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What are the Eigenvalues of a Sum of (Non-Commuting) Random Symmetric Matrices? : A "Quantum Information" Inspired Answer.
Alan EdelmanRamis Movassagh
July 14, 2011FOCM Random Matrices
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Example Resultp=1 classical probabilityp=0 isotropic convolution (finite free probability)
We call this “isotropic entanglement”
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Simple Question
The eigenvalues of
where the diagonals are random, and randomly ordered. Too easy?
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Another Question
where Q is orthogonal with Haar measure. (Infinite limit = Free probability)
The eigenvalues of
T
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Quantum Information Question
where Q is somewhat complicated. (This is the general sum of two symmetric matrices)
The eigenvalues of
T
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Preview to the Quantum Information Problem
mxm nxn mxm nxn
Summands commute, eigenvalues addIf A and B are random eigenvalues are classical sum of random variables
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Closer to the true problem
d2xd2 dxd dxd d2xd2
Nothing commutes, eigenvalues non-trivial
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Actual Problem Hardness =(QMA complete)
di-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!
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Intuition on the eigenvectors
Classical Quantum Isostropic
Kronecker Product of Haar Measures for A and B
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Moments?
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Matching Three Moments Theorem
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A first try:Ramis “Quantum Agony”
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The Departure Theorem
A “Pattern Match”
Hardest to Analyze
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The Istropically Entangled Approximation
But this one is hard
The kurtosis
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The convolutions
• Assume A,B diagonal. Symmetrized ordering.
• A+B:
• A+Q’BQ:
• A+Qq’BQq
(“hats” indicate joint density is being used)
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The Slider Theorem
p only depends on the eigenvectors! Not the eigenvalues
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Wishart
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Summary