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Gap of Ignorance
Philosophy of Isotropic Entanglement
Theory of Isotropic Entanglement
The Method of Isotropic Entanglement
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Isotropic Entanglement
(Density of States of Quantum Spin Systems)
Ramis Movassagh1 and Alan Edelman2
1 Department
of Mathematics, Northeastern University
of Mathematics, M.I.T.
2 Department
Fields Institute, July 2013
Ramis Movassagh
arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of Ignorance
Philosophy of Isotropic Entanglement
Theory of Isotropic Entanglement
The Method of Isotropic Entanglement
More Numerical results
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
Ramis Movassagh
arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of Ignorance
Philosophy of Isotropic Entanglement
Theory of Isotropic Entanglement
The Method of Isotropic Entanglement
More Numerical results
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.
Ramis Movassagh
arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of Ignorance
Philosophy of Isotropic Entanglement
Theory of Isotropic Entanglement
The Method of Isotropic Entanglement
More Numerical results
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, 040501 (2011)
Ramis Movassagh
arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of Ignorance
Philosophy of Isotropic Entanglement
Theory of Isotropic Entanglement
The Method of Isotropic Entanglement
More Numerical results
Sums of non-commuting Hamiltonians
Figure: Qudit Chain
Hk,k+1 :
d2 × d2
Generic local terms =⇒ Quantum Spin Glasses.
N−1
H=
∑ Id
k=1
k−1
⊗ Hk,k+1 ⊗ Id N−k−1 .
Ramis Movassagh
arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of Ignorance
Philosophy of Isotropic Entanglement
Theory of Isotropic Entanglement
The Method of Isotropic Entanglement
More Numerical results
Non-zero elements of H
H = Σ4k=1 Hk,k+1
0
100
200
300
400
500
600
700
800
900
1000
0
100
200
300
400
Ramis Movassagh
500
600
nz = 53248
700
800
900
1000
arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of Ignorance
Philosophy of Isotropic Entanglement
Theory of Isotropic Entanglement
The Method of Isotropic Entanglement
More Numerical results
Why not just add eigenvalues?
Id×d ⊗ Ad×d
Ramis Movassagh
+ Bd×d ⊗ Id×d
arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of Ignorance
Philosophy of Isotropic Entanglement
Theory of Isotropic Entanglement
The Method of Isotropic Entanglement
More Numerical results
Why not just add eigenvalues?
Id×d ⊗ Ad×d
Id×d ⊗ Ad 2 ×d 2
Ramis Movassagh
+ Bd×d ⊗ Id×d
+ Bd 2 ×d 2 ⊗ Id×d
arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of Ignorance
Philosophy of Isotropic Entanglement
Theory of Isotropic Entanglement
The Method of Isotropic Entanglement
More Numerical results
Interactions: H = ∑N−1
I
⊗
H
⊗
I
= Hodd + Heven
k,k+1
k=1
Eigenvectors of odds: QA = Q1 ⊗ Q3 ⊗ · · · ⊗ QN−2 ⊗ Id×d
Qk : d 2 × d 2 matrix of eigenvectors of Hk,k+1
Ramis Movassagh
arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of Ignorance
Philosophy of Isotropic Entanglement
Theory of Isotropic Entanglement
The Method of Isotropic Entanglement
More Numerical results
Interactions: H = ∑N−1
I
⊗
H
⊗
I
= Hodd + Heven
k,k+1
k=1
Eigenvectors of odds: QA = Q1 ⊗ Q3 ⊗ · · · ⊗ QN−2 ⊗ Id×d
Eigenvectors of evens: QB = Id×d ⊗ Q2 ⊗ Q4 ⊗ · · · ⊗ QN−1
Qk : d 2 × d 2 matrix of eigenvectors of Hk,k+1
Ramis Movassagh
arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of Ignorance
Philosophy of Isotropic Entanglement
Theory of Isotropic Entanglement
The Method of Isotropic Entanglement
More Numerical results
H = Hodd + Heven = QA AQA−1 + QB BQB−1
Change bases such that Hodd is diagonal. Therefore,
H = A + Qq−1 BQq
Qq ≡ (QB )−1 QA
Ramis Movassagh
∼N
random parameters
arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of Ignorance
Philosophy of Isotropic Entanglement
Theory of Isotropic Entanglement
The Method of Isotropic Entanglement
More Numerical results
Classical sum
Ramis Movassagh
arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of Ignorance
Philosophy of Isotropic Entanglement
Theory of Isotropic Entanglement
The Method of Isotropic Entanglement
More Numerical results
Isotropic (Free) sum
Ramis Movassagh
arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of Ignorance
Philosophy of Isotropic Entanglement
Theory of Isotropic Entanglement
The Method of Isotropic Entanglement
More Numerical results
Isotropic, Quantum, and Classical
Ramis Movassagh
arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of Ignorance
Philosophy of Isotropic Entanglement
Theory of Isotropic Entanglement
The Method of Isotropic Entanglement
More Numerical results
Quantum as a “sliding” sum of classical and iso
Ramis Movassagh
arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of Ignorance
Philosophy of Isotropic Entanglement
Theory of Isotropic Entanglement
The Method of Isotropic Entanglement
More Numerical results
Local terms: Wishart matrices
N=5, d=2, r=4, trials=500000
0.05
Isotropic Approximation
0.04
Density
Isotropic Entanglement (IE)
0.03
0.02
Classical Approximation
0.01
0
0
10
20
Ramis Movassagh
30
Eigenvalue
40
50
60
arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of Ignorance
Philosophy of Isotropic Entanglement
Theory of Isotropic Entanglement
The Method of Isotropic Entanglement
More Numerical results
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.
