Since 2002 Perimeter Institute has been recording seminars, conference talks, and public outreach events using video cameras installed in our lecture theatres. Perimeter now has 7 formal presentation spaces for its many scientific conferences, seminars, workshops and educational outreach activities, all with advanced audio-visual technical capabilities. Recordings of events in these areas are all available On-Demand from this Video Library and on Perimeter Institute Recorded Seminar Archive (PIRSA). PIRSA is a permanent, free, searchable, and citable archive of recorded seminars from relevant bodies in physics. This resource has been partially modelled after Cornell University's arXiv.org.
In this talk, I describe two cases in which questions in quantum information theory have lead me to random matrices.
In the first case, analyzing a protocol for quantum cryptography lead us to the following question: what is the largest eigenvalue of a sum of p random product states in (C^d)^{otimes k}, where k and p/d^k are fixed while d grows?
We associate to any unoriented graph a random pure quantum state, obtained by randomly rotating a tensor product of Bell states.
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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.
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}.
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.
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.
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.
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.
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