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Quantum Spin Systems from the point a view of

Quantum Information Theory

- Frank Verstraete, Ignacio Cirac
- Max-Planck-Institut für Quantenoptik

Overview

- Entanglement verus correlations in quantum spin

systems - Localizable entanglement
- Diverging entanglement length for gapped quantum

systems - Valence bond states / Projected entangled pair

states (PEPS) - In Spin chains
- In quantum information theory
- Coarse-graining (RG) of PEPS
- PEPS as variational ground states
- Illustration RG DMRG
- Extending DMRG to periodic boundary conditions,

time-evolution, finite-T - Using quantum parallelism for simulating 1-D

quantum spin glasses - Simulation of 2-D quantum spin systems
- Conclusion

Motivation

- Many interesting phenomena in condensed matter

occur in regime with strong correlations (e.g.

quantum phase transitions) - Hard to describe ground states due to

exponentially large Hilbert space - Powerful tool study of 2-point correlation

functions (?length scale) - Central object of study in Quantum Information

Theory entanglement or quantum correlations - It is a resource that is the essential ingredient

for e.g. quantum cryptography and quantum

computing - Quantifies quantum nonlocality
- Can QIT shed new light on properties of strongly

correlated states as occurring in condensed

matter?

Entanglement versus correlations

- Consider the ground state of e.g. a 1-D

quantum Heisenberg Hamiltonian - Natural question in Statistical Mechanics what

are the associated correlation functions? - correlation functions of the form

play central

role related to thermodynamic properties, to

cross sections, detect long-range order and

quantum phase transitions, define length scale - Natural question in QIT what is the amount of

entanglement between separated spins (qubits) in

function of their distance?

Quantum Repeater

Briegel, Dür, Cirac, Zoller 98

- Spin Hamiltonians could also effectively describe

a set of e.g. coupled cavities used as a quantum

repeater - The operationally motivated measure is in this

case how much entanglement is there between the

first atom and the last one?

Entanglement in spin systems

- Simplest notion of entanglement would be to study

mixed state entanglement between reduced density

operators of 2 spins

- Problem does not reveal long-range effect

(Osborne and Nielsen 02, Osterloch et al. 02) - Natural definition of entanglement in spin

systems from the resource point a view

localizable entanglement (LE) - Consider a state , then the LE

is variationally defined as the maximal amount of

entanglement that can be created / localized, on

average, between spins i and j by doing LOCAL

measurements on the other spins - Entanglement length
- quantifies the distance at which useful

entanglement can be created/localized, and

hence the quality of a spin chain if used as a

quantum repeater/channel

Verstraete, Popp, Cirac 04

Entanglement versus correlations

- LE quantifies quantum correlations that can be

localized between different spins how is this

related to the classical correlations studied

in quantum statistical mechanics? - Theorem the localizable entanglement is always

larger than or equal to the connected 2-point

correlation functions - Consequences
- Correlation length is a lower bound to the

Entanglement length long-range correlations

imply long-range entanglement - Ent. Length is typically equal to Corr. Length

for spin ½ systems - LE can detect new phase transitions when the

entanglement length is diverging but correlation

length remains finite - When constructing a quantum repeater between e.g.

cavities, the effective Hamiltonian should be

tuned to correspond to a critical spin chain

Verstraete, Popp, Cirac 04

Illustration the spin-1 AKLT-model

- All correlation functions decay exponentially
- The symmetric subspace is spanned by 3 Bell

states, and hence this ground state can be used

as a perfect quantum repeater - Diverging entanglement length but finite

correlation length - LE detects new kind of
- long range order
- Antiferromagnetic spin chain
- is a perfect quantum channel

Verstraete, Martin-Delgado, Cirac 04

Generalizing the AKLT-state PEPS

- Every state can be represented as a Projected

Entangled Pair State (PEPS) as long as D is large

enough - Extension to mixed states take Completely

Positive Maps (CPM) instead of projectors - 1-D PEPS reduce to class of finitely correlated

states / matrix product states (MPS) in

thermodynamic limit (N!1) when P1P2L P1 - Systematic way of constructing translational

invariant states - MPS become dense in space of all states when D!1
- yield a very good description of ground states of

