A smooth entropy approach to quantum hypothesis testing and the classical capacity of quantum channels

Abstract

We use the smooth entropy approach to treat the problems of binary quantum hypothesis testing and the transmission of classical information through a quantum channel. We provide lower and upper bounds on the optimal type II error of quantum hypothesis testing in terms of the smooth max-relative entropy of the two states representing the two hypotheses. Using then a relative entropy version of the Quantum Asymptotic Equipartition Property (QAEP), we can recover the strong converse rate of the i.i.d. hypothesis testing problem in the asymptotics. On the other hand, combining Stein's lemma with our bounds, we obtain a stronger ($\ep$-independent) version of the relative entropy-QAEP. Similarly, we provide bounds on the one-shot $\ep$-error classical capacity of a quantum channel in terms of a smooth max-relative entropy variant of its Holevo capacity. Using these bounds and the $\ep$-independent version of the relative entropy-QAEP, we can recover both the Holevo-Schumacher-Westmoreland theorem about the optimal direct rate of a memoryless quantum channel with product state encoding, as well as its strong converse counterpart.

v4: Title changed, improved bounds, both direct and strong converse rates are covered, a new Discussion section added. 20 pages

References in corpus (14)

• The Quantum Chernoff Bound
• Coding Theorem and Strong Converse for Quantum Channels
• One-Shot Classical-Quantum Capacity and Hypothesis Testing
• A Hierarchy of Information Quantities for Finite Block Length Analysis of Quantum Tasks
• The Chernoff lower bound for symmetric quantum hypothesis testing
• Error Exponent in Asymmetric Quantum Hypothesis Testing and Its Application to Classical-Quantum Channel coding
• Second-order asymptotics for quantum hypothesis testing
• Finite blocklength converse bounds for quantum channels
• Noisy channel coding via privacy amplification and information reconciliation
• Smooth Renyi Entropies and the Quantum Information Spectrum
• The Converse Part of The Theorem for Quantum Hoeffding Bound
• Coding Theorem for a Class of Quantum Channels with Long-Term Memory
• Generalized relative entropies and the capacity of classical-quantum channels
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• Optimal quantum networks and one-shot entropies
• Second-Order Asymptotics for the Classical Capacity of Image-Additive Quantum Channels
• Resource theory of asymmetric distinguishability
• Second-order asymptotics for source coding, dense coding and pure-state entanglement conversions
• Quantum Achievability Proof via Collision Relative Entropy
• On Composite Quantum Hypothesis Testing
• Decoding quantum information via the Petz recovery map
• Non-Asymptotic Classical Data Compression with Quantum Side Information
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• Communication, Dynamical Resource Theory, and Thermodynamics
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• The variance of relative surprisal as single-shot quantifier
• One-shot quantum error correction of classical and quantum information
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• Overcoming entropic limitations on asymptotic state transformations through probabilistic protocols
• Complexity-constrained quantum thermodynamics
• Quantum Network Discrimination
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• Adversarial quantum channel discrimination
• Asymptotic Equipartition Theorems in von Neumann algebras
• Fast suppression of classification error in variational quantum circuits
• One-shot and asymptotic classical capacity in general physical theories
• Tight relations and equivalences between smooth relative entropies
• The entropy of quantum causal networks