Quantum channels

Course Overview

The first part of the course is an introduction to the theory of quantum channels in the finite-dimensional setting of quantum information theory. We discuss the various mathematically equivalent representations of quantum channels, focus on some important subclasses of channels, and make various connections to information-theoretic aspects of quantum information theory.

In the second part of the course we turn to the theory of quantum Markov chains. We first discuss the quantum relative entropy and its fundamental property, the data-processing inequality, and give a proof of this inequality that naturally leads to equality conditions and the concept of recovery channels. Specializing this analysis to the partial trace, we obtain the strong subadditivity property of the von Neumann entropy, as well as a natural definition of quantum Markov chains. We then review a structure theorem for quantum Markov chains, the fundamental differences to classical Markov chains, and—time permitting—discuss the notion of approximate quantum Markov chains.

Table of Contents

• Basics of finite-dimensional quantum information theory:
  Quantum systems and quantum states, Measurements, Composite systems and entanglement, Distance measures
• Representations of quantum channels:
  Isometric picture, Unitary evolution, Kraus representation, Linear operator representation
• Classes of quantum channels:
  Mixed-unitary and unital channels, Entanglement-breaking channels, Symmetric and antidegradable channels, PPT channels
• Unital channels and majorization:
  Majorization for real vectors, Majorization for Hermitian operators, Schur-Horn Theorem
• Covariant quantum channels:
  Definition, properties, examples, Holevo information and minimum output entropy
• Quantum relative entropy:
  Definition and operational interpretation, Joint convexity and data-processing inequality
• Equality conditions for data-processing:
  Petz's proof of the equality condition, Formulation in terms of recovery channels
• Quantum Markov chains: Strong subadditivity of von Neumann entropy, Quantum conditional mutual information and its operational interpretations, Structure theorem for exact quantum Markov chains
• Approximate quantum Markov chains:
  Classical vs. quantum setting, Approximate recovery channels

Course material

Lecture notes

Literature

• Stephane Attal: Quantum channels, Lecture notes.
• Ben Ibinson et al.: Robustness of quantum Markov chains, Communications in Mathematical Physics 277.2 (2008), pp. 289–304.
• Michael A. Nielsen and Denes Petz: A Simple Proof of the Strong Subadditivity Inequality, Quantum Information & Computation 5.6 (2005), pp. 507–513.
• Denes Petz: Quantum Information Theory and Quantum Statistics. Springer, 2008.
• Denes Petz: Monotonicity of quantum relative entropy revisited, Reviews in Mathematical Physics 15.01 (2003), pp. 79–91.
• Mary Beth Ruskai: Inequalities for quantum entropy: A review with conditions for equality, Journal of Mathematical Physics 43.9 (2002), pp. 4358–4375.
• David Sutter: Approximate quantum Markov chains, Vol. 28. SpringerBriefs in Mathematical Physics, 2018.
• John Watrous: Theory of Quantum Information, Cambridge University Press, 2018.
• Mark M. Wilde: Quantum Information Theory, Cambridge University Press, 2017.
• Michael M. Wolf: Quantum Channels & Operations: Guided Tour, Lecture notes, 2012.