multipartite entanglement measures from matrix and tensor product states ching-yu huang feng-li lin...
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Multipartite Entanglement Measures from Matrix and Tensor Product States
Ching-Yu Huang Feng-Li Lin Department of Physics, National Taiwan Normal University
arXiv:0911.4670
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Outline
• Quantum Phase Transition• Entanglement measure• Matrix/Tensor Product States• Entanglement scaling• Quantum state RG transformation
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Quantum Phase transition
• Phase transition due to tuning of the coupling constant.
• How to characterize quantum phase transition?
a. Order parameters
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Frustrated systems
• These systems usually have highly degenerate ground states.
• This implies that the entropy measure can be used to characterize the quantum phase transition.
• Especially, there are topological phases for some frustrated systems (no order parameters)
?
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Entanglement measure
• The degeneracy of quantum state manifests as quantum entanglement.
• The entanglement measure can be used to characterize the topological phases.
• However, there is no universal entanglement measure of many body systems.
• Instead, the entanglement entropy like quantities are used.
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Global entanglement• Consider pure states of N particles . The
global multipartite entanglement can be quantified by considering the maximum fidelity.
• The larger indicates that the less entangled.
• A well-defined global measure of entanglement is
【 Wei et al. PRA 68, 042307(2003) 】
maxmax
max
2maxlog)( E
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Global entanglement density and its h derivative for the ground state of three systems at N. Ising ; anisotropic XY model; XX model.
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Our aim
• We are trying to check if the entanglement measure can characterize the quantum phase transition in 2D spin systems.
• Should rely on the numerical calculation• Global entanglement can pass the test but is
complicated for numerical implementation.• One should look for more easier one.
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Matrix/Tensor Product States
• The numerical implementation for finding the ground states of 2D spin systems are based on the matrix/tensor product states.
• These states can be understood from a series of Schmidt (bi-partite) decomposition. It is QIS inspired.
• The ground state is approximated by the relevance of entanglement.
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1D quantum state representation
1
1
... 11 1
| ... | ...n
n
d d
i i ni i
c i i
31 2
1 1 1 1 2 2 2 3 1
1 1
[3] [ ][1] [2][1] [2]...
...
.... n
n n
n
i n ii ii ic
Γ NΓ1 Γ 2 Γ3
λ1 λ2 λN-1
NdNdDparameters <<» 2# • Representation is efficient• Single qubit gates involve only local update• Two-qubit gates reduces to local updating
Virtual dimension = D
Physical dimension = d
【 Vidal et al. (2003) 】
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``Solve” for MPS/TPS
• In old days, the 1D spin system is numerically solved by the DMRG(density matrix RG).
• For translationally invariant state, one can solve the MPS by variational methods.
• More efficiently by infinite time - evolving block decimation (iTEBD) method.
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Time evolution• Real time• imaginary time
• Trotter expansion
0 (-iHt) exp| t
( ) / 2[ ] ( )i F G T iF iG Te e e O T
Γ1 Γ 2 Γ3
λ1 λ2
Γ4 Γ5
λ3 λ4
F F
Γ1 Γ 2 Γ3
λ1 λ2
Γ4 Γ5
λ3 λ4
G G
(-Ht) exp
(-Ht) exp|
0
0
【 Vidal et al. (2004) 】
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Γ (si) Γ (sj)
λi λkλk+1
U
ΘSVD
Γ’(si) Γ’(sj)
λi λ’k λk+1
X (si) Y (sj)
λ'kλ-1iλi
λk+1 λk+1
Evolution quantum state
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Global entanglement from matrix product state for 1D system
• Matrix product state (MPS) form
• The fidelity can be written by transfer matrix
where
Nppp
d
ppp
pppAAATr N
N
..., )...( 211...,
21
21
)...( 321 ngggg TTTTTr
is i
ii
iig sBsAT )()( ][][
【 Q.-Q. Shi, R. Orus, J.-O. Fjaerestad, H.-Q. Zhou, arXiv:0901.2863 (2009) 】
ANA1 A2 A3
BNB1 B2 B3
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Quantum spin chain and entanglement
• XY spin chain
• Phase diagram: oscillatory (O) ferromagnetic (F) paramagnetic (P)
)2
1
2
1( 11
1
zi
yi
yi
xi
xi
N
iXY hH
-0.5 0.5 1.50
0.25
0.5
0.75
1
h
γ
OF p
Ising
XY
【 Wei et al. PRA 71, 060305(2003) 】
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• A second-order quantum phase transition as h is tuned across a critical value h = 1.
• Magnetization along the x direction . (D= 16)
0.96 0.98 1.00 1.02 1.04 1.06 1.08 1.100.0
0.2
0.4
0.6
0.8
mx
h
D D D D
1)( ** hh
0 2 4 6 8 10 12 14 160
0.2
0.4
0.6
0.8
1
1.2
hc
D
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• For a system of N spin 1/2. the separable statesrepresented by a matrix
• The global entanglement has a nonsingular maximum at h=1.1.
• The von Neuman entanglement entropy also has similar feature.
