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The capacity of a quantum channel for simultaneous transmission of classical and quantum information

I. Devetak, P. W. Shor

TL;DR

This work derives a universal capacity region for simultaneous transmission of classical and quantum information over a quantum channel, unifying the classical and quantum capacities through a regularized expression $S({\cal N}) = \bigcup_{l=1}^{\infty} \frac{1}{l} S^{(1)}({\cal N}^{\otimes l})$. It introduces three equivalent information-processing scenarios (Ia, Ib, II) and shows achievability by piggybacking classical information on quantum codes using random $\rho$-type coding and the Holevo–Schumacher–Westmoreland framework, with the coherent-information bound $I(A\rangle BX)$ governing the quantum part. A converse is established via fidelity and information-quantity bounds, culminating in a single-region description that is proven to be concave and to reduce to known capacities $C({\cal N})$ and $Q({\cal N})$ under appropriate limits; generalized dephasing channels admit a single-letter optimization, yielding a strictly concave trade-off curve. The paper also discusses extensions to public-private capacities, static correlations, and joint source-channel coding, highlighting the piggybacking technique as a versatile tool and outlining open questions for incorporating classical communication costs and other extensions. Overall, it provides a foundational, albeit technically intricate, framework for understanding how classical and quantum resources can be simultaneously transmitted over quantum channels.

Abstract

An expression is derived characterizing the set of admissible rate pairs for simultaneous transmission of classical and quantum information over a given quantum channel, generalizing both the classical and quantum capacities of the channel. Although our formula involves regularization, i.e. taking a limit over many copies of the channel, it reduces to a single-letter expression in the case of generalized dephasing channels. Analogous formulae are conjectured for the simultaneous public-private capacity of a quantum channel and for the simultaneously 1-way distillable common randomness and entanglement of a bipartite quantum state.

The capacity of a quantum channel for simultaneous transmission of classical and quantum information

TL;DR

This work derives a universal capacity region for simultaneous transmission of classical and quantum information over a quantum channel, unifying the classical and quantum capacities through a regularized expression . It introduces three equivalent information-processing scenarios (Ia, Ib, II) and shows achievability by piggybacking classical information on quantum codes using random -type coding and the Holevo–Schumacher–Westmoreland framework, with the coherent-information bound governing the quantum part. A converse is established via fidelity and information-quantity bounds, culminating in a single-region description that is proven to be concave and to reduce to known capacities and under appropriate limits; generalized dephasing channels admit a single-letter optimization, yielding a strictly concave trade-off curve. The paper also discusses extensions to public-private capacities, static correlations, and joint source-channel coding, highlighting the piggybacking technique as a versatile tool and outlining open questions for incorporating classical communication costs and other extensions. Overall, it provides a foundational, albeit technically intricate, framework for understanding how classical and quantum resources can be simultaneously transmitted over quantum channels.

Abstract

An expression is derived characterizing the set of admissible rate pairs for simultaneous transmission of classical and quantum information over a given quantum channel, generalizing both the classical and quantum capacities of the channel. Although our formula involves regularization, i.e. taking a limit over many copies of the channel, it reduces to a single-letter expression in the case of generalized dephasing channels. Analogous formulae are conjectured for the simultaneous public-private capacity of a quantum channel and for the simultaneously 1-way distillable common randomness and entanglement of a bipartite quantum state.

Paper Structure

This paper contains 11 sections, 7 theorems, 61 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Scenario Ia (subspace transmission)
  3. Scenario Ib (entanglement transmission)
  4. Scenario II (entanglement generation)
  5. Main result
  6. Proof of theorem 1
  7. Remarks on related problems
  8. Discussion
  9. Proof of concavity of $S^{(1)}({\cal N})$
  10. The capacity region for dephasing channels
  11. Proof of the cardinality bound

Key Result

Theorem 1

The simultaneous capacity regions of ${\cal N}$ for the various scenarios Ia, Ib and II are all equal and given by where $S^{(1)}({\cal N})$ is the union, over all $\sigma^{X A B}$ arising from the channel ${\cal N}$, of the $(r, R)$ pairs obeying Furthermore, in computing $S^{(1)}({\cal N})$ one only needs to consider random variables $X$ defined on some set ${\cal X}$ of cardinality $|{\cal X}

Figures (2)

  • Figure 1: A generic trade-off curve for the simultaneous (classical, quantum) capacity region (solid line). The dashed-dotted line represents the time-sharing inner bound. The dashed line is the outer bound which follows from the observation that the transmitted quantum subspace may always be used to encode classical information at 1 bit/qubit. The continuation to the negative $R$ axis (see text) is shown for scenario II (solid) and scenario I (dashed).
  • Figure 2: The trade-off curve for the dephasing qubit channel with dephasing parameter 0.2 (i.e., the channel obtained by applying the identity operator with probability 0.9 and $\sigma_z$ with probability 0.1). In the left-hand plot, the trade-off curve is plotted with a solid line and the time-sharing bound with a dashed line. The right-hand plot gives the difference between the optimal strategy and time-sharing.

Theorems & Definitions (7)

  • Theorem 1
  • Lemma 2
  • Lemma 3
  • Proposition 4
  • Proposition 5
  • Proposition 6: HSW Theorem
  • Lemma 7