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Quantum Spin Systems from the point a view of Quantum Information Theory


Title: Quantum spin systems from the perspective of quantum information theory Author: Admin Last modified by: Admin Created Date: 6/6/2004 5:49:09 PM – PowerPoint PPT presentation

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Title: Quantum Spin Systems from the point a view of Quantum Information Theory

Quantum Spin Systems from the point a view of
Quantum Information Theory
  • Frank Verstraete, Ignacio Cirac
  • Max-Planck-Institut für Quantenoptik

  • Entanglement verus correlations in quantum spin
  • Localizable entanglement
  • Diverging entanglement length for gapped quantum
  • 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

  • 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
  • Quantifies quantum nonlocality
  • Can QIT shed new light on properties of strongly
    correlated states as occurring in condensed

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
  • 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
  • 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
  • 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 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
  • 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

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
  • 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
  • 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
  • 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

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
  • 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

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
  • 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
  • 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
  • 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

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
  • 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

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
  • 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

  • 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
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