Representation-theoretic methods in quantum information theory

Course Overview

In this course we study symmetries in quantum information theory using tools from representation theory. Two fundamental symmetry groups in quantum information are the symmetric group, acting by permuting subsystems in a tensor product of identical Hilbert spaces, and the unitary group, acting diagonally on a tensor product space. Schur-Weyl duality establishes a close relationship between these two representations, giving rise to a useful decomposition of the representation space into irreducible representations. This structure result allows us to succinctly describe invariant objects and characterize optimal information-processing protocols in the presence of permutation and unitary symmetries.

The first half of the course starts with a brief review of the basics of quantum information theory and representation theory. We then discuss the representation theory of the symmetric and unitary groups and how they relate to each other via Schur-Weyl duality. These findings can be applied to characterize symmetric quantum states such as Werner and isotropic states.

In the second half of the course we apply these representation-theoretic methods to various topics and tasks in quantum information theory, including the quantum theory of types, de Finetti theorems, approximate cloning, spectrum estimation, and universal quantum source compression.

Table of Contents

• Basics of quantum information theory (review)
• Basics of representation theory (review)
• Representation theory of the symmetric and unitary groups (review)
• Schur-Weyl duality
• Werner states, isotropic states, covariant quantum channels
• Permutation invariance and de Finetti theorems
• Quantum theory of types
• Spectrum estimation
• Approximate cloning
• Universal source compression

Course material

Lecture notes

Literature

• Judith M. Alcock-Zeilinger, The Special Unitary Group, Birdtracks, and Applications in QCD, Lecture notes, 2018.
• Matthias Christandl, The structure of bipartite quantum states – insights from group theory and cryptography, PhD thesis, University of Cambridge, 2006.
• Matthias Christandl, Graeme Mitchison, The Spectra of Density Operators and the Kronecker Coefficients of the Symmetric Group, Communications in Mathematical Physics 261.3, pp. 789–797 (2006).
• Imre Csiszár, János Körner, Information theory: coding theorems for discrete memoryless systems, New York, New York: Academic Press, 1981.
• Jeongwan Haah et al., Sample-optimal tomography of quantum states, IEEE Transactions on Information Theory 63.9, pp. 5628–5641 (2017).
• Aram W. Harrow, Applications of coherent classical communication and the Schur transform to quantum information theory, PhD thesis, Massachusetts Institute of Technology, 2005.
• Aram W. Harrow, The church of the symmetric subspace, arXiv preprint, 2013.
• Masahito Hayashi, Exponents of quantum fixed-length pure state source coding, Physical Review A 66, 032321 (2002).
• M. Keyl, R. Werner, Estimating the spectrum of a density operator, Physical Review A 64, 052311 (2001).
• Anthony W. Knapp, Representation Theory of Semisimple Groups: An Overview Based on Examples, Princeton, NJ: Princeton University Press, 2016.
• Jean-Pierre Serre, Linear Representations of Finite Groups, Graduate Texts in Mathematics, New York: Springer, 1977.
• Richard Stanley, Enumerative Combinatorics. Volume 2, Cambridge: Cambridge University Press, 1999.
• Constantin Teleman, Representation Theory, Lecture notes, 2005.
• Michael Walter, Symmetry and Quantum Information, Lecture notes, 2018.
• John Watrous, The Theory of Quantum Information, Cambridge: Cambridge University Press, 2018.
• Reinhard Werner, Optimal Cloning of Pure States, Physical Review A 58, 1827 (1998).
• Mark M. Wilde, Quantum information theory, 2nd edition, Cambridge: Cambridge University Press, 2016.