### Presentations text content in What are the Eigenvalues of a Sum of

What are the Eigenvalues of a Sum of (Non-Commuting) Random Symmetric Matrices? : A "Quantum Information" Inspired Answer.

Alan Edelman

Ramis

Movassagh

July 14, 2011

FOCM

Random Matrices

Slide2Example Resultp=1 classical probabilityp=0 isotropic convolution (finite free probability)

We call this “isotropic

entanglement”

Slide3Simple Question

The

eigenvalues of

where the diagonals are random, and randomly ordered. Too easy?

Slide4Another Question

where Q is orthogonal with Haar measure. (Infinite limit = Free probability)

The

eigenvalues of

T

Slide5Quantum Information Question

where Q is somewhat complicated. (This is the general sum of two symmetric matrices)

The

eigenvalues of

T

Slide6Preview to the Quantum Information Problem

mxm

nxn

mxm nxn

Summands commute,

eigenvalues

add

If A and B are random

eigenvalues

are

classical sum of random variables

Slide7Closer to the true problem

d

2

xd2 dxd

dxd d2xd2

Nothing commutes,

eigenvalues

non-trivial

Slide8Actual Problem Hardness =(QMA complete)

d

i-1xdi-1 d2xd2 dN-i-1xdN-i-1

The Random matrix could be

Wishart

,

Gaussian Ensemble, etc (Ind Haar Eigenvectors)The big matrix is dNxdN

Interesting Quantum Many Body System Phenomena tied to this overlap!

Slide9Intuition on the eigenvectors

Classical Quantum Isostropic

Kronecker

Product of

Haar

Measures for A and B

Slide10Moments?

Slide11Matching Three Moments Theorem

Slide12A first try:Ramis “Quantum Agony”

Slide13The Departure Theorem

A “Pattern Match”

Hardest to Analyze

Slide14The Istropically Entangled Approximation

But this one is hard

The kurtosis

Slide15The convolutions

Assume A,B diagonal. Symmetrized ordering.A+B:A+Q’BQ:A+Qq’BQq

(“hats” indicate joint density is being used)

Slide16The Slider Theorem

p only depends on the eigenvectors! Not the

eigenvalues

Slide17Wishart

Slide18Bernoulli ±1

Slide19Slide20

Summary

Slide21Slide22

## What are the Eigenvalues of a Sum of

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