Transcript 20040823153016201-148651

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
 
H   Si  Si 1
i
• Natural question in Statistical Mechanics: what are the associated
correlation functions?





– correlation functions of the form Cij   i   j   i  j
play central
role: related to thermodynamic properties, to cross sections, detect longrange 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 Eij   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:
Ei ,i n  e n / E
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: longrange 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

Singlet

2
1
H   Si .Si 1   Si .Si 1
3 i
i
Proj. in sym. subspace
• 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
I i i
D
i 1
Pi
Pi+1
Pi+2
Pi+3
Pi+4
Pi+5
Pi+6
Map : H D  H D  H d
• Every state can be represented as a Projected Entangled Pair State (PEPS) as
long as D is large enough
Ex.: 5 qubit state
P ( 2 3  2)
P(2  2)
9
P ( 2 3  2)
• 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
Fannes, Nachtergaele, Werner 92
thermodynamic limit (N!1) when P1=P2=L =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)
• 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
c
(NdD ) 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
Proj. P in phys. subspace
• Optimal measurement basis in context of LE is determined by the
basis that maximizes the entanglement of assistance of the
operator   P† P
Eass    max  
 pi  i
E    D det  A
i
 p E  
1/ D
i
i
DiVincenzo, Fuchs, Mabuchi, Smolin, Thapliyal, Uhlmann 98
(LE with more common entanglement measures
can be calculated using combined DMRG/Monte
Carlo method )
This is indeed the measurement that will optimize the quality of
entanglement swapping
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
00  11
GHZ
P  0 00  1 11
5-qubit ECC
H  00  01  10  11
P  0 00  1 11
Measurement/Teleportation based
quantum computation
H  U  I
• Implementing local unitary U:
 

H  00  01  10  11
 000  111  
• Implementing phase gate:
i
x
  xj   zk

 000  111  
    U ph 
i
x
  xj   zk
• 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
P  0 00...0  1 11...1
H  00  01  10  11
– 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
The Hamiltonian defines a hyperplane
in (2s+1)2 dim. space
• 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
PEPS as variational trial states for
ground states of spin systems
Pi
Pi+1
Pi+2
Pi+3
Pi+4
Pi+5
• All expectation values and hence the energy E   vbs H  vbs
quadratic in the variables Pk
• Strategy for minimizing energy for N-spin state:
Pi+6
are multi-
– Fix all projectors Pi except the jth
– Both the energy and the norm  vbs  vbs 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 (j§1)th particle and repeat previous steps (sweep) until convergence
Verstraete, Porras, Cirac 04
Illustration 1
• Wilson’s Renormalization Group (RG) for Kondo-effect:
J
P0
P1
l1
J
P0
l1
J
P1
P2
L
P0
P1
P2
l2
l3
l4
P3
P4
l6
l5
P5
P6
• 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 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:
P1
P3
-4
10
10
P5
DMRG (PBC)
DM
10
P4
-6
RG
(O
Ne
w
-8
(P
< mSi S i+1> / E0 - 1
E 0 /|E0|
10
P2
BC
)
BC
)
mi
-10
0
20
40
60
L
PN
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.)
Pi+1
Pi
e
e
 iH( i 1,i )t
Pi+2
 iH( i ,i 1)t
Pi+3
e
e
 iH( i 1,i  2 )t
Pi+4
 iH( i  2 ,i  3 )t
Pi+5
e
e
Pi+6
 iH( i  4 ,i 5 )t
 iH( i  3,i  4 )t
e
e
 iH( i  6 ,i  7 )t
 iH( i 5,i  6 )t
•Basic trick: variational dimensional reduction of PEPS
–Given a PEPS |Di of dimension D, find the one |cD’i of dimension D’< D such
that || |Di-|cD’i ||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 |Di parameterized by the D£D matrices Ai, find the one |cD’i
parameterized by D’£D’matrices Bi (D’< D) such as to minimize







  c  Tr   B1i B1i   B2i B2i L  BNi BNi 

i

i
i



 

 2Tr   B1i A1i   B2i A2i  L  BNi ANi   cst
 i
  i

 i
– Fixing all Bi but one to be optimized, this leads to an optimization of the form
xy Heffx-xy y , with solution: Heffx=y/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:
• Ancilla’s 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 ancilla’s encode the randomness
N
N
i 1
i 1
H spinglass   S i  S i 1   ( 1) r ( i ) Siz
N
N
i 1
i 1
H simulation  I anc   S i  S i 1   Siz,anc  Siz
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
P11
P12
P13
P14
P15
P21
P22
P23
P24
P25
P31
P32
P33
P34
P35
P41
P42
P31
P44
P45
• 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
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
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X
X
X
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X
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X
X
X
X
X
X
X
X
X
X
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 4£4, 10£10 and 20£20
– We used bonds of dimension 2,3,4; the error seems to decay exponentially in D
– Note that we get mean field if D=1
– The number of variational parameters scales as ND4 and we expect the same
accuracy as 1-D DMRG with dimension of bonds D2
2
2
0
1

Heisenberg
10x10
Frustrated
-2
0
20x20
0
-1
25

50
Frustrated
Heisenberg
-2 4x4
0 4x4: 36.623
25

10x10: 2.353 (D=2); 2.473 (D=3)
20x20: 2.440 (D=2); 2.560 (D=3)
50
Wilson’s 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 Wilson’s
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