Quantum Sphere-Packing Bounds with Polynomial Prefactors

Abstract

We study lower bounds on the optimal error probability in classical coding over classical-quantum channels at rates below the capacity, commonly termed quantum sphere-packing bounds. Winter and Dalai have derived such bounds for classical-quantum channels; however, the exponents in their bounds only coincide when the channel is classical. In this paper, we show that these two exponents admit a variational representation and are related by the Golden-Thompson inequality, reaffirming that Dalai's expression is stronger in general classical-quantum channels. Second, we establish a sphere-packing bound for classical-quantum channels, which significantly improves Dalai's prefactor from the order of subexponential to polynomial. Furthermore, the gap between the obtained error exponent for constant composition codes and the best known classical random coding exponent vanishes in the order of $o(\log n/n)$, indicating our sphere-packing bound is almost exact in the high rate regime. Finally, for a special class of symmetric classical-quantum channels, we can completely characterize its optimal error probability without the constant composition code assumption. The main technical contributions are two converse Hoeffding bounds for quantum hypothesis testing and the saddle-point properties of error exponent functions.

References in corpus (7)

• The Quantum Chernoff Bound
• The Chernoff lower bound for symmetric quantum hypothesis testing
• On the strong converses for the quantum channel capacity theorems
• Fundamental bound on the reliability of quantum information transmission
• Moderate deviation analysis for classical communication over quantum channels
• The Renyi Capacity and Center
• The Sphere Packing Bound via Augustin's Method

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• Simple and Tighter Derivation of Achievability for Classical Communication over Quantum Channels
• Discrimination of quantum states under locality constraints in the many-copy setting
• Refined Strong Converse for the Constant Composition Codes
• Reliability Function of Classical-Quantum Channels
• An invitation to the sample complexity of quantum hypothesis testing
• Reliability Function of Quantum Information Decoupling via the Sandwiched Rényi Divergence
• Reliable Simulation of Quantum Channels: the Error Exponent
• On the Existence of the Augustin Mean
• Quantum $f$-divergences via Nussbaum-Szkoła Distributions and Applications to $f$-divergence Inequalities
• Strong Converse Exponent for Entanglement-Assisted Communication
• Quantum channel coding: Approximation algorithms and strong converse exponents
• The Mutual Information In The Vicinity of Capacity-Achieving Input Distributions