Smooth Renyi Entropies and the Quantum Information Spectrum

Abstract

Many of the traditional results in information theory, such as the channel coding theorem or the source coding theorem, are restricted to scenarios where the underlying resources are independent and identically distributed (i.i.d.) over a large number of uses. To overcome this limitation, two different techniques, the information spectrum method and the smooth entropy framework, have been developed independently. They are based on new entropy measures, called spectral entropy rates and smooth entropies, respectively, that generalize Shannon entropy (in the classical case) and von Neumann entropy (in the more general quantum case). Here, we show that the two techniques are closely related. More precisely, the spectral entropy rate can be seen as the asymptotic limit of the smooth entropy. Our results apply to the quantum setting and thus include the classical setting as a special case.

References in corpus (8)

• Coding Theorem and Strong Converse for Quantum Channels
• Quantum cryptography with finite resources: unconditional security bound for discrete-variable protocols with one-way post-processing
• An Information-Spectrum Approach to Classical and Quantum Hypothesis Testing for Simple Hypotheses
• General formulas for fixed-length quantum entanglement concentration
• Quantum Coding Theorems for Arbitrary Sources, Channels and Entanglement Resources
• Entanglement Cost for Sequences of Arbitrary Quantum States
• Smooth Rényi Entropy of Ergodic Quantum Information Sources
• Entanglement cost and distillable entanglement of symmetric states

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• Duality Between Smooth Min- and Max-Entropies
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• A Generalization of Quantum Stein's Lemma
• Von Neumann entropy from unitarity
• Single-shot Quantum State Merging
• One-Shot Classical Data Compression with Quantum Side Information and the Distillation of Common Randomness or Secret Keys
• Optimal quantum networks and one-shot entropies
• Smooth Max-Information as One-Shot Generalization for Mutual Information
• Second-order asymptotics for source coding, dense coding and pure-state entanglement conversions
• A smooth entropy approach to quantum hypothesis testing and the classical capacity of quantum channels
• Generalized Entropies
• Min- and Max-Entropy in Infinite Dimensions
• Applications of position-based coding to classical communication over quantum channels
• Relative Thermalization
• Equivalence of Statistical Mechanical Ensembles for Non-Critical Quantum Systems
• Multiparty Quantum Mutual Information: An alternative definition
• Tight Exponential Analysis for Smoothing the Max-Relative Entropy and for Quantum Privacy Amplification
• Coherence measure: Logarithmic coherence number
• The nonequilibrium cost of accurate information processing
• Entanglement Theory and the Quantum Simulation of Many-Body Physics
• Macroscopic Thermodynamic Reversibility in Quantum Many-Body Systems
• Epsilon-smooth measure of coherence
• Quantum reading capacity: General definition and bounds
• A solution of the generalised quantum Stein's lemma
• Towards Quantum Cybernetics
• Sequential, successive, and simultaneous decoders for entanglement-assisted classical communication
• Capacity with energy constraint in coherent state channel
• Smooth Metric Adjusted Skew Information Rates
• Asymptotic Reversibility of Thermal Operations for Interacting Quantum Spin Systems via Generalized Quantum Stein's Lemma
• Non-Asymptotic Analysis of Privacy Amplification via Renyi Entropy and Inf-Spectral Entropy
• Beyond i.i.d. in the Resource Theory of Asymmetry: An Information-Spectrum Approach for Quantum Fisher Information
• Universal coding for transmission of private information
• Rényi and von Neumann entropies of thermal state in Generalized Uncertainty Principle-corrected harmonic oscillator
• Directed quantum communication
• Asymptotic Convertibility of Entanglement: A General Approach to Entanglement Concentration and Dilution
• One-shot Marton inner bound for classical-quantum broadcast channel
• The eight-port homodyne detector: the effect of imperfections on quantum random number generation and on detection of quadratures
• Max- Relative Entropy of Entanglement, alias Log Robustness
• Asymptotic Equipartition Theorems in von Neumann algebras
• One-shot source coding with coded side information available at the decoder
• Quantum Uncertainty and Entropy
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