Ramis Movassagh
arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of Ignorance
Philosophy of Isotropic Entanglement
Theory of Isotropic Entanglement
The Method of Isotropic Entanglement
More Numerical results
The Departure Theorem
The Departure Theorem
n h
io
2
Tr A4 +4A3 Q −1 BQ+4A2 Q −1 B 2 Q+4AQ −1 B 3 Q+2(AQ−1 BQ) +B 4
h
io
2
4
m4q = 1N E Tr A4 +4A3 Qq−1 BQq +4A2 Qq−1 B 2 Qq +4AQq−1 B 3 Qq +2(AQ−1
q BQq ) +B
m4iso =
d
1
E
d Nn
m4c =
1
dN
E{Tr[A4 +4A3 B+4A2 B 2 +4AB 3 +2(AB)2 +B 4 ]}
Ramis Movassagh
arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of Ignorance
Philosophy of Isotropic Entanglement
Theory of Isotropic Entanglement
The Method of Isotropic Entanglement
More Numerical results
Quantum agony
Ramis Movassagh
arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of Ignorance
Philosophy of Isotropic Entanglement
Theory of Isotropic Entanglement
The Method of Isotropic Entanglement
More Numerical results
Resolving the agony
Lemma: Only these matter
1
m ETr
=
[(H (3) ⊗Id N−2 )(I⊗H (4) ⊗Id N−3 )(H (3) ⊗Id N−2 )(I⊗H (4) ⊗Id N−3 )]
1
d3
n o
(3)
(3)
(4)
(4)
E Hi3 i4 ,j3 j4 Hi3 p4 ,j3 k4 E Hj4 i5 ,k4 k5 Hi4 i5 ,p4 k5 ,
Ramis Movassagh
arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of Ignorance
Philosophy of Isotropic Entanglement
Theory of Isotropic Entanglement
The Method of Isotropic Entanglement
More Numerical results
Quantum as a convex combination of classical and iso
Use fourth moments to form a hybrid theory
γ2q = pγ2c + (1 − p) γ2iso
(•)
γ2
is found from the fourth moments
Ramis Movassagh
arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of Ignorance
Philosophy of Isotropic Entanglement
Theory of Isotropic Entanglement
The Method of Isotropic Entanglement
More Numerical results
The Slider Theorem
Theorem
(The Slider Theorem) The quantum kurtosis lies in between the
classical and the iso kurtoses, γ2iso ≤ γ2q ≤ γ2c . Therefore there exists
a 0 ≤ p ≤ 1 such that γ2q = pγ2c + (1 − p) γ2iso . Further,
limN→∞ p = 1.
Ramis Movassagh
arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of Ignorance
Philosophy of Isotropic Entanglement
Theory of Isotropic Entanglement
The Method of Isotropic Entanglement
More Numerical results
Universality of p
Corollary
(Universality) p 7→ p (N, d , β ), namely, it is independent of the
distribution of the local terms.
p Vs. N
1
0.8
d=2
d=3
d=4
d=∞
p
0.6
0.4
0.2
0
1
2
10
10
N
Here β = 1. Therefore, p only depends on eigenvectors!
