# Perimeter Institute Quantum Discussions

This series consists of weekly discussion sessions on foundations of quantum Theory and quantum information theory. The sessions start with an informal exposition of an interesting topic, research result or important question in the field. Everyone is strongly encouraged to participate with questions and comments.

## Seminar Series Events/Videos

Currently there are no upcoming talks in this series.

## The entropy power inequality for quantum systems

Wednesday May 16, 2012
Speaker(s):

When two independent analog signals, $X$ and $Y$ are added together giving $Z=X+Y$, the entropy of $Z$, $H(Z)$, is not a simple function of the entropies $H(X)$ and $H(Y)$, but rather depends on the details of $X$ and $Y$'s distributions.  Nevertheless, the entropy power inequality (EPI), which states that $e^{2H(Z)} \geq e^{2H(X)} + e^{2H(Y)}$, gives a very tight restriction on the entropy of $Z$.

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## Quantum Reed-Muller Codes and Magic State Distillation in All Prime Dimensions

Wednesday May 02, 2012
Speaker(s):

Joint work with Earl Campbell (FU-Berlin) and Hussain Anwar (UCL)   Magic state distillation is a key component of some high-threshold schemes for fault-tolerant quantum computation [1], [2]. Proposed by Bravyi and Kitaev [3] (and implicitly by Knill [4]), and improved by Reichardt [4], Magic State Distillation is a method to broaden the vocabulary of a fault-tolerant computational model, from a limited set of gates (e.g.

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## Universal Low-rank Matrix Recovery from Pauli Measurements

Wednesday Apr 04, 2012
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We study the problem of reconstructing an unknown matrix M, of rank r and dimension d, using O(rd poly log d) Pauli measurements. This has applications to compressed sensing methods for quantum state tomography.  We give a solution to this problem based on the restricted isometry property (RIP), which improves on previous results using dual certificates. In particular, we show that almost all sets of O(rd log^6 d) Pauli measurements satisfy the rank-r RIP.

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## Negative Quasi-Probability Representation is a Necessary Resource for Quantum Computation

Wednesday Mar 21, 2012
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Abstract The magic state model of quantum computation gives a recipe for universal quantum computation using perfect Clifford operations and repeat preparations of a noisy ancilla state. It is an open problem to determine which ancilla states enable universal quantum computation in this model. Here we show that for systems of odd dimension a necessary condition for a state to enable universal quantum computation is that it have negative representation in a particular quasi-probability representation which is a discrete analogue to the Wigner function.

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## A Quantum Information Approach to Statistical Physics

Wednesday Jan 25, 2012

I will first present a theorem based on the Decoupling Theorem of [1] which gives sufficient and necessary conditions for a quantum channel (CPTPM) being such that it yields the same output for almost all possible inputs. This theorem allows us to reproduce and generalize results oft [2,3], in which cornerstones of statistical physics are derived from first principles of quantum mechanics, in a very natural and easy way. Specifically, we express them in a way which allows to apply results about random 2-qubit interactions [4].

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## Topological Liquid Nucleation Induced by Vortex-vortex Interactions in Kitaev's Honeycomb Model

Wednesday Oct 26, 2011
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We provide a microscopic understanding of the nucleation of topological quantum liquids that arise due to interactions between non-Abelian anyons. With the pairwise anyon interactions typically showing RKKY-type oscillations in sign, but decaying exponentially with distance, we show that the character of the nucleated phase is fully determined by anyon interactions beyond nearest neighbor exchange. We investigate this issue in the context of Kitaev's honeycomb lattice model.

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## Crossing Tsirelson's Bound with Super Non-localStates

Wednesday Oct 12, 2011
Speaker(s):

We construct a class of entangled supersymmetric states which is used as a non-local resource in the CHSH game. This class of super entangled states is more non-local then maximally entangled states if the supersymmetric degrees of freedom are accessible to measurement.
Consequently, we show that the winning probability for the CHSH game is greater than cos2(pi/8) corresponding to an expected value greater than Tsirelson's bound.

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## The Curious Nonexistence of Gaussian 2-designs

Monday Jul 18, 2011
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Continuous-variable SICPOVMS seem unlikely to exist, for a variety of reasons. But that doesn't rule out the possibility of other 2-designs for the continuous-variable Hilbert space L2(R). In particular, it would be nice if the coherent states -- which form a rather nice 1-design -- could be generalized in some way to get a 2-design comprising *Gaussian* states. So the question is: "Can we build a 2-design out of Gaussian states?". The answer is "No, but in a very surprising way!" Like coherent states, Gaussian states have a natural transitive symmetry group.

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## Under what conditions quantum systems thermalize?

Wednesday May 18, 2011
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This talk presents sufficient conditions for equilibration and thermalization of subsystems within closed many body quantum systems. That is, we identify when the local properties of the equilibrium state resemble those of a thermal state. With this aim, the recent progress in this field is reviewed and we introduce a novel perturbation technique for a realistic weak coupling between the subsystem and its environment. Unlike the standard perturbation theory, our technique is robust in the thermodynamic limit.

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## Beyond bosons and fermions: how to detect and use anyons

Wednesday Apr 27, 2011
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One of the key features of the quantum Hall effect (QHE) is the fractional charge and statistics of quasiparticles. Fractionally charged anyons accumulate non-trivial phases when they encircle each other. In some QHE systems an unusual type of particles, called non-Abelian anyons, is expected to exist. When one non-Abelian particle makes a circle around another anyon this changes not only the phase but even the direction of the quantum-state vector in the Hilbert space. This property makes non-Abelian anyons promising for fault-tolerant quantum computation.

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## LECTURES ON-DEMAND

### Gabriela González, Louisiana State University

Speaker: Gabriela Gonzalez