Finite-Length Information Theory for Quantum Systems
This course introduces finite-length information theory for quantum systems from an operational perspective. Binary hypothesis testing provides the guiding framework: many of these tasks can be reduced to, bounded by, or understood through the problem of distinguishing two statistical models. Starting with one-shot formulations and finite-length bounds, we will connect entropies, divergences, and coding rates to concrete performance limits, and examine how the familiar asymptotic laws emerge as the available resources grow.
Students will acquire an understanding of:
Quantum states as statistical models, their distinguishability, and the role of quantum channels in information processing.
Classical and quantum binary hypothesis testing, including one-shot bounds and asymptotic error exponents.
The operational meaning of entropies and divergences, and the data-processing principle.
Fundamental limits of classical and quantum source compression and classical-quantum channel coding with finite resources.
Program
Unit 1. Quantum systems as statistical models: states and distinguishability measures.
Unit 2. Hypothesis testing: one-shot classical and quantum binary hypothesis testing, exact formulations, and finite-length bounds.
Unit 3. Quantum channels and information processing: physical transformations and data processing for distinguishability measures.
Unit 4. Quantum divergences: Rényi entropies and divergences, their quantum extensions, and structural tools behind data processing.
Unit 5. Asymptotic hypothesis testing: many-copy state discrimination and the operational meaning of relative entropy and Rényi divergences through error exponents.
Unit 6. Source compression: classical and quantum compression from a one-shot and asymptotic hypothesis-testing perspective.
Unit 7. Classical-quantum channel coding: finite-blocklength and asymptotic limits for transmitting classical information through quantum states.
Course material
Class notes with detailed derivations, available through Aula Global.
Problem sets on quantum states and hypothesis testing, channels and divergences, source compression, and classical-quantum channel coding.