Isotropic Entanglement

Transcription

1 Isotropic Entanglement (Density of States of Quantum Spin Systems) Ramis Movassagh 1 and Alan Edelman 2 1 Department of Mathematics, Northeastern University 2 Department of Mathematics, M.I.T. Fields Institute, July 2013

2 The eigenvalue distribution: Motivation Synonymous: (Energy) Spectrum; eigenvalue distribution, density of states, level densities etc. First step for all eigenvalue problems (e.g. quantum mechanics) of sums of matrices Physical: Partition function and therefore the thermodynamics of QMBS

3 The eigenvalue distribution: Motivation Synonymous: (Energy) Spectrum; eigenvalue distribution, density of states, level densities etc. First step for all eigenvalue problems (e.g. quantum mechanics) of sums of matrices Physical: Partition function and therefore the thermodynamics of QMBS Goal: Given the geometry, local spin states, and type of local interaction, capture the spectrum of the H.

4 Complexity issues Note: Generally the Spectrum of QMBS is hard to find exactly (QMA-complete): F.G.S.L. Brandao s Thesis (2008). B. Brown, S. T. Flammia, N. Schuch (2010), Computational Difficulty of Computing the Density of States. Phys. Rev. Lett. 107, (2011)

5 Sums of non-commuting Hamiltonians Figure: Qudit Chain H k,k+1 : d 2 d 2 Generic local terms = Quantum Spin Glasses. N 1 H = k=1 I d k 1 H k,k+1 I d N k 1.

6 Gap of Ignorance Non-zero elements of H H = Σ4k=1 Hk,k Ramis Movassagh nz = arxiv: ; Phys. Rev. Lett. 107, (2011)

7 Why not just add eigenvalues? I d d A d d + B d d I d d

8 Why not just add eigenvalues? I d d A d d + B d d I d d I d d A d 2 d 2 + B d 2 d 2 I d d

9 Interactions: H = N 1 k=1 ( I Hk,k+1 I ) = H odd + H even Eigenvectors of odds: Q A = Q 1 Q 3 Q N 2 I d d Q k : d 2 d 2 matrix of eigenvectors of H k,k+1

10 Interactions: H = N 1 k=1 ( I Hk,k+1 I ) = H odd + H even Eigenvectors of odds: Q A = Q 1 Q 3 Q N 2 I d d Eigenvectors of evens: Q B = I d d Q 2 Q 4 Q N 1 Q k : d 2 d 2 matrix of eigenvectors of H k,k+1

11 H = H odd + H even = Q A AQA 1 + Q BBQB 1 Change bases such that H odd is diagonal. Therefore, H = A + Qq 1 BQ q Q q (Q B ) 1 Q A N random parameters

12 Classical sum

13 Isotropic (Free) sum

14 Isotropic, Quantum, and Classical

15 Quantum as a sliding sum of classical and iso

16 Local terms: Wishart matrices N=5, d=2, r=4, trials= Isotropic Approximation 0.04 Density Classical Approximation Isotropic Entanglement (IE) Eigenvalue

17 The action starts at the fourth moment Theorem (The Matching Three Moments Theorem) The first three moments of the quantum, iso and classical sums are equal.

18 The Departure Theorem The Departure Theorem { m4 iso= 1 d N { E [ m q 4 = 1 d N E Tr Tr [A 4 +4A 3 Q 1 BQ+4A 2 Q 1 B 2 Q+4AQ 1 B 3 Q+2(AQ 1 BQ) 2 +B 4]} A 4 +4A 3 Q 1 q BQ q +4A 2 Qq 1 B 2 Q q +4AQq 1 B 3 Q q +2(AQ 1 q BQ q) 2 +B 4]} m c 4 = 1 d N E{Tr[A4 +4A 3 B+4A 2 B 2 +4AB 3 +2(AB) 2 +B 4 ]}

19 Quantum agony

20 Resolving the agony Lemma: Only these matter 1 m ETr[(H(3) I d N 2)(I H (4) I d N 3)(H (3) I d N 2)(I H (4) I d N 3)] ( ( = {E 1 H (3) d 3 i 3 i 4,j 3 j 4 H (3) i 3 p 4,j 3 k 4 )E H (4) j 4 i 5,k 4 k 5 H (4) i 4 i 5,p 4 k 5 )},

21 Quantum as a convex combination of classical and iso Use fourth moments to form a hybrid theory γ q 2 = pγc 2 + (1 p)γ iso 2 γ ( ) 2 is found from the fourth moments

22 The Slider Theorem Theorem (The Slider Theorem) The quantum kurtosis lies in between the classical and the iso kurtoses, γ2 iso γ q 2 γc 2. Therefore there exists a 0 p 1 such that γ q 2 = pγc 2 + (1 p)γiso 2. Further, lim N p = 1.

