Random Matrix Techniques in Quantum Information - 2010

We introduce a class of probability spaces whose objects are infinite graphs and whose probability distributions are obtained as limits of distributions for finite graphs. The notions of Hausdorff and spectral dimension for such ensembles are defined and some results on their value in koncrete examples, such as random trees, will be described.

Understanding NP-complete problems is a central topic in computer science. This is why adiabatic quantum optimization has attracted so much attention, as it provided a new approach to tackle NP-complete problems using a quantum computer. The efficiency of this approach is limited by small spectral gaps between the ground and excited states of the quantum computer's Hamiltonian. We show that the statistics of the gaps can be analyzed in a novel way, borrowed from the study of quantum disordered systems in statistical mechanics. It turns out that due to a phenomenon similar to Anderson localization, exponentially small gaps appear close to the end of the adiabatic algorithm for large random instances of NP-complete problems. We show that this effect makes adiabatic quantum optimization fail, as the system gets trapped in one of the numerous local minima. We will also discuss recent developments including the effect of the exponential number of solutions and Hamiltonian path change. Joint work with Boris Altshuler and Hari Krovi Based on arXiv:0908.2782 and arXiv:0912.0746

It is a fundamental, if elementary, observation that to obliterate the quantum information in n qubits by random unitaries, an amount of randomness of at least 2n bits is required. If the randomisation condition is relaxed to perform only approximately, we obtain two answers, depending on the norm used to compare the ideal and the approximation. Using the ''naive'' norm brings down the cost to n bits, while under the more appropriate complete norm it is still essentially 2n. After reviewing these facts and some constructions, we go on to explore the quantum information theoretical uses of the two notions of erasure. Most prominently, for a given quantum channel and its complementary channel, complete erasure is dual to correctability of the quantum noise; while approximate erasure is dual to the decodability of another task of quantum information, dubbed ''quantum identification''

In this talk we will give an overview of how different probabilistic and quantum probabilistic techniques can be used to find Bell inequalities with large violation. This will include previous result on violation for tripartite systems and more recent results with Palazuelos on probabilities for bipartite systems. Quite surprisingly the latest results are the most elementary, but lead to some rather surprsing independence of entropy and large violation.

Entangled (i.e., not separable) quantum states play fundamental roles in quantum information theory; therefore, it is important to know the ''size'' of entanglement (and hence separability) for various measures, such as, Hilbert-Schmidt measure, Bures measure, induced measure, and $\alpha$-measure. In this talk, I will present new comparison results of $\alpha$-measure with Bures measure and Hilbert-Schmidt measure. Employing these comparison results to the subsets of separable states and of states with positive partial transpose, we show that the probability of separability is very small, and the well-known Peres-Horodecki PPT Criterion as a tool to detect separability is imprecise for (even moderate) large dimension of Hilbert space. This talk is based on my papers: J. Math. Phys. 50 (2009) 083502, and J. Phys. A: Math. Theor. in press.

One of the major problems hindering progress in quantum many body systems is the inability to describe the spectrum of the Hamiltonian. The spectrum corresponds to the energy spectrum of the problem and is of out-most importance in accounting for the physical properties of the system. A perceived difficulty is the exponential growth of the Hamiltonian with the number of particles involved. Therefore, even for a modest number of particles, direct computation appears intractable. This work offers a new method, using free probability and random matrix theory, of approximating the spectrum of generic frustrated Hamiltonians of arbitrary size with local interactions. In addition, we show a number of numerical experiments that demonstrate the accuracy of this method.

A famous result in classical probability - Hin\v{c}in's Theorem - establishes a bijection between infinitely divisible probability distributions and limits of infinitesimal triangular arrays of independent random variables. Analogues of this result have been proved by Bercovici and Pata for scalar-valued {\em free probability}. However, very little is known for the case of operator-valued distributions, when the field of scalars is replaced by a $C^*$-algebra; essentially the only result known in full generality that we are aware of is Voiculescu's operator-valued central limit theorem. In this talk we will use a recent breakthrough in the description of infinite divisibility of operator-valued distributions achieved by Popa and Vinnikov to prove a Hin\v{c}in-type theorem for operator-valued free random variables and to formulate a free - to - conditionally free Bercovici-Pata bijection. Time permitting, we will discuss in more detail relaations between the operator-valued free, Boolean and monotone central limits. This is joint work with Mihai V. Popa and Victor Vinnikov.

Limit laws and large deviations for the empirical measure of the singular values for ensembles of non-Hermitian matrices can be obtained based on explicit distributions for the eigenvalues. When considering the eigenvalues, however, the situation changes dramatically, and explicit expressions for the joint distribution of eigenvalues are not available (except in very special cases). Nevertheless, in some situations the limit of the empirical measure of eigenvalues (as a measure supported in the complex plane) can be computed, and it exhibits interesting features. I will describe some results along these line, part of a joint work with Alice Guionnet and Manjunath Krishnapur.

We associate to any unoriented graph a random pure quantum state, obtained by randomly rotating a tensor product of Bell states. Marginals of such states define new ensembles of density matrices, which we study in the asymptotical regime of large Hilbert spaces. Limit eigenvalue distributions are computed, as well as average von Neumann entropies and purities. Fuss-Catalan distributions are identified as limits of the eigenvalue distributions of particular marginals. Finally, we discuss area laws for these random states. This is joint work with Benoit Collins and Karol Zyczkowski.