Alan Edelman Ramis Movassagh July 14, 2011 FOCM Random Matrices

Slides:

Advertisements

Similar presentations

Solving Hard Problems With Light Scott Aaronson (Assoc. Prof., EECS) Joint work with Alex Arkhipov vs.

Advertisements

Quantum Computing MAS 725 Hartmut Klauck NTU

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Matrix – Basic Definitions Chapter 3 Systems of Differential.

Prof. Ramin Zabih (CS) Prof. Ashish Raj (Radiology) CS5540: Computational Techniques for Analyzing Clinical Data.

GRADIENT ALGORITHMS for COMMON LYAPUNOV FUNCTIONS Daniel Liberzon Univ. of Illinois at Urbana-Champaign, U.S.A. Roberto Tempo IEIIT-CNR, Politecnico di.

1.5 Elementary Matrices and a Method for Finding

Slides by Olga Sorkine, Tel Aviv University. 2 The plan today Singular Value Decomposition  Basic intuition  Formal definition  Applications.

Eigenvalues and eigenvectors

Random Matrices Hieu D. Nguyen Rowan University Rowan Math Seminar

CS Pattern Recognition Review of Prerequisites in Math and Statistics Prepared by Li Yang Based on Appendix chapters of Pattern Recognition, 4.

Lecture 18 Eigenvalue Problems II Shang-Hua Teng.

Visual Recognition Tutorial1 Random variables, distributions, and probability density functions Discrete Random Variables Continuous Random Variables.

4. The Postulates of Quantum Mechanics 4A. Revisiting Representations

1 Quantum NP Dorit Aharonov & Tomer Naveh Presented by Alex Rapaport.

The Multivariate Normal Distribution, Part 2 BMTRY 726 1/14/2014.

Modern Navigation Thomas Herring

Review of Probability.

Linear Algebra With Applications by Otto Bretscher. Page The Determinant of any diagonal nxn matrix is the product of its diagonal entries. True.

: Appendix A: Mathematical Foundations 1 Montri Karnjanadecha ac.th/~montri Principles of.

The Erdös-Rényi models

Random Matrix Theory Numerical Computation and Remarkable Applications Alan Edelman Mathematics Computer Science & AI Labs Computer Science & AI Laboratories.

Fields Institute Talk Note first half of talk consists of blackboard – see video: – then I.

Physics 3 for Electrical Engineering

Therorem 1: Under what conditions a given matrix is diagonalizable ??? Jordan Block REMARK: Not all nxn matrices are diagonalizable A similar to (close.

Mathematical Preliminaries. 37 Matrix Theory Vectors nth element of vector u : u(n) Matrix mth row and nth column of A : a(m,n) column vector.

What are the Eigenvalues of a Sum of (Non-Commuting) Random Symmetric Matrices? : A "Quantum Information" inspired Answer. Alan Edelman Ramis Movassagh.

Ch 2. Probability Distributions (1/2) Pattern Recognition and Machine Learning, C. M. Bishop, Summarized by Yung-Kyun Noh and Joo-kyung Kim Biointelligence.

N– variate Gaussian. Some important characteristics: 1)The pdf of n jointly Gaussian R.V.’s is completely described by means, variances and covariances.

Class 27: Question 1 TRUE or FALSE: If P is a projection matrix of the form P=A(A T A) -1 A T then P is a symmetric matrix. 1. TRUE 2. FALSE.

Linear algebra: matrix Eigen-value Problems Eng. Hassan S. Migdadi Part 1.

Lyapunov exponents of products of free operators Vladislav Kargin (CIMS)

What is the determinant of What is the determinant of

Direct Methods for Sparse Linear Systems Lecture 4 Alessandra Nardi Thanks to Prof. Jacob White, Suvranu De, Deepak Ramaswamy, Michal Rewienski, and Karen.

Review of Probability. Important Topics 1 Random Variables and Probability Distributions 2 Expected Values, Mean, and Variance 3 Two Random Variables.

