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Optimal signal ensembles

Benjamin Schumacher, Michael D. Westmoreland

TL;DR

This work investigates the quantum-channel capacity for transmitting classical information by optimizing the Holevo bound χ over ensembles of signal states drawn from a given output set 𝒜. Utilizing the quantum relative entropy as a geometric distance, it establishes existence and structure of χ-optimal ensembles, showing they can be taken as pure (extremal) inputs and that all nonzero members share a maximal distance to the average state. A key result is a min–max characterization χ* = min_ρ max_{ρ₀} ${ m D}( ho₀||ρ)$, along with a maximal-distance property ${ m D}( ho_k||ρ^*) = χ^*$ for optimal ensembles, and bounds on how χ shifts when the ensemble is perturbed. The findings provide a geometric framework for understanding and computing the capacity of quantum channels, and show that non-orthogonal inputs can outperform orthogonal ones in certain noisy channels.

Abstract

Classical messages can be sent via a noisy quantum channel in various ways, corresponding to various choices of signal states of the channel. Previous work by Holevo and by Schumacher and Westmoreland relates the capacity of the channel to the properties of the signal ensemble. Here we describe some properties characterizing the ensemble that maximizes the capacity, using the relative entropy "distance" between density operators to give the results a geometric flavor.

Optimal signal ensembles

TL;DR

This work investigates the quantum-channel capacity for transmitting classical information by optimizing the Holevo bound χ over ensembles of signal states drawn from a given output set 𝒜. Utilizing the quantum relative entropy as a geometric distance, it establishes existence and structure of χ-optimal ensembles, showing they can be taken as pure (extremal) inputs and that all nonzero members share a maximal distance to the average state. A key result is a min–max characterization χ* = min_ρ max_{ρ₀} , along with a maximal-distance property for optimal ensembles, and bounds on how χ shifts when the ensemble is perturbed. The findings provide a geometric framework for understanding and computing the capacity of quantum channels, and show that non-orthogonal inputs can outperform orthogonal ones in certain noisy channels.

Abstract

Classical messages can be sent via a noisy quantum channel in various ways, corresponding to various choices of signal states of the channel. Previous work by Holevo and by Schumacher and Westmoreland relates the capacity of the channel to the properties of the signal ensemble. Here we describe some properties characterizing the ensemble that maximizes the capacity, using the relative entropy "distance" between density operators to give the results a geometric flavor.

Paper Structure

This paper contains 6 sections, 13 equations, 1 figure.

Table of Contents

  1. Communication via quantum channels
  2. Relative entropy
  3. The optimal signal ensemble
  4. Changing the ensemble
  5. Properties of optimal ensembles
  6. Acknowledgements

Figures (1)

  • Figure 1: Bloch sphere diagram for amplitude damping. The highest value of $\chi$ for a set of orthogonal input signals is attained by an equally weighted mixture of $\left | \rightarrow \right \rangle$ and $\left | \leftarrow \right \rangle$, but the non-orthogonal input signals $\left | \phi_0 \right \rangle$ and $\left | \phi_1 \right \rangle$ yield a larger value of $\chi$.