1. Measures of Entanglement at Quantum Phase Transitions
M. Roncaglia
Condensed Matter Theory Group in Bologna
G. Morandi
F. Ortolani
E. Ercolessi
C. Degli Esposti Boschi
L. Campos Venuti
S. Pasini
Intro
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Entanglement is a resource for:
teleportation
dense coding
quantum cryptography
quantum computation
QUBITS
Spin chains are natural candidates
as quantum devices
Strong quantum fluctuations in low-dimensional quantum systems at T=0
The Entanglement can give another perspective for understanding Quantum Phase Transitions
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Entanglement is a property of a state, not of an Hamiltonian. But the GS of strongly correlated quantum systems are generally entangled. A B
Direct product states
Nonzero correlations at T=0 reveal entanglement
2-qubit states
Product states
Maximally entangled
(Bell states)
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Block entropy B A
Reduced density matrix for the subsystem A
Von Neumann entropy
For a 1+1 D critical system
Off-critical
CFT with central charge c
l= block size
[ See P.Calabrese and J.Cardy, JSTAT P06002 (2004).]
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Renormalization Group (RG)
RG flow UV
fixed point IR
c-theorem:
(Zamolodchikov, 1986)
RG flow UV
fixed point
Massive theory (off critical)
Block entropy saturation
Irreversibility of RG trajectories
Loss of entanglement
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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)
[ S.Gu, S.Deng, Y.Li, H.Lin, PRL 93, (2004).]
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Bond-charge Hubbard model
(half-filling, x=1)
Critical points: U=-4, U=0
Negativity
Mutual information
Some indicators show
singularities at transition points, while others don’t.
[ A.Anfossi et al., PRL 95, (2005).]
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Ising model in transverse field
Critical point: l=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).
[ A.Osterloh, et al., Nature 416, 608 (2002).]
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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
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Ising model in transverse field
Critical point
2D classical Ising model
CFT with central charge c=1/2
Jordan-Wigner transformation
Exactly solvable fermion model
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Near the transition (h=1):
S1 has the same
singularity as
Local (single site) entropy:
Local measures of entanglement based on the 2-site density
matrix depend on 2-point functions
Nearest-neighbour concurrence
inherits logarithmic singularity
Accidental cancellation of the leading singularity may occur,
as for the concurrence at distance 2 sites
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Seeking for QPT point
Alternative: FSS of magnetization
Standard route: PRG
First excited state needed
C. Hamer, M. Barber, J. Phys. A: Math. Gen. (1981) 247.
Exact scaling function in the
critical region
Crossing points:
Shift
term
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Quantum phase transitions (QPT’s)
Let