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The purpose of this review article is to present some of the latest developments using random techniques, and in particular, random matrix techniques in quantum information theory. Our review is a blend of a rather exhaustive review, combined with more detailed examples -- coming from research projects in which the authors were involved. We focus on two main topics, random quantum states and random quantum channels. We present results related to entropic quantities, entanglement of typical states, entanglement thresholds, the output set of quantum channels, and violations of the minimum output entropy of random channels.
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We compute critical properties of a general class of quantum spin chains which are quadratic in the Fermi operators and can be solved exactly under certain symmetry constraints related to the classical compact groups $U(N)$, $O(N)$ and $Sp(2N)$. In particular we calculate critical exponents $s$, $ u$ and $z$, corresponding to the energy gap, correlation length and dynamic exponent respectively. We also compute the ground state correlators $leftlangle sigma^{x}_{i} sigma^{x}_{i+n} rightrangle_{g}$, $leftlangle sigma^{y}_{i} sigma^{y}_{i+n} rightrangle_{g}$ and $leftlangle prod^{n}_{i=1} sigma^{z}_{i} rightrangle_{g}$, all of which display quasi-long-range order with a critical exponent dependent upon system parameters. Our approach establishes universality of the exponents for the class of systems in question.
In this work we build a theoretical framework for the transport of information in quantum systems. This is a framework aimed at describing how out of equilibrium open quantum systems move information around their state space, using an approach inspired by transport theories. The main goal is to build new mathematical tools, together with physical intuition, to improve our understanding of non-equilibrium phenomena in quantum systems. In particular, we are aiming at unraveling the interplay between dynamical properties and information-theoretic features. The main rationale here is to have a framework that can imitate, and potentially replicate, the decades-long history of success of transport theories in modeling non-equilibrium phenomena.
This article begins with a brief review of random matrix theory, followed by a discussion of how the large-$N$ limit of random matrix models can be realized using operator algebras. I then explain the notion of Brown measure, which play the role of the eigenvalue distribution for operators in an operator algebra. I then show how methods of partial differential equations can be used to compute Brown measures. I consider in detail the case of the circular law and then discuss more briefly the case of the free multiplicative Brownian motion, which was worked out recently by the author with Driver and Kemp.
We prove that for any infinite-dimensional quantum channel the entropic disturbance (defined as difference between the $chi$-quantity of a generalized ensemble and that of the image of the ensemble under the channel) is lower semicontinuous on the natural set of its definition. We establish a number of useful corollaries of this property, in particular, we prove the continuity of the output $chitextrm{-}$quantity and the existence of $chi$-optimal ensemble for any quantum channel under the energy-type input constraint.
We show that the Davies generator associated to any 2D Kitaevs quantum double model has a non-vanishing spectral gap in the thermodynamic limit. This validates rigorously the extended belief that those models are useless as self-correcting quantum memories, even in the non-abelian case. The proof uses recent ideas and results regarding the characterization of the spectral gap for parent Hamiltonians associated to Projected Entangled Pair States in terms of a bulk-boundary correspondence.
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