The Random Matrix Technique of Ghosts and Shadows Alan Edelman Dept of Mathematics Computer Science and AI Laboratories Massachusetts Institute of Technology.

EXAMPLE 1 Add and subtract matrices

Progress in the method of Ghosts and Shadows for Beta Ensembles Alan Edelman (MIT) Alex Dubbs (MIT) and Plamen Koev (SJS) Aug 10, 2012 IMS Singapore Workshop.

Random Processes Gaussian and Gauss-Markov processes Power spectrum of random processes and white processes.

Spectral and Wavefunction Statistics (I) V.E.Kravtsov, Abdus Salam ICTP.

Similar diagonalization of real symmetric matrix

10/22/2014PHY 711 Fall Lecture 241 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 24: Rotational.

Linear Algebra Chapter 6 Linear Algebra with Applications -Gareth Williams Br. Joel Baumeyer, F.S.C.

Ch 2. Probability Distributions (1/2) Pattern Recognition and Machine Learning, C. M. Bishop, Summarized by Joo-kyung Kim Biointelligence Laboratory,

Learning Theory Reza Shadmehr Distribution of the ML estimates of model parameters Signal dependent noise models.

Density matrix and its application. Density matrix An alternative of state-vector (ket) representation for a certain set of state-vectors appearing with.

Characteristic Polynomial Hung-yi Lee. Outline Last lecture: Given eigenvalues, we know how to find eigenvectors or eigenspaces Check eigenvalues This.

Tutorial 6. Eigenvalues & Eigenvectors Reminder: Eigenvectors A vector x invariant up to a scaling by λ to a multiplication by matrix A is called.

Linear Algebra With Applications by Otto Bretscher.

Sampling and Sampling Distributions

Standard Errors Beside reporting a value of a point estimate we should consider some indication of its precision. For this we usually quote standard error.

Twirling states to simplify separability

Jordan Block Under what conditions a given matrix is diagonalizable ??? Therorem 1: REMARK: Not all nxn matrices are diagonalizable A similar to.

EIGEN … THINGS (values, vectors, spaces … )

PHY 711 Classical Mechanics and Mathematical Methods

Review of Probability Theory

Systems of First Order Linear Equations

4. The Postulates of Quantum Mechanics 4A. Revisiting Representations

Postulates of Quantum Mechanics

Derivative of scalar forms

Summarizing Data by Statistics

Ilan Ben-Bassat Omri Weinstein

Linear Algebra Lecture 3.

ENM 500 Linear Algebra Review

Lecture 13: Singular Value Decomposition (SVD)

The Multivariate Normal Distribution, Part I

MAT 2401 Linear Algebra 7.2 Diagonalization

Physics 319 Classical Mechanics

Linear Algebra: Matrix Eigenvalue Problems – Part 2

Marios Mattheakis and Pavlos Protopapas

Presentation transcript:

Alan Edelman Ramis Movassagh July 14, 2011 FOCM Random Matrices 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

Example Result p=1  classical probability p=0 isotropic convolution (finite free probability) We call this “isotropic entanglement”

Simple Question The eigenvalues of where the diagonals are random, and randomly ordered. Too easy?

Another Question The eigenvalues of where Q is orthogonal with Haar measure. (Infinite limit = Free probability)

Quantum Information Question The eigenvalues of T where Q is somewhat complicated. (This is the general sum of two symmetric matrices)

Preview 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

Closer to the true problem d2xd2 dxd dxd d2xd2 Nothing commutes, eigenvalues non-trivial

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!

Intuition on the eigenvectors Classical Quantum Isostropic Kronecker Product of Haar Measures for A and B

Moments?

Matching Three Moments Theorem

A first try: Ramis “Quantum Agony”

The Departure Theorem A “Pattern Match” Hardest to Analyze

The Istropically Entangled Approximation The kurtosis But this one is hard

The convolutions Assume A,B diagonal. Symmetrized ordering. A+B: A+Q’BQ: A+Qq’BQq (“hats” indicate joint density is being used)

The Slider Theorem p only depends on the eigenvectors! Not the eigenvalues

Wishart

Summary