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

1
Quantum Spin Systems from the point a view of
Quantum Information Theory

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

2
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

3
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?

4
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?

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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?

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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
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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
8
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
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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
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• PEPS in higher dimensions

Verstraete, Cirac 04
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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

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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
)
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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

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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
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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
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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

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• 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
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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
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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
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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)

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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)
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• 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

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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

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• 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

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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
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• 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
27

• 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

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10x10
20x20
4x4
4x4 36.623 10x10 2.353 (D2) 2.473
(D3) 20x20 2.440 (D2) 2.560 (D3)
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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

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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