spectrum of reduced density matrix: bethe ansatz vs valence...

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Spectrum of Reduced Density Matrix: BetheAnsatz vs Valence Bond Solid states

Vladimir Korepin

C.N. Yang Institute for Theoretical Physics

⊲⊳

Vladimir Korepin Operator Structures in Quantum Information

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Reduced Density Matrix

Was introduced by Dirac in 1930.Proceedings of Cambridge Philosophical Society, volume 26.Matrix elements of reduced density matrix arecorrelation functions.Miwas and Jimbo used reduced density matrix in the book.It is also useful in quantum information and entanglement.Brief introduction.


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Entropy of a subsystem of a pure system

In classical case the following simple theorem is valid:If the entropy of a system is zero, then there is no entropy inany subsystem:

�

In quantum case the entropy of a subsystem of a pure systemcan be positive. This means that the subsystem is entangledwith the rest of the system. It is an important resource forquantum control.


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Example

Two spins 1/2: |ΨE,B〉 = 1√2

(| ↑〉E | ↓〉B − | ↓〉E | ↑〉B)

Density matrix of one spin is proportional to identical matrix

ρB = trE

(

|ΨE,B〉〈ΨE,B |)

=12

I2

SE,B = 0 but SB = ln(2) > 0

Cannot happen for classical subsystems.This is a way to distinguish quantum system form classical .


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Quantum Control

Measurement of subsystem E can be organized to project onone of basis states |ψE

j0〉. The wave function will turn into

|ΨE,BM 〉 = |ψE

j0 〉⊗

|ψBj0 〉

with probability |a2j0|.

This changes the state of another subsystem B.E and B entangled: one can control one system by another.This is quantum control useful for building quantum devices.


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Application to XX Model

HXX = −∞∑

n=−∞σx

nσxn+1 + σy

nσyn+1 + hσz

n , |h| ≤ 2

Solved by Lieb , Schultz & Mattis 1961

The |gs〉 is unique

Entanglement of a block of L sequential spins with the restof the ground state [environment].


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Entanglement

|gs〉 = |B ∪ E〉density matrix of the ground state state

ρ = |gs〉〈gs|

Density matrix of the block is the reduced density matrix.

ρB = TrEρ

Individual entries of ρB are correlation functions.


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Density matrix and correlations

In each lattice site j we have four ’Pauli’ matrices σaj , here

a = 0, x , y , z with σ0 = I.

ρB = 2−L∑

ajj=1,...,L

L⊗

j=1

σaj

j

〈gs|L∏

j=1

σaj

j |gs〉

For gap-full models limit of large block exists.


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Quantum Entanglement

Measures of entanglement for a pure state

Von Neumann Entropy of subsystem [block] :

S(ρB) = −TrB (ρB ln ρB)

Rényi Entropy of subsystem [block]:

S(ρB, α) =ln(

TrBραB

)

1 − α

α is a parameter.Also known in literature as zeta function or replica trick.


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Calculation of Entropy

Method of calcualtions of for correlation functions is based onDeterminant Representation and Riemann-Hilbert ProblemFredholm integral operators of a special form: integrableintegral operators. Developed by A.R. Its, A.G. Izergin, V.E.Korepin, N.M. Bogoliubov Quantum Inverse Scattering Methodand Correlation Functions.Cambridge University Press, Cambridge 1993Same method work for evaluation of the entropy


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Von Neumann Entropy

Asymptotic L → ∞ :

S(ρB) → 13

ln L +16

ln

[

1 −(

h2

)2]

+13

ln 2 + Υ1 + O(

1L

)

(ln L ) term was discovered Holzhey, Larsen, Wilczek 1994Sub-leading terms: B.-Q. Jin, V.E. Korepin in 2004

Υ1 = −∫

∞

0dt

{

e−t

3t+

1

t sinh2(t/2)− cosh(t/2)

2 sinh3(t/2)

}

≈ 0.495


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Rényi Entropy

Asymptotic of Rényi entropy is

S(ρB , α) →(

1 + α−1

6

)

ln

2L

√

1 −(

h2

)2

+ Υ{α}

B.-Q. Jin, V.E. Korepin 2003, see arXiv:quant-ph/0304108Later reproduced by J. Cardy and P. Calabrese from conformalfiled theory, arXiv:hep-th/0405152


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XY model

Gap-full models

HXY = −+∞∑

n=−∞(1 + γ)σx

nσxn+1 + (1 − γ)σy

nσyn+1 + hσz

n

Lieb , Schultz & Mattis 1961

Phases1a. Moderate field: 2

√

1 − γ2 < h < 21b. Weak field: 0 ≤ h < 2

√

1 − γ2

2. Strong field: h > 2


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Phase diagram

h

γ

0

1

2

Case 1B

Case 1A

Case 2


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Von Neumann Entropy

Entropy of a large block has a limit:

S(ρB) =12

∫

∞

1ln

(

θ3(

β(λ) + τ2

)

θ3(

β(λ) − τ2

)

θ23

(

στ2

)

)

dλ

β(λ) =1

2πilnλ+ 1λ− 1

, τ = iI(k ′)I(k)

, k =

√

1 − (h/2)2 − γ2

1 − (h/2)2

θ3 the Jacobi theta-function; (k ′)2 + k2 = 1

Entropy is constant on the ellipsis: k = const

Complete elliptic integral of the first kind I(k) =∫ 1

0dx√

(1−x2)(1−x2k2)


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references

First analytical expression for limiting entropy of the large blockof spin in the ground state of XY spin chain on infinite latteceIts, Jin, Korepin: September 3 of 2004, seehttp://arxiv.org/abs/quant-ph/0409027

Shortly after on 16 Oct 2004 Ingo Peschel simplified ourexpression , see http://arxiv.org/abs/cond-mat/0410416


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Rényi Entropy

Limiting entropy in the region 0 < h < 2 is:

S(ρB , α) = 16

α1−α

ln(

k ′

k2

)

+ 13

11−α

ln(

θ2(0,qα)θ4(0,qα)

θ23(0,qα)

)

+ 13 ln 2

τ = iI(k ′)I(k)

≡ iτ0, q = eπiτ

F. Franchini, A. R. Its, B.-Q. Jin, V. E. Korepin 2006


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Spectrum of the density matrix of very large block

pm are eigenvalues of the limiting density matrix.They form exact geometric sequence:

pm = p0e−πτ0m, m = 0, 1, 2, . . .∞, τ0 =I(k ′)

I(k)> 0

Here m is a label of eigenvalue. The maximum eigenvalue

p0 = (kk ′/4)1/6 exp

[ π

12τ0

]

is unique

Degeneracy increases as eigenvalue diminishes:

Degeneracy → (192)−1/4

(m)−3/4 eπ

√m/3, m → ∞

A. R. Its, F.Franchini, V.E.Korepin and L.A. Takhtajan 2009


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Affleck, Kennedy, Lieb and Tasaki

Spin chain consists of N spin-1’s in the bulk and two spin-1/2on the boundary: ~Sk is spin-1 and ~sb is spin-1/2.

H =

N−1∑

k=1

(

~Sk~Sk+1 +

13

(~Sk~Sk+1)

2 +23

)

+ π0,1 + πN,N+1.

12~Sk~Sk+1 + 1

6(~Sk~Sk+1)

2 + 13 is a projector on a state of spin 2.

The π is a projector on a state with spin 3/2:

π0,1 =23

(

1 + ~s0~S1

)

, πN,N+1 =23

(

1 + ~sN+1~SN

)

.


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Ground state is VBS

Kirillov-Korepin set up 1989. Ground state is unique:

s s s s s s s s s s s s s s��

��

��

��

��

��

a dot is spin-12 ; circle means symmetrisation [makes spin 1]. A

line is a anti-symmetrisation (| ↑↓〉 − | ↓↑〉) . Each projectorannihilates the ground state (no frustration). The ground stateis called VBS. Exact formula for correlation function at anydistance is:

34< ~Sx

~S1 >=

(

−13

)x

= p(x)


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Entropy of entanglement of VBS

AKLT again. Fan, Korepin and Roychowdhury 2004 calculatedexactly the entropy of the block of the size x on a finite lattice:

S(x) = −3(1−p(x))4 log

(

1−p(x)4

)

−

−1+3p(x)4 log

(

1+3p(x)4

)

It does not depend on the size of the lattice.For large block

S(x) → ln 4 as x → ∞


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Eigenvalues of the density matrix

Fan, Korepin and Roychowdhury 2004 PRL:the density matrix of finite block of x spins on the finite latticehas only 4 non-zero eigenvalues. One

p0 =1 + 3p(x)

4and three

p1,2,3 =1 − p(x)

4

Does not depend on the length of the lattice.


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Eigenvectors of the Density matrix

Consider ’block’ Hamiltonian

HB =

(k+1)∈block∑

k∈block

(

~Sk~Sk+1 +

13(~Sk

~Sk+1)2 +

23

)

.

It describes interaction of spins 1 inside of the block. Theground state is quadruple degenerated: these are eigenvectorsof the density matrix corresponding non-zero eigenvalues .

ρB = limβ→∞

(

e−βHB

Tre−βHB

)


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Generalizations

Together with Hosho Katsura and Ying Xu these results for VBSwe generalized to:

For higher spins s the entropy S = 2 ln(s + 1)

Other Lie algebras: for SU(n) for smallest non-trivialrepresentation the entropy S = 2 ln n

For AKLT on a connected graph we established rank of thedensity matrix.


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AKLT on hexagonal Lattice

Spin 3/2 is present in each site. Only nearest neighborsinteract. Density of the Hamiltonian is projector on spin 3

HAKLT =∑

<kl>

P3(Sk ,Sl)

The • will represent spin 1/2. Symmetrization of product ofthree spins 1/2 will give spin 3/2. Solid circle will representsymmetrization.


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VBS on hexagonal Lattice

Figure: • is spin 1/2, solid circle is symmetrization. Part of straightline is antisymmtrization. Doted circle is the block.


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Entropy of entanglement on hexagonal Lattice

Collaboration with Anatol Kirillov and Hosho Katsura.Only 2L eigenvalues of the block density matrix not vanish.Here L is the length of the boundary .The entropy of the block is

SB → (coeff) L + o(L−1), as L → ∞

In agreement with area law.

0 < coeff ≤ ln 2

z


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For Further Reading I

Korepin V E, Izergin G, Bogoliubov N MQuantum Inverse Scattering Method, Correlation Functionsand Algebraic Bethe Ansatz.Cambridge University Press, Cambridge 1993

Jin B -Q & Korepin V EQuantum Spin Chain, Toeplitz Determinants andFisher-Hartwig Conjecture.Jour. Stat. Phys. 116 79-95 2004

Its A R, Jin B -Q & Korepin V EEntropy of XY Spin Chain and Block Toeplitz Determinants.Fields Institute Communications, Universality andRenormalization [editors Bender I & Kreimer D] 50 151,2007


spectrum of reduced density matrix: bethe ansatz vs valence...

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