)1sin0(cos1 ii
iNi e
iii
ii eBB sin)1( , cos)0( ][][
0.0 0.5 1.0 1.5 2.0 2.50.00
0.02
0.04
0.06
0.08
0.10
GE
h
Ising XY
0.0 0.5 1.0 1.5 2.00.0
0.2
0.4
0.6
0.8
1.0
1.2
M
h
Ising Mx Ising Mz XY Mx XY Mz
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Generalization to 2D problem
• The easy part promote 1D MPS to 2D Tensor product state (TPS)
• The different partHow to perform imaginary time evolution a. iTEBDHow to efficiently calculate expectation value a. TRG
【 Levin et al. (2008), Xiang et al. (2008) , Wen et al. (2008) 】
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2D Tensor product state (TPS) • Represent wave-function by the tensor
network of T tensors
NNppp
d
ppp
pppTTTTr NN
NN
,1,21,11...,
..., )...( ,1,21,1
,1,21,1
p
l
d
u
r
Virtual dimension = D
Physical dimension = d
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Global entanglement from matrix product state for 2D system
• The fidelity can be written by transfer matrix
where
• It is difficult to calculate tensor trace (tTr), so we using the “TRG” method to reduce the exponentially calculation to a polynomial calculation.
)...( 321 ngggg TTTTtTr
A4 A3
A1 A2
B1
B4
B2
B3
Ť4
Ť2
Ť3
Ť1
Double Tensor
is i
ii
iig sBsAT )()( ][][
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The TRG method Ⅰ
• First, decomposing the rank-four tensor into two rank-three tensors.
u
S2 d
S4
S3
l
uS1
T T
T T
rd
lγ
γ r
,3
,1
;
lulu
rdrd
Rlurd
VS
US
VUM
,2
,4
,
dldl
urur
Rdlur
VS
US
VUM
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The TRG method Ⅱ• The second step is to build a new rank-four
tensor.• This introduces a coarse-grained square lattice.
γ1
γ2
γ3
T’
γ4
γ1 S1
S4 S3
S2
a1
a2
a3
a4
γ2
γ3γ4
432343214121
4321
4321aa
4aa
3aa
2aa
1
,,,,,, SSSS
aaaa
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The TRG method Ⅲ
• Repeat the above two steps, until there are only four sites left. One can trace all bond indices to find the fidelity of the wave function.
TA
TC
TB
TD
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Global entanglement for 2D transverse Ising
• Ising model in a transverse magnetic field
• The global entanglement density v.s. h, the entanglement has a nonsingular maximum at h=3.25.
• The entanglement entropy
0 1 2 3 4 50.0
0.2
0.4
0.6
0.8
1.0
1.2
h
Mz GE S
BP
S1
xi
zj
zi
ji
hH ,
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Global entanglement for 2D XXZ model
• XXZ model in a uniform z-axis magnetic field
• A first-order spin-flop quantum phase transition from Neel to spin-flipping phase occurs at some critical field hc.
• Another critical value at hs = 2(1 + △), the fully polarized state is reached.
1 2 3 4 5 60.0
0.2
0.4
0.6
0.8
1.0
<M
>
h
<Mx> <Mz>
<Mzs>
0 1 2 3 4 5 60.0
0.1
0.2
0.3
0.4
0.5
0.6
h
GE S
BP
S1
zi
zj
zi
yj
yi
xj
xi
ji
hH )(,
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Scaling of entanglement
L spins
ρL SL
Lc
SLL 2log
3 |1|log
6 22/ hc
S NL
【 Vidal et al. PRL,90,227902 (2003) 】
Entanglement entropy for a reduced block in spin chains
At Quantum Phase Transition Away from Quantum Phase Transition
LLL TrS ln
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Block entanglement v.s. Quantum state RG
• Numerically, it is involved to calculate the block entanglement because the block trial state is complicated.
• Entanglement per block of size 2L = entanglement per site of the L-th time quantum state RG(merging of sites).
• We can the use quantum state RG to check the scaling law of entanglement.
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1D Quantum state transformationQuantum state RG 【 Verstraete et al. PRL 94, 140601
(2005) 】
• Map two neighboring spins to one new block spin
• Identify states which are equivalent under local unitary operations.
By SVD
RG
qpD
pq AAA
1
)()( :
~
p q lRG
α β γ α γ
l)(
)(),min(
1
)()( V
~22
lpql
Dd
l
pq UA
lllRGp VAA
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• For MPS representation for a fixed value of D, this entropy saturates at a distance L ~ Dκ (kind of correlation length)
• Could we attain the entropy scaling at critical point from MPS?【 Latorre et al. PRB,78,024410 (2008) 】
At quantum point
1 2 5 10 20 30 405060 70 800.30
0.35
0.40
0.45
0.50
1 2 5 10 20 30 405060 70 800.0
0.5
1.0
1.5
2.0
2.5
3.0
SL
L
D D D
D D D
SL
L
Near quantum point
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• MPS support of entropy obeys scaling law!!
1 2 3 4-0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
)ln(182.0
DDS log6
1)(
Dlog*182.0~
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2D Scaling of entanglement
• In 2D system, we want to calculate the block entropy and block global entanglement.
Could we find the scaling behavior like 1D system?
)ln1( LcLSLLxL
Area law
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2D Quantum state transformation
• Map one block of four neighboring spins to one new block spin.
• By TRG and SVD method
RG
s1 s2
s4 s3s
RG
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The failure
• We find that the resulting block entanglement decreases as the block size L increases.
• This failure may mean that we truncate the bond dimension too much in order to keep D when performing quantum state RG transformation, so that the entanglement is lost even when we increase the block size.
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Summary
• Using the iTEBD and TRG algorithms, we find the global entanglement measure.- 1D XY model- 2D Ising model- 2D XXZ model
• The scaling behaviors of entanglement measures near the quantum critical point.
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Thank You ForYour Attention!