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Random Matrix Theory in Quantum Information

Lecturer

Dr Mario Kieburg, University of Melbourne

Synopsis

This course introduces specific concepts that lie on the intersection of contemporary research in Random Matrix Theory (RMT) and Quantum Information (QI). It will aim for a derivation Page’s result of the entanglement entropy of generic pure quantum states in a bipartite system. Despite the narrow goal the course will introduce some major concepts in the two areas of RMT and QI. In this way, it is aimed at a broader audience from mathematics and physics where the students can learn and deepen their knowledge in several techniques. The restriction to this specific goal allows for suitable time to prepare the basics for understanding Page’s result which is nowadays known as the Page curve in QI. It mainly says that a Haar distributed normalised vector in a bipartite Hilbert space is almost surely maximally entangled. Although this statement is true for a general system, the course will illustrate the ideas with spin systems.

Course Overview

From the physics and information theoretical perspective I will revise and introduce the ideas of:

  • Hilbert spaces and inner products
  • tensors and the tensor (Kronecker) product
  • a pure quantum state in a Hilbert space and bipartite systems
  • density matrices and their properties
  • definition of the mean and the covariance
  • entanglement and entanglement entropy

From the mathematical side of RMT I will explain the concepts and techniques:

  • what are random matrices, especially the complex Wishart-Laguerre ensemble
  • the Haar measure of compact Lie groups, particularly of the unitary group, and the uniform measure on spheres
  • the computation of the Jacobian with the help of the Riemannian metric when performing the singular value decomposition of a complex random matrix
  • the method of orthogonal polynomials, particularly the Laguerre polynomials, and proving some main results such as the Andreief identity and the Christoffel-Darboux formula

A tentative schedule is:

  1. Revision of vector spaces, inner products and the definition of Hilbert spaces
  2. Introduction of different representations of matrices such as in block form as a sum of dyadic products
  3. Discussion and proof of the spectral decomposition theorem for matrices
  4. Introducing tensors and the tensor product and deriving some rules for tensors (such as the determinant and the trace)
  5. Introducing the idea of bi-partite systems and the partial trace
  6. Introducing the notions of pure and mixed quantum states and the definition of density matrices
  7. Derivation of some properties of density matrices and the introduction to entanglement
  8. Introducing the idea of entropy and what the entanglement entropy is
  9. Introduction of the volume measure and its relation the Riemannian metric
  10. Computation of the Jacobian in a volume measure via the Riemannian metric
  11. Revision of what a group is and introducing the concept of a Lie group
  12. Discussing group invariant volume measures and the Haar measure for compact groups
  13. Discussion of cosets of groups and the induced Haar measures on those as well as the uniform measure on a sphere
  14. Definition of a random matrix and the introduction to the complex Wishart-Laguerre ensemble
  15. Introducing the concept of the averaged level density and the Green function.
  16. Computation of the Jacobian of the singular value decomposition of a complex random matrix with the help of the Riemannian metric
  17. The idea of k-point correlation functions
  18. Introduction of the concept of orthogonal polynomials, particularly for the Wishart-Laguerre ensemble with the help of Laguerre polynomials
  19. Proving the generalised Andreief identity
  20. Proving the Christoffel-Darboux formula and discussing the idea of determinantal point processes
  21. Revising the ideas of the moments and cumulants (especially mean and variance) of observables.
  22. The fixed trace ensemble for reduced density matrices and its relation to the complex Wigner-Laguerre ensemble
  23. Computation of the mean of the entanglement entropy of reduced density matrices resulting from uniformly distributed pure states (the Page curve)
  24. Computation of the variance of the entanglement entropy of reduced density matrices resulting from uniformly distributed pure states
  25. The log-gas picture
  26. An introduction to functional derivatives
  27. Proving Tricomi’s formula
  28. The concept of the macroscopic level density and computing the Marcenko-Pastur distribution with the help of Tricomi’s formula and the universality of Page’s result

Note, that only the Riemannian metric and its relation to the volume measure is needed from Differential Geometry. Thus, no other concepts from that area will be introduced.

Prerequisites

The essential prerequisites comprise:

  • Complex Analysis (MAST30021, undergraduate)
  • Vector Calculus (MAST20009, undergraduate)
  • either one of the following undergraduate subjects Statistical Physics (PHYC30017)/Probability (MAST20004)/Methods of Mathematical Physics (MAST30031).

Recommended prerequisites are:

  • Metric and Hilbert Spaces (MAST30026, undergraduate)
  • Quantum Physics (PHYC30018, undergraduate)
  • Group Theory and Linear Algebra (MAST20022, undergraduate)

Assessment

  • Final assessment details to be confirmed

Attendance requirements

  • For those completing the subject for their own knowledge/interest, final attendance requirements to be confirmed

Resources/pre-reading

  • TBC

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Dr Mario Kieburg, University of Melbourne

Dr. Mario Kieburg has defended his PhD in Mathematical Physics at the University Duisburg-Essen (Germany) in 2010. He went as Feodor-Lynen Fellow of the Alexander von Humboldt Foundation to Stony Brook University (USA, NY) in 2011-2013. From 2013-2019, he worked as an Honorary Lecturer at Bielefeld University (Germany) where he habilitated in 2015. Since 2019, he is a Lecturer at the School of Mathematics ad Statistics at the University of Melbourne. His research is focused on Random Matrix Theory with applications in Quantum Field Theories, Quantum Information and Time Series Analysis.