Download Quantum entropy and its use

Quantum entropy and its use
IMPRS/TopMath Frühlingsschule 2017
Abtei Frauenwörth, Chiemsee
March 29–31, 2017
List of topics
Robert König and Simone Warzel
[email protected] [email protected]
Abstract
Entropy is a central quantity in information theory, probability and physics. This spring school
will focus on fundamental concepts and basic operational interpretations of entropy with a particular
focus on applications to quantum mechanics. The goal is provide a thorough overview ranging from
fundamental concepts to present-day research problems in the field.
A first introductory part of the spring school is devoted to the concept of entropy in classical
information theory. We will study data processing, channels and their capacities and Shannon’s
channel coding theorem. Historically, entropy was introduced by Boltzmann in the context of classical
statistical mechanics. The mathematical key to understanding the equivalence of statistical ensembles
are the laws of large numbers and large deviation principles. We will study these concepts on the
basis of two examples: the asymptotic equipartition property in information theory and Cramer’s
large deviation principle with an application to statistical mechanics.
The second, major part of the spring school focuses on basic concepts in quantum information
theory. After a short review of fundamental notions of quantum mechanics, we will focus on information transmission over quantum channels. We discuss major results in quantum (Shannon) theory:
the Holevo-Schumacher-Westmoreland theorem which concerns the classical capacity of a quantum
channel, as well as the Lloyd-Shor-Devetak theorem for the quantum capacity. We also consider the
remarkable phenomenon of (super)activation of capacities, a purely quantum effect.
In the last part of the school, we will discuss the role of entropy in many-body physics, including,
in particular, area laws for mutual information, and statements about the approximability by tensor
network states.
Format: The format of the school is as follows. Each participant gives a talk about a preselected
topic and there will be plenty of room for discussions and questions concerning background and open
questions. Talks will be prepared in advance. We will be available to assist with the preparation.
Prerequisites: A keen interest in the emergence of quantum effects in operational settings.
Some familiarity with basic notions of quantum mechanics is desirable.
Registration: by email to [email protected]. Spots are limited and will be assigned on
a first-come-first-served basis.
1
Talks 1 and 2: Information measures and asymptotic equipartition. Entropy, conditional
entropy, relative entropy, mutual information, data processing, Fano’s inequality. Asympotic
equipartition property, data compression, typical sets. Joint typicality, joint asymptotic equipartition property. [1, Chapter 2], [1, Chapter 3 and Section 8.6]
Talk 3: Channel capacity. Channels as doubly stochastic maps, examples. Motivation & Definition of capacity. Shannon’s channel coding theorem: achievability and converse. [1, Chapter 8]
Talk 4: Entropy in statistical mechanics. Gibbs-Boltzmann entropy, Example: identical localized particles, equivalence of ensembles, Cramer’s large deviation theorem. [2, Chapters 20
and 22]
Talk 5: Fundamentals of quantum information. Quantum states, density operators, measurements (positive operator valued measures), Born’s rule, classical probability theory as a special
case. Tensor product (composite systems). Binary hypothesis testing and variational (trace)
distance: Helstrom’s result, fidelity, purification, Uhlmann’s theorem, entanglement fidelity. [3,
Lectures 1,2,4], [4, Section 3.1.2, 3.1.3, 4.1.1, 4.1.2]
Talk 6: Quantum channels. Completely positive trace-preserving maps. Kraus- and Stinespring
representation. Examples: unitary conjugation, convex combinations of unitaries, depolarizing/dephasing/amplitude damping channel, partial trace, adjoining an ancilla. Choi-Jamiolkowski
isomorphism. One-parameter semigroups and generators. [5, Lecture 15], [3, Lecture 8], [4,
Section 5.1.1, 5.3.1, 5.3.2].
Talks 7: Data processing and entropy inequalities. von Neumann entropy, Klein’s inequality
(concavity), monotonicity under unital maps, additivity for tensor product states, subadditivity
of entropy as data processing, strong subadditivity, proof via Lieb’s triple matrix extension of
the Golden-Thompson inequality [6, Section 5.4].
Talk 8: Classical capacity of a quantum channel. Accessible information (Holevo-χ-quantity),
Holevo bound. Pretty good measurements, Holevo-Schumacher-Westmoreland theorem [5, Lecture 15], [7]. Non-additivity of Holevo-quantity [8].
Talk 9 & 10: Quantum capacity of a quantum channel. Coherent information. Decoupling,
Lloyd-Shor-Devetak-Theorem. [5, Lecture 15], [7]. Zero-capacity channels: PPT and antidegradable channels, superactivation [9, 10].
Talk 11 & 12: Area laws. Tensor network states, Schmidt rank, entanglement entropy, matrix
product states [11]. Axiomatic derivation of Gibbs states [12], area law for mutual information [13]
References
[1] T. Cover and J. Thomas, Elements of Information Theory.
Wiley, 2005, available via OPAC.
[2] T. C. Dorlas, Statistical Mechanics: Fundamentals and Model Solutions.
IOP, 1999.
[3] M. Wolf, “Quantum Effects, Lecture Notes WS 2013/14,” 2013, available here.
[4] T. Heinosaari and M. Ziman, “Guide to Mathematical Concepts of Quantum Theory,” Acta
Physica Solvaca, vol. 58, pp. 487–674, 2008, available here.
[5] D. Reeb, “From Classical to Quantum Information Theory, Lecture Notes, SS 2016,” University
of Hannover, available here.
[6] D. Petz, Quantum Information Theory and Quantum Statistics.
Springer, 2008.
[7] J. Preskill, “Quantum Shannon Theory,” 2016, available here.
[8] M. B. Hastings, “Superadditivity of communication capacity using entangled inputs,” Nature
Physics, vol. 5, pp. 255–257, 2009.
[9] G. Smith and J. Yard, “Quantum communication with zero-capacity channels,” Science,
vol. 321, no. 5897, pp. 1812–1815, 2008. [Online]. Available: http://science.sciencemag.org/
content/321/5897/1812
[10] G. Smith and J. A. Smolin, “Detecting incapacity of a quantum channel,” Phys. Rev. Lett., vol.
108, p. 230507, Jun 2012. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevLett.
108.230507
[11] N. Schuch, M. M. Wolf, K. G. H. Vollbrecht, and J. I. Cirac, “On entropy growth and the
hardness of simulating time evolution,” New Journal of Physics, vol. 10, no. 3, p. 033032, 2008.
[Online]. Available: http://stacks.iop.org/1367-2630/10/i=3/a=033032
[12] V. Gorini, A. Frigerio, and M. Verri, “Quantum gibbs states and the zeroth law of thermodynamics,” Lecture Notes in Mathematics, January 1985.
2
[13] M. M. Wolf, F. Verstraete, M. B. Hastings, and J. I. Cirac, “Area laws in quantum systems:
Mutual information and correlations,” Phys. Rev. Lett., vol. 100, p. 070502, Feb 2008. [Online].
Available: http://link.aps.org/doi/10.1103/PhysRevLett.100.070502
3