1-D systems (DMRG)

Ex. 5 qubit state

Fannes, Nachtergaele, Werner 92

- PEPS in higher dimensions

Verstraete, Cirac 04

Basic properties of PEPS

- Correlation functions for 1-D PEPS can easily be

calculated by multiplying transfer

matrices of dimension D2 - Number of parameters grows linearly in number of

particles (NdDc) with c coordination number of

lattice - 2-point correlations decay exponentially
- Area law entropy of a block of spins is

proportional to its surface

Localizable Entanglement of VBS

- Optimal measurement basis in context of LE is

determined by the basis that maximizes the

entanglement of assistance of the operator - This is indeed the measurement that will

optimize the quality of entanglement swapping

Proj. P in phys. subspace

DiVincenzo, Fuchs, Mabuchi, Smolin, Thapliyal,

Uhlmann 98

(LE with more common entanglement measures can be

calculated using combined DMRG/Monte Carlo method

)

VBS in QIT

- VBS play a crucial role in QIT all

stabilizer/graph/cluster states are simple VBS

with qubit bonds - Gives insight in their decoherence properties,

entropy of blocks of spins ... - Examples
- GHZ
- 5-qubit ECC

Measurement/Teleportation based quantum

computation

- Implementing local unitary U
- Implementing phase gate
- As Pauli operators can be pulled through Uph ,

this proves that 2- and 3-qubit measurements on a

distributed set of singlets allows for universal

QC

Gottesman and Chuang 99 Verstraete and Cirac 03

Measurement based quantum computation

- Can joint measurements be turned into local ones

at the expense of initially

preparing a highly entangled state? - Yes interpret logical qubits and singlets as

virtual qubits and bonds of a 2-D VBS - Local measurements on physical qubits correspond

to Bell/GHZ-measurements on virtual ones needed

to implement universal QC - This corresponds exactly to the cluster-state

based 1-way computer of Raussendorf and Briegel,

hence unifying the different proposals for

measurement based QC

Raussendorf and Briegel 01 Verstraete and Cirac

03 Leung, Nielsen et al. 04

Spin systems basic properties

- Hilbert space scales exponentially in number of

spins - Universal ground state properties
- Entropy of block of spins / surface of block

(holographic principle) - Correlations of spins decay typically

exponentially with distance (correlation length) - The N-particle states with these properties form

a tiny subspace of the exponentially large

Hilbert space

- Ground states are extreme points of a convex set
- Problem of finding ground state energy of all

nearest-neighbor transl. invariant Hamiltonians

is equivalent to characterizing the convex set of

n.n. density operators arising from transl.

invariant states - Finitely Correlated States / Matrix Product

States / Projected Entangled Pair States provide

parameterization that only grows linearly with

number of particles but captures these desired

essential features very well

The Hamiltonian defines a hyperplane in

(2s1)2 dim. space

PEPS as variational trial states for ground

states of spin systems

- All expectation values and hence the energy

are multi-quadratic in

the variables Pk - Strategy for minimizing energy for N-spin state
- Fix all projectors Pi except the jth
- Both the energy and the norm

are quadratic functions of the variable Pj and

hence the minimal energy by varying Pi can be

obtained by a simple generalized eigenvalue

problem - Heff and N are function of the Hamiltonian

and all other projectors, and can efficiently be

calculated by the transfer matrix method - Move on to the (j1)th particle and repeat

previous steps (sweep) until convergence

Verstraete, Porras, Cirac 04

Illustration 1

- Wilsons Renormalization Group (RG) for

Kondo-effect - RG calculates effective Hamiltonian by projecting

out high energy modes the effective Hamiltonian

is spanned by a set of PEPS - Very successful for impurity problems,

demonstrating validity of PEPS-ansatz

L

Illustration 2 DMRG

White 92

- Most accurate method for determining ground

states properties of 1-D spin chains (e.g.

