QUANTUM f-DIVERGENCES AND ERROR CORRECTION
- 1 August 2011
- journal article
- Published by World Scientific Pub Co Pte Ltd in Reviews in Mathematical Physics
- Vol. 23 (7), 691-747
- https://doi.org/10.1142/s0129055x11004412
Abstract
Quantum f-divergences are a quantum generalization of the classical notion of f-divergences, and are a special case of Petz' quasi-entropies. Many well-known distinguishability measures of quantum states are given by, or derived from, f-divergences. Special examples include the quantum relative entropy, the Rényi relative entropies, and the Chernoff and Hoeffding measures. Here we show that the quantum f-divergences are monotonic under substochastic maps whenever the defining function is operator convex. This extends and unifies all previously known monotonicity results for this class of distinguishability measures. We also analyze the case where the monotonicity inequality holds with equality, and extend Petz' reversibility theorem for a large class of f-divergences and other distinguishability measures. We apply our findings to the problem of quantum error correction, and show that if a stochastic map preserves the pairwise distinguishability on a set of states, as measured by a suitable f-divergence, then its action can be reversed on that set by another stochastic map that can be constructed from the original one in a canonical way. We also provide an integral representation for operator convex functions on the positive half-line, which is the main ingredient in extending previously known results on the monotonicity inequality and the case of equality. We also consider some special cases where the convexity of f is sufficient for the monotonicity, and obtain the inverse Hölder inequality for operators as an application. The presentation is completely self-contained and requires only standard knowledge of matrix analysis.Keywords
Other Versions
This publication has 31 references indexed in Scilit:
- From ƒ-Divergence to Quantum Quasi-Entropies and Their UseEntropy, 2010
- The Chernoff lower bound for symmetric quantum hypothesis testingThe Annals of Statistics, 2009
- A matrix convexity approach to some celebrated quantum inequalitiesProceedings of the National Academy of Sciences, 2009
- Error exponent in asymmetric quantum hypothesis testing and its application to classical-quantum channel codingPhysical Review A, 2007
- Generalized cutoff rates and Renyi's information measuresIEEE Transactions on Information Theory, 1995
- The proper formula for relative entropy and its asymptotics in quantum probabilityCommunications in Mathematical Physics, 1991
- On an inequality of Lieb and ThirringLetters in Mathematical Physics, 1990
- SUFFICIENCY OF CHANNELS OVER VON NEUMANN ALGEBRASThe Quarterly Journal of Mathematics, 1988
- The “transition probability” in the state space of a ∗-algebraReports on Mathematical Physics, 1976
- Über konvexe MatrixfunktionenMathematische Zeitschrift, 1936