Fields Institute Talk
description
Transcript of Fields Institute Talk
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Fields Institute Talk
• Note first half of talk consists of blackboard– see video:
http://www.fields.utoronto.ca/video-archive/2013/07/215-1962
– then I did a matlab demot=1000000; i=sqrt(-1);figure(1);hold offfor p=10.^[-3:.2:3] % Florent's two coin tosses a=pi+angle(-1/p+randn(t,1)+i*randn(t,1)); r=2*cos(a/4); % Draw the symmetrized density [x,y]=hist([-r r],linspace(-2,2,99)); bar(y,x/sum(x)/(y(2)-y(1))); title(['p= ' num2str(p)]); pause(0.1) end
– and finally these slides show up around 34 minutes in
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Example Resultp=1 classical probabilityp=0 isotropic convolution (finite free probability)
We call this “isotropic entanglement”
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Complicated Roadmap
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Complicated Roadmap
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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
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 Isotropic
Intertwined Kronecker Product of Haar Measures
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Example Resultp=1 classical convolutionp=0 isotropic convolution
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First three moments match theorem
• It is well known that the first three free cumulants match the first three classical cumulants
• Hence the first three moments for classical and free match
• The quantum information problem enjoys the same matching!
• Three curves have the same mean, the same variance, the same skewness!
• Different kurtoses (4th cumulant/var2+3)
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Fitting the fourth moment
• Simple idea• Worked better than we expected• Underlying mathematics guarantees more
than you would expect– Better approximation– Guarantee of a convex combination between
classical and iso
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Illustration
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Roadmap
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The Problem
Let H=
di-1xdi-1 d2xd2 dN-i-1xdN-i-1
Compute or approximate
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di-1 d2 dN-i-1
The Problem
Let H=
The Random matrix has known joint eigenvalue density & independent eigenvectors distributed with β-Haar measure .
β=1 random orthogonal matrixβ=2 random unitary matrixβ=4 random symplectic matrixGeneral β: formal ghost matrix
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Easy Step
H=
= (odd terms i=1,3,…) + (even terms i=2,4,…)
Eigenvalues of odd (even) terms add= Classical convolution of probability densities(Technical note: joint densities needed to preserve all the information)
Eigenvectors “fill” the proper slots
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Complicated Roadmap
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Eigenvectors of odd (even)
(A) Odd(B) Even
Quantify how we are in between Q=I and the full Haar measure
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The same mean and variance as Haar
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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 Istropically Entangled Approximation
But this one is hard
The kurtosis
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A first try:Ramis “Quantum Agony”
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The Entanglement
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The Slider Theorem
p only depends on the eigenvectors! Not the eigenvalues
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More pretty pictures
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p vs. Nlarge N: central limit theorem
large d, small N: free or isowhole 1 parameter family in between
The real world? Falls on a 1 parameter family
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Wishart
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Wishart
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Wishart
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Bernoulli ±1
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Roadmap