototypical application, we derive closed-form analytical expressions for several one-mode Gaussian channels that are fundamental to optical communication. Extending recent work [Lami et al., arXiv:2408.07067 (2024)] to infinite-dimensional systems, we further endow the reverse relative entropy of entanglement with an exact operational interpretation in entanglement testing, and show that it characterises the rate of entanglement distillation under non-entangling operations. These findings offer a new perspective on entanglement as a resource and sharpen the theoretical benchmarks for future quantum optical networks.

Paper Structure

This paper contains 27 sections, 12 theorems, 150 equations, 1 figure.

Table of Contents

  1. The Challenge with Capacities.
  2. A Link to Entanglement Theory.
  3. The Error Exponent of Quantum Communication.
  4. Gaussian Reverse Relative Entropy of Entanglement.
  5. Thermal Attenuator Channel.
  6. Thermal Amplifier Channel.
  7. Additive-Noise Channel.
  8. Operational Interpretations.
  9. Outlook.
  10. Acknowledgments.
  11. Notation and Preliminaries
  12. Infinite-Dimensional Quantum Systems
  13. Reverse Relative Entropy of Entanglement
  14. The Error Exponent of Entanglement Distillation and Entanglement Testing
  15. Generalised Sanov Theorem of Entanglement Testing
  16. ...and 12 more sections

Key Result

Proposition 1

For the class of teleportation-simulable channels, the reverse relative entropy of entanglement provides an upper bound on their error exponent of two-way assisted quantum communication. Specifically, we have for a channel $\mathcal{N}$ that acts on $m$ bosonic modes that where $\mathrm{SEP}$ denotes the set of separable states and the quasi-Choi state, $\rho_\mathcal{N}(r) \coloneqq (\mathcal{N}

Figures (1)

  • Figure 1: (a) A quantum channel $\mathcal{N}$ mapping an input state $\sigma$ to an output state $\mathcal{N}(\sigma)$. (b) The Choi state $\rho_{\mathcal{N}} = (\mathcal{N} \otimes I )(\Phi)$ of the channel $\mathcal{N}$, where $\Phi$ is a maximally entangled state. (c) The channel $\mathcal{N}$ is said to be teleportation-simulable if its action on any input state $\sigma$ can be reproduced by means of an LOCC protocol supplemented by the Choi state $\rho_{\mathcal{N}}$ distributed between sender and receiver. The quantum communication scenario studied in this work can be recast in terms of how well one can distill $m$ copies of approximately maximally entangled states $\approx_{\varepsilon}\Phi^{\otimes m}$ from $n \gg 1$ copies $\rho^{\otimes n}_{\mathcal{N}}$ of the Choi state $\rho_{\mathcal{N}}$ of a channel $\mathcal{N}$, for which two paradigms can be considered: (d) aiming for quantity, i.e. maximising the number of output copies $m$, or (e) aiming for quality, i.e. maximising the exponent at which the error $\varepsilon$ decays as $n \rightarrow \infty$. This work investigates the latter scenario.

Theorems & Definitions (35)

  • Proposition 1
  • Proposition 2
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 2.1
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4
  • Remark 2.5
  • ...and 25 more