Heisenberg chains, Hubbard, ) - PEPS-approach proves the variational nature of

DMRG - Numerical effort to find ground state is related

to the amount of entanglement in a block of spins

(Osborne and Nielsen 02, Vidal et al. 03)

DMRG and periodic boundary conditions

- DMRG with periodic instead of open boundary

conditions

Exactly translational invariant states are

obtained, which seems to be important for

describing dynamics Computational cost ND5

versus ND3 (OBC)

- Further extensions
- Variational way of Calculating Excitations and

dynamical correlation functions / structure

factors using PEPS - Variational time evolution algorithms (see also

Vidal et al.)

- Basic trick variational dimensional reduction of

PEPS - Given a PEPS yDi of dimension D, find the one

cDi of dimension Dlt D such that yDi-cDi

2 is minimized - This can again be done efficiently in a similar

iterative way, yielding a variational and hence

optimal way of treating time-evolution within the

class of PEPS

Variational Dimensional Reduction of PEPS

- Given a PEPS yDi parameterized by the DD

matrices Ai, find the one cDi parameterized by

DDmatrices Bi (Dlt D) such as to minimize - Fixing all Bi but one to be optimized, this leads

to an optimization of the form xy

Heffx-xy y , with solution Heffxy/2

iterating this leads to global optimum - The error of the truncation can exactly be

calculated at each step! - In case of OBC more efficient due to

orthonormalization - In the case of OBC, the algorithms of Vidal,

Daley et al., White et al. are suboptimal but a

factor of 2-3 times faster a detailed comparison

should be made

- Finite-T DMRG imaginary/real time evolution of a

PEPS-purification - Ancillas can also be used to describe quantum

spin-glasses due to quantum parallelism, one

simulation run allows to simulate an exponential

amount of different realizations the ancillas

encode the randomness

Simulation of 2-D quantum systems

- Standard DMRG approach trial state of the form

Problems with this approach dimension of bonds

must be exponentially large - area theorem -

only possibility to get large correlations

between vertical nearest neighbors

We propose trial PEPS states that have bonds

between all nearest neighbors, such that the area

theorem is fulfilled by construction and all

neighbors are treated on equal footing

- The energy of such a state is still a

multi-quadratic function of all variable, and

hence the same iterative variational principle

can be used - The big difference the determination of Heff

and N is not obtained by multiplying matrices,

but contracting tensors - This can be done using the variational

dimensional reduction discussed before note that

the error in the truncation is completely

controlled

No (sign) problem with frustrated

systems! Possible to devise an infinite

dimensional variant

- Alternatively, the ground state can be found by

imaginary time evolution on a pure 2-D PEPS - This can be implemented by Trotterization the

crucial ingredient is again the variational

dimensional reduction the computational cost

scales linearly in the number of spins D10 - The same algorithm can of course be used for

real-time evolution and for finding thermal

states. - Dynamical correlation functions can be calculated

as in the 1-D PEPS case - We have done simulations with the Heisenberg

antiferromagnetic interaction and a frustrated

version of it on 44, 1010 and 2020 - We used bonds of dimension 2,3,4 the error seems

to decay exponentially in D - Note that we get mean field if D1
- The number of variational parameters scales as

ND4 and we expect the same accuracy as 1-D DMRG

with dimension of bonds D2

10x10

20x20

4x4

4x4 36.623 10x10 2.353 (D2) 2.473

(D3) 20x20 2.440 (D2) 2.560 (D3)

Wilsons RG on the level of states

Coarse-graining PEPS

- Goal coarse-graining of PEPS-ground states
- This can be done exactly, and leads to a fixed

point exponentially fast the fixed points are

scale-invariant. This procedure is equivalent to

Wilsons numerical RG procedure - The fixed point of the generic case consists of

the virtual subsystems becoming real, and where

the ME-states are replaced with states with some

entropy determined by the eigenvectors of the

transfer matrix note that no correlations are

present - A complete classification of fixed points in case

of qubit bonds has been made special cases

correspond to GHZ, W, cluster and some other

exotic states in QIT

Conclusion

- PEPS give a simple parameterization of

multiparticle entanglement in terms of bipartite

entanglement and projectors - Examples of PEPS Stabilizer, cluster, GHZ-states

- QIT-approach allows to generalize numerical RG

and DMRG methods to different settings, most

notably to higher dimensions