23 Universality of p Corollary (Universality) p p (N,d,β), namely, it is independent of the distribution of the local terms. 1 p Vs. N 0.8 p d=2 d=3 d=4 d= N Here β = 1. Therefore, p only depends on eigenvectors!

24 Local terms: Wishart matrices N=11, d=2, r=4, trials= p = 0 p = 1 Exact Diagonalization I.E. Theory: p = 0.73 Density Eigenvalue

25 Suppose you have the first four moments Density moment matching methods vs. IE: binomial local terms Exact Diagonalization I.E. Theory: p = Pearson Gram Charlier Eigenvalue

26 Suppose you have the first four moments moment matching methods vs. IE: Wishart local terms Density Exact Diagonalization I.E. Theory: p = Pearson Gram Charlier Eigenvalue

27 Summary: Method of Isotropic Entanglement H 1,2 H N-1,N H 2,3 H N-2,N-1 A } B + + add random variables Classical convolution of densities } add random variables Classical convolution of densities } + Several options for adding random variables Isotropic (or free) convolution dν Iso = dν A Iso dν B Isotropic p = 0 Isotropically Entangled (1-p) dν Iso + p dν C p Quantum convolution dν q = dν A q dν B Isotropically Entangled Classical convolution dν C = dν A dν B Classical p = 1

28 Others got excited about it too! d-mat.dis-nn] 27 Feb 2012 Error analysis of free probability approximations to the density of states of disordered systems Jiahao Chen, Eric Hontz, Jeremy Moix, Matthew Welborn, and Troy Van Voorhis Department of Chemistry, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA Alberto Suárez Departamento de Ingeniería Informática, Escuela Politécnica Superior, Universidad Autónoma de Madrid, Ciudad Universitaria de Cantoblanco, Calle Francisco Tomás y Valiente, 11 E Madrid, Spain Ramis Movassagh and Alan Edelman Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA Theoretical studies of localization, anomalous diffusion and ergodicity breaking require solving the electronic structure of disordered systems. We use free probability to approximate the ensembleaveraged density of states without exact diagonalization. We present an error analysis that quantifies the accuracy using a generalized moment expansion, allowing us to distinguish between different approximations. We identify an approximation that is accurate to the eighth moment across all noise strengths, and contrast this with the perturbation theory and isotropic entanglement theory. PACS numbers: An, k Disordered materials have long been of interest for their two PDFs w (x) and w (x) with finite cumulants k1, k2,... unique physics such as localization Phys. [1, Rev. 2], anomalous Lett. dif-109fusion and k1, k2,..., and moments (2012) µ1, µ2,... and µ1, µ2,... respectively, [3, 4] and ergodicity breaking [5]. Their properties have been exploited for applications as diverse as quantum we can define a formal differential operator which transforms w into w and is given by [18, 20] dots [6, 7], magnetic nanostructures [8], disordered metals [9, 10], and bulk heterojunction photovoltaics [11 13]. Â # " kn kn d n w (x) = exp w (x). (1) Despite this, theoretical studies are complicated by the need n! dx n=1 to calculate the electronic structureramis of the respective Movassagh systems arxiv: ; Phys. Rev. Lett. 107, (2011)

29 Question for you: Why do we do so well? m 5 = 1 m ETr (A 5 +5A 4 Q T BQ +5A 3 Q T B 2 Q +5A 2 Q T B 3 Q +5A(AQ T BQ ) 2 + 5(AQ T BQ ) 2 Q T BQ +5AQ T B 4 Q +B 5 ) (1)

30 Question for you: Why do we do so well? ETr {...Q 1 B 1 QA 1 Q 1 B 1 Q... } ETr {...Qq 1 B 1 Q q A 1 Qq 1 B 1 Q q... } ETr {...B 1 A 1 B 1... }. (AQ e.g. ETr{ 1 BQ ) } { k (AQ ) } 1 k ETr q BQ q ETr {(AB) k} K > 2

31 Lastly... ThankQ