Ramis Movassagh
arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of Ignorance
Philosophy of Isotropic Entanglement
Theory of Isotropic Entanglement
The Method of Isotropic Entanglement
More Numerical results
Local terms: Wishart matrices
N=11, d=2, r=4, trials=2800
0.03
p=0
p=1
Exact Diagonalization
I.E. Theory: p = 0.73
0.025
Density
0.02
0.015
0.01
0.005
0
0
10
20
30
40
Ramis Movassagh
50
60
Eigenvalue
70
80
90
100
110
arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of Ignorance
Philosophy of Isotropic Entanglement
Theory of Isotropic Entanglement
The Method of Isotropic Entanglement
More Numerical results
Suppose you have the first four moments
4−moment matching methods vs. IE: binomial local terms
2.5
Exact Diagonalization
I.E. Theory: p = 0.046
Pearson
Gram−Charlier
Density
2
1.5
1
0.5
0
−3
−2
−1
Ramis Movassagh
0
Eigenvalue
1
2
3
arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of Ignorance
Philosophy of Isotropic Entanglement
Theory of Isotropic Entanglement
The Method of Isotropic Entanglement
More Numerical results
Suppose you have the first four moments
4−moment matching methods vs. IE: Wishart local terms
0.05
Exact Diagonalization
I.E. Theory: p = 0.42817
Pearson
Gram−Charlier
Density
0.04
0.03
0.02
0.01
0
0
10
20
Ramis Movassagh
30
40
Eigenvalue
50
60
70
arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of Ignorance
Philosophy of Isotropic Entanglement
Theory of Isotropic Entanglement
The Method of Isotropic Entanglement
More Numerical results
Summary: Method of Isotropic Entanglement
H 1,2
H N-1,N
H N-2,N-1
H 2,3
B
}
}
A
+
+
add random variables ≡
Classical convolution of densities
add random variables ≡
Classical convolution of densities
}
+
Several options for adding
random variables
Isotropic (or free) convolution ≡
dνIso = dνA ⊞IsodνB
Isotropic
p=0
Isotropically Entangled ≡
(1-p) dνIso + p dνC
p
Quantum convolution ≡ dνq = dνA ⊞qdνB ≈
Classical convolution ≡
dνC = dνA ∗ dνB
Classical
p=1
Isotropically Entangled
Ramis Movassagh
arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of Ignorance
Philosophy of Isotropic Entanglement
Theory of Isotropic Entanglement
The Method of Isotropic Entanglement
More Numerical results
Others got excited about it too!
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 Ingenierı́a Informática, Escuela Politécnica Superior,
Universidad Autónoma de Madrid, Ciudad Universitaria de Cantoblanco,
Calle Francisco Tomás y Valiente, 11 E-28049 Madrid, Spain
d-mat.dis-nn] 27 Feb 2012
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: 71.23.An, 71.23.-k
Disordered materials have long been of interest for their
unique physics such as localization [1, 2], anomalous diffusion [3, 4] and ergodicity breaking [5]. Their properties
have been exploited for applications as diverse as quantum
dots [6, 7], magnetic nanostructures [8], disordered metals [9, 10], and bulk heterojunction photovoltaics [11–13].
Despite this, theoretical studies are complicated by the need
to calculate the electronic structureRamis
of the respective
systems
Movassagh
two PDFs w (x ) and w̃ (x ) with finite cumulants k , k , . . .
1 2
Phys. Rev. Lett. 109,
(2012)
and k̃1036403
, k̃2 , . . . , and moments
µ1 , µ2 , . . . and µ̃1 , µ̃2 , . . . respec-
tively, we can define a formal differential operator which
transforms w̃ into w and is given by [18, 20]
"
✓
◆ #
•
kn k̃n
d n
w̃ (x ) .
(1)
w (x ) = exp Â
n!
dx
n =1
arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of Ignorance
Philosophy of Isotropic Entanglement
Theory of Isotropic Entanglement
The Method of Isotropic Entanglement
More Numerical results
Question for you: Why do we do so well?
2
1
ETr A5 +5A4 Q•T BQ• +5 A3 Q•T B 2 Q• +5 A2 Q•T B 3 Q• +5A(AQT
m5 = m
• BQ• ) +
2
T 4
5
T
5(AQT
• BQ• ) Q• BQ• +5 AQ• B Q• +B
Ramis Movassagh
(1)
arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of Ignorance
Philosophy of Isotropic Entanglement
Theory of Isotropic Entanglement
The Method of Isotropic Entanglement
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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 Qq A≥1 Qq−1 B ≥1 Qq . . . ≤
ETr . . . B ≥1 A≥1 B ≥1 . . .
.
e.g. ETr
n
AQ −1 BQ
k o
≤ ETr
K >2
Ramis Movassagh
n
AQq−1 BQq
k o
o
n
≤ ETr (AB)k
arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of Ignorance
Philosophy of Isotropic Entanglement
Theory of Isotropic Entanglement
The Method of Isotropic Entanglement
More Numerical results
Lastly...
ThankQ
Ramis Movassagh
arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)