Open Systems & Quantum Information Milano, 10 Marzo 2006
Measures of Entanglement at Quantum Phase Transitions
Measures of Entanglement at Quantum Phase Transitions
M. Roncaglia
G. Morandi F. OrtolaniE. Ercolessi
C. Degli Esposti BoschiL. Campos Venuti
S. Pasini
Condensed Matter Theory Group in BolognaCondensed Matter Theory Group in Bologna
Open Systems & Quantum Information Milano, 10 Marzo 2006
• Entanglement is a resource for:
teleportation dense coding quantum cryptography quantum computation
QUBITS Spin chains are natural candidates as quantum devices
• The Entanglement can give another perspective for understanding Quantum Phase Transitions
• Strong quantum fluctuations in low-dimensional quantum systems at T=0
Open Systems & Quantum Information Milano, 10 Marzo 2006
• Entanglement is a property of a state, not of an Hamiltonian. But the GS of strongly correlated quantum systems are generally entangled.
• Direct product states
• 2-qubit states 1001
2
1
11002
1
A BBA
0 jiji BABA
• Nonzero correlations at T=0 reveal entanglement
Maximally entangled(Bell states)
11,00
Product states
Open Systems & Quantum Information Milano, 10 Marzo 2006
Block entropyB A
AAAAS logTr
BA Tr
• Reduced density matrix for the subsystem A
• Von Neumann entropy
• For a 1+1 D critical system
CFT with central charge c
lSA log3
c log
6
cAAS
Off-critical
[ See P.Calabrese and J.Cardy, JSTAT P06002 (2004).]l= block size
Open Systems & Quantum Information Milano, 10 Marzo 2006
RG flow
UVfixed point
IRfixed point
• c-theorem: IRUV cc
(Zamolodchikov, 1986)
Loss of entanglement
Renormalization Group (RG)
Irreversibility of RG trajectories
RG flow
UVfixed point
• Massive theory (off critical) Block entropy saturation
Open Systems & Quantum Information Milano, 10 Marzo 2006
[ S.Gu, S.Deng, Y.Li, H.Lin, PRL 93, 86402 (2004).]
• Local Entropy: when the subsystem A is a single site.
• Applied to the extended Hubbard model
• The local entropy depends only on the average double occupancy
• The entropy is maximal at the phase transition lines (equipartition)
Open Systems & Quantum Information Milano, 10 Marzo 2006
[ A.Anfossi et al., PRL 95, 056402 (2005).]
• Bond-charge Hubbard model (half-filling, x=1)
• Negativity
2/1)(1 AT
ABABN
• Mutual information
)()()( ABBA SSSI
• Critical points: U=-4, U=0
• Some indicators show singularities at transition points, while others don’t.
Open Systems & Quantum Information Milano, 10 Marzo 2006
[ A.Osterloh, et al., Nature 416, 608 (2002).]
Ising model in transverse field
• Critical point: =1
• The concurrence measures the entanglement between two sites after having traced out the remaining sites.
• The transition is signaled by the first derivative of the concurrence, which diverges logarithmically (specific heat).
Open Systems & Quantum Information Milano, 10 Marzo 2006
11100100 dcba
bcadC 2
yj
yiijC *)(
Concurrence
For a 2-qubit pure state the concurrence is (Wootters, 1998)
if
• Is maximal for the Bell states and zero for product states
For a 2-qubit mixed state in a spin ½ system
22
12
1 zj
zi
zj
zi
yj
yi
xj
xiij σσσσσσσσ=C
;,,0max ijijij CC=C
Open Systems & Quantum Information Milano, 10 Marzo 2006
Ising model in transverse field
][1
zi
xi
xi
hHi
1h
1h
1zP
1h Critical point
2D classical Ising modelCFT with central charge c=1/2
Exactly solvable fermion model
Jordan-Wigner transformation
Open Systems & Quantum Information Milano, 10 Marzo 2006
Local (single site) entropy:
1111 ln2
1ρρTr=Sσσ+I=ρ zz
Near the transition (h=1):
1ln 112
hhσ z
S1 has the same
singularity aszσ
Nearest-neighbour concurrence inherits logarithmic singularity 1ln 1 hhσσ ji
Local measures of entanglement based on the 2-site density matrix depend on 2-point functions
Accidental cancellation of the leading singularity may occur, as for the concurrence at distance 2 sites
1ln112
1 22222, hhσσσσσσC z
izi
yi
yi
xi
xiii
Open Systems & Quantum Information Milano, 10 Marzo 2006
Alternative: FSS of magnetization
100,20,30,=N
2
1
121/8lnln
12
N
π+γ+π+N
π
h+
π=σ C
z
2
2 1
61
~
N
π+=hN
Crossing points:
Exact scaling function in the critical region ξ<N
C. Hamer, M. Barber, J. Phys. A: Math. Gen. (1981) 247.
Standard route: PRG
M,hM=N,hN~~
NENE=Nh, 01
First excited state needed
Shiftterm
Seeking for QPT point
Open Systems & Quantum Information Milano, 10 Marzo 2006
gVHgH 0
)(
cgg
cc
sg
sggggOV )sgn()()(
Letg
cg
Quantum phase transitions (QPT’s)
• First order: discontinuity in
cggge
(level crossing)
• Second order:cgg
n
n
g
e
diverges for some 2n
)()( )()( gegege sr
1)1( d
• GS energy:
• At criticality the correlation length diverges
scaling hypothesis
• Differentiating w.r.t. g
Open Systems & Quantum Information Milano, 10 Marzo 2006
• The singular term appears in every reduced density matrix containing the sites connected by .
)(sg
OV
• Local algebra hypothesis: every local quantity can be expanded in terms of the scaling fields permitted by the symmetries.
• Any local measure of entanglement contains the singularity of the most relevant term.
• The best suited operator for detecting and classifying QPT’s is V , that naturally contains . Moreover, FSS at criticality
/)sgn( /)()(LLggLOLV
cs
g
s
• Warning: accidental cancellations may occur depending on the specific functional form next to leading singularity
)(sg
O )( L
00
Open Systems & Quantum Information Milano, 10 Marzo 2006
N
i
z
i
z
i
z
i
y
i
y
i
x
i
x
i SSSSSSSJH1
2
111 ][ )(
Spin 1 D model
D
=Ising-like D = single ion
2ziS=
D
e
In this case
Phase Diagram
• Symmetries: U(1)xZ2
Around the c=1 line:
KH x 4cos2
1 22(sine-Gordon)
cDD Criticalexponents
)2/( KK )2/(1 K
Open Systems & Quantum Information Milano, 10 Marzo 2006
[ L.Campos Venuti, et. al., PRA 73, 010303(R) (2006).]
Crossing effect
Derivative ccz DDDD~S sgn
2
0.64=
Single-site entropy
2.59=λ 42.29=Dc0.82=
cczi
zi ~SS sgn1
The same for
• What about local measures of entanglement?
zzzz=S 1ln12/ln1 2ziS=z
Using symmetries:
• Two-sites density matrix contains the same leading singularity)2(
ij
Open Systems & Quantum Information Milano, 10 Marzo 2006
Localizable Entanglement
• LE is the maximum amount of entanglement that can be localized on two q-bits by local measurements.
i j
s
sss
ij EpL )(|max}{
N+2 particle state
Nsss ,,1 • Maximum over all local measurement basis
ssp = probability of getting
sE | is a measure of entanglement
[ F.Verstraete, M.Popp, J.I.Cirac, PRL 92, 27901 (2004).]
(concurrence)
Open Systems & Quantum Information Milano, 10 Marzo 2006
LE = max of correlation LE = string correlations
[ L. Campos Venuti, M. Roncaglia, PRL 94, 207207 (2005).]
Ising model Quantum XXZ chain
MPS (AKLT)
Dλ
Calculating the LE requires finding an optimal basis, which is a formidable task in general
However, using symmetries some maximal (optimal) basis are easily found and the LE takes a manageable form
Spin 1/2 Spin 1
totN SiL exp1 1
• The LE shows that spin 1 are perfect quantum channels but is insensitive to phase transitions.
jiijL max
• :The lower bound is attained
CE
Open Systems & Quantum Information Milano, 10 Marzo 2006
A spin-1 model: AKLT
N
ijiji SSSSJH
1
2 ])(3
1[
• Infinite entanglement length but finite correlation length
• Actually in S=1 case LE is related to string correlation
0explim1
1
k
k
jlljString
SSiSO
|| jk
=Bell state
000000Typical configurations
Optimal basis: 2/11,0
Open Systems & Quantum Information Milano, 10 Marzo 2006
Conclusions
References:L.Campos Venuti, C.Degli Esposti Boschi, M.Roncaglia, A.Scaramucci, PRA 73, 010303(R) (2006).L.Campos Venuti and M. Roncaglia, PRL 94, 207207 (2005).
• Localizable Entanglement It is related to some already known correlation functions. It promotes S=1 chains as perfect quantum channels.
• The most natural local quantity is , where g is the driving parameter across the QPT.
ge /
• it shows a crossing effect• it is unique and generally applicable
Advantages:
• Low-dimensional systems are good candidates for Quantum Information devices.
• Several local measures of entanglement have been proposed recently for the detection and classification of QPT. (nonsystematic approach)
• Open problem: Hard to define entanglement for multipartite systems, separating genuine quantum correlations and classical ones.
• Apart from accidental cancellations all the scaling properties of local entanglement come from the most relevant (RG) scaling operator.
Open Systems & Quantum Information Milano, 10 Marzo 2006
The End