Quantum Information Theory and Quantum Statistics

Peter Burton

Frank Hansen

The geometrical formulation of quantum statistics and the theory of measures of quantum information are intimately connected by the introduction of the notion of metric adjusted skew information.We survey the area with a focus on the ‘representing operator monotone functions’. In particular, we exhibit a new order structure that renders the set of metric adjusted skew informations into a lattice with a maximal element. The Wigner–Yanase–Dyson skew informations (with parameter p) are increasing with respect to this order structure for 0 < p ≤ 1/2 and decreasing for 1/2 ≤ p < 1 with maximum in the Wigner–Yanase skew information. Introduction The geometrical formulation of quantum statistics and the theory of measures of quantum information are two distinct theories with separate motivations and histories, and for a long time they did not seem to be related in any way. Early contributions which view quantum information in statistical terms were given by Hasegawa and Petz (Hasegawa and Petz 1996), and by Luo (Luo 2003a, Luo 2003b, Luo 2005). Today the two theories are largely fused with the notion of metric (or metrically) adjusted skew information and a common set of tools and techniques involving certain operator monotone functions and their representations. The geometrical approach to statistics reveals its fundamental nature when we try to generalise classical statistics to the quantum setting. The key to obtaining quantisation of the Fisher information is to consider it as a Riemannian metric with a certain behaviour in the presence of noise.

Herbert S. Green

Bouchra El Alaoui, Abdallah Slaoui, R. Ahl Laamara

Information theory, introduced by Claude Shannon in 1948, provides a mathematical model, predominantly probabilistic, for addressing the challenges associated with measuring the amount of information in a message and efficiently and reliably storing the message (compression and protection). This theory has always been intricately connected to communication theory. Information theory not only relies on communication theory but also draws from fields such as statistics, statistical physics, and inference theory. In this chapter, a concise overview is presented, covering the mathematical foundations of classical and quantum information, their manipulation and associated qualifications. The fundamental concepts of classical information theory, including Shannon entropy, joint entropy, conditional entropy, and mutual information, are introduced. Additionally, basic concepts related to quantum information and quantum correlation measurements are presented.

Harish Parthasarathy

This book deals with certain important problems in Classical and Quantum Information Theory Quantum Information Theory, A Selection of Matrix Inequalities Stochastic Filtering Theory Applied to Electromagnetic Fields and Strings Wigner-distributions in Quantum Mechanics Quantization of Classical Field Theories Statistical Signal Processing Quantum Field Theory, Quantum Statistics, Gravity, Stochastic Fields and Information Problems in Information Theory It will be very helpful for students of Undergraduate and Postgraduate Courses in Electronics, Communication and Signal Processing. Print edition not for sale in South Asia (India, Sri Lanka, Nepal, Bangladesh, Pakistan or Bhutan).