Spectrum of Reduced Density Matrix: vs Valence Bond Solid states

Vladimir Korepin

C.N. Yang Institute for Theoretical Physics

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Vladimir Korepin Operator Structures in ENTROPY Reduced Density Matrix

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

Vladimir Korepin Operator Structures in Quantum Information ENTROPY 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 in any subsystem:

 In quantum case the entropy of a subsystem of a pure system can be positive. This means that the subsystem is entangled with the rest of the system. It is an important resource for quantum control.

Vladimir Korepin Operator Structures in Quantum Information ENTROPY Example

Two spins 1/2: ΨE,B = 1 ( ) | i √2 | ↑iE | ↓iB − | ↓iE | ↑iB Density matrix of one spin is proportional to identical matrix

1 ρ = tr ΨE,B ΨE,B = I B E | ih | 2 2   SE,B = 0 but SB = ln(2) > 0 Cannot happen for classical subsystems. This is a way to distinguish quantum system form classical .

Vladimir Korepin Operator Structures in Quantum Information ENTROPY No factorization

Two systems in general. One can choose the orthonormal basis in both spaces so that:

d ΨE,B = a ψE ψB and a2 = 1 | i j | j i | j i | j | Xj=1 O Xj

Theorem: If SB > 0 then d > 1. The subsystems E and B are entangled.

Vladimir Korepin Operator Structures in Quantum Information ENTROPY Quantum Control

Measurement of subsystem E can be organized to project on one of basis states ψE . The wave function will turn into | j0 i ΨE,B = ψE ψB | M i | j0 i | j0 i O with probability a2 . j0 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.

Vladimir Korepin Operator Structures in Quantum Information ENTROPY Application to XX Model

∞ H = σx σx + σy σy + hσz , h 2 XX − n n+1 n n+1 n | |≤ n= X−∞

Solved by Lieb , Schultz & Mattis 1961 The gs is unique | i Entanglement of a block of L sequential spins with the rest of the [environment].

Vladimir Korepin Operator Structures in Quantum Information ENTROPY Entanglement

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

ρ = gs gs | ih | Density matrix of the block is the reduced density matrix.

ρB = TrE ρ

Individual entries of ρB are correlation functions.

Vladimir Korepin Operator Structures in Quantum Information ENTROPY Density matrix and correlations

a In each lattice site j we have four ’Pauli’ matrices σj , here a = 0, x, y, z with σ0 = I.

L L L aj aj ρ = 2− σ gs σ gs B  j  h | j | i aj j=1 j=1 j=X1,...,L O Y   For gap-full models limit of large block exists.

Vladimir Korepin Operator Structures in Quantum Information ENTROPY

Measures of entanglement for a pure state Von Neumann Entropy of subsystem [block] :

S(ρ )= Tr (ρ ln ρ ) B − B B B Rényi Entropy of subsystem [block]:

α ln TrBρ S(ρ , α)= B B 1 α −  α is a parameter. Also known in literature as zeta function or replica trick.

Vladimir Korepin Operator Structures in Quantum Information ENTROPY Calculation of Entropy

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

Vladimir Korepin Operator Structures in Quantum Information ENTROPY Von Neumann Entropy

Asymptotic L : →∞ 1 1 h 2 1 1 S(ρ ) ln L + ln 1 + ln 2 +Υ + O B → 3 6 − 2 3 1 L "   #   (ln L ) term was discovered Holzhey, Larsen, Wilczek 1994 Sub-leading terms: B.-Q. Jin, V.E. Korepin in 2004

∞ e−t 1 cosh(t/2) Υ1 = dt + 2 3 0.495 − 0 3t t sinh t 2 − 2 sinh t 2 ≈ Z ( ( / ) ( / ) )

Vladimir Korepin Operator Structures in Quantum Information ENTROPY Rényi Entropy

Asymptotic of Rényi entropy is

1 2 1 + α− h α S(ρB, α) ln 2L 1 +Υ{ } → 6  s − 2        B.-Q. Jin, V.E. Korepin 2003, see arXiv:quant-ph/0304108 Later reproduced by J. Cardy and P. Calabrese from conformal filed theory, arXiv:hep-th/0405152

Vladimir Korepin Operator Structures in Quantum Information ENTROPY XY model

Gap-full models

+ ∞ H = (1 + γ)σx σx + (1 γ)σy σy + hσz XY − n n+1 − n n+1 n n= X−∞

Lieb , Schultz & Mattis 1961 Phases 1a. Moderate field: 2 1 γ2 < h < 2 − 1b. Weak field: 0 h < 2 1 γ2 p≤ − 2. Strong field: h > 2 p

Vladimir Korepin Operator Structures in Quantum Information ENTROPY Phase diagram

γ

Case 1A 1

Case 1B Case 2

0 2 h

Vladimir Korepin Operator Structures in Quantum Information ENTROPY Boundary between 1a and 1b

Barouch-McCoy circle h = 2 1 γ2 − GS = GS + GS p | i | 1i | 2i

GS = [cos(θ) n + sin(θ) n ] , | 1i | ↑ i | ↓ i n lattice ∈ Y GS = [cos(θ) n sin(θ) n ] | 2i | ↑ i− | ↓ i n lattice ∈ Y cos2(2θ) = (1 γ)/(1 + γ) − Partially ordered state.

Vladimir Korepin Operator Structures in Quantum Information ENTROPY Von Neumann Entropy

Entropy of a large block has a limit:

∞ τ τ 1 θ3 β(λ)+ 2 θ3 β(λ) 2 S(ρB)= ln − dλ 2 θ2 στ Z1 3  2 ! 

1 λ + 1 I(k ) 1 (h/2)2 γ2 β(λ)= ln , τ = i ′ , k = − − 2πi λ 1 I(k) s 1 (h/2)2 − − 2 2 θ3 the Jacobi theta-function; (k ′) + k = 1 Entropy is constant on the ellipsis: k = const 1 dx Complete elliptic integral of the first kind I(k)= 0 √(1 x2)(1 x2k 2) − − R

Vladimir Korepin Operator Structures in Quantum Information ENTROPY references

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

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

Vladimir Korepin Operator Structures in Quantum Information ENTROPY Rényi Entropy

Limiting entropy in the region 0 < h < 2 is: ′ α α S(ρ , α)= 1 α ln k + 1 1 ln θ2(0,q )θ4(0,q ) + 1 ln 2 B 6 1 α k 2 3 1 α θ2(0,qα) 3 − − 3     I(k ) τ = i ′ iτ , q = eπiτ I(k) ≡ 0

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

Vladimir Korepin Operator Structures in Quantum Information ENTROPY Spectrum of the density matrix of very large block

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

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

Here m is a label of eigenvalue. The maximum eigenvalue

′ 1 6 π p = (kk /4) / exp τ is unique 0 12 0 h i 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

Vladimir Korepin Operator Structures in Quantum Information ENTROPY Affleck, Kennedy, Lieb and Tasaki

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

N 1 − 1 2 H = S~ S~ + (S~ S~ )2 + + π + π . k k+1 3 k k+1 3 0,1 N,N+1 Xk=1   1 ~ ~ 1 ~ ~ 2 1 2 Sk Sk+1 + 6 (Sk Sk+1) + 3 is a projector on a state of spin 2. The π is a projector on a state with spin 3/2: 2 2 π = 1 + ~s S~ , π = 1 + ~s S~ . 0,1 3 0 1 N,N+1 3 N+1 N    

Vladimir Korepin Operator Structures in Quantum Information ENTROPY Ground state is VBS

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

      s ss ss ss ss ss ss s 1      a dot is spin- 2 ; circle means symmetrisation [makes spin 1]. A line is a anti-symmetrisation ( ) . Each projector annihilates the ground state (no| ↑↓i frustration). − | ↓↑i The ground state is called VBS. Exact formula for correlation function at any distance is: 3 1 x < S~ x S~ >= = p(x) 4 1 −3  

Vladimir Korepin Operator Structures in Quantum Information ENTROPY Entropy of entanglement of VBS

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

3(1 p(x)) 1 p(x) S(x)= − log − − 4 4 − 1 3p x 1 3p x  + ( ) log + ( ) − 4 4   It does not depend on the size of the lattice.For large block

S(x) ln 4 as x → →∞

Vladimir Korepin Operator Structures in Quantum Information ENTROPY Eigenvalues of the density matrix

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

1 + 3p(x) p = 0 4 and three

1 p(x) p = − 1,2,3 4

Does not depend on the length of the lattice.

Vladimir Korepin Operator Structures in Quantum Information ENTROPY Eigenvectors of the Density matrix

Consider ’block’ Hamiltonian

(k+1) block ∈ 1 2 H = S~ S~ + (S~ S~ )2 + . B k k+1 3 k k+1 3 k block   ∈X It describes interaction of spins 1 inside of the block. The ground state is quadruple degenerated: these are eigenvectors of the density matrix corresponding non-zero eigenvalues .

βH e− B ρB = lim β Tre βHB →∞  − 

Vladimir Korepin Operator Structures in Quantum Information ENTROPY Generalizations

Together with Hosho Katsura and Ying Xu these results for VBS we generalized to: For higher spins s the entropy S = 2 ln(s + 1) Other Lie algebras: for SU(n) for smallest non-trivial representation the entropy S = 2 ln n For AKLT on a connected graph we established rank of the density matrix.

Vladimir Korepin Operator Structures in Quantum Information ENTROPY AKLT on hexagonal Lattice

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

HAKLT = P3(Sk , Sl ) The will represent spin 1/2. Symmetrization of product of • three spins 1/2 will give spin 3/2. Solid circle will represent symmetrization.

Vladimir Korepin Operator Structures in Quantum Information ENTROPY VBS on hexagonal Lattice

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

Vladimir Korepin Operator Structures in Quantum Information ENTROPY 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

1 S (coeff) L + o(L− ), as L B → →∞ In agreement with area law.

0 < coeff ln 2 ≤

z

Vladimir Korepin Operator Structures in Quantum Information ENTROPY For Further Reading I

Korepin V E, Izergin G, Bogoliubov N M Quantum Inverse Scattering Method, Correlation Functions and Algebraic Bethe Ansatz. Cambridge University Press, Cambridge 1993 Jin B -Q & Korepin V E Quantum Spin Chain, Toeplitz Determinants and Fisher-Hartwig Conjecture. Jour. Stat. Phys. 116 79-95 2004 Its A R, Jin B -Q & Korepin V E Entropy of XY Spin Chain and Block Toeplitz Determinants. Fields Institute Communications, Universality and [editors Bender I & Kreimer D] 50 151, 2007

Vladimir Korepin Operator Structures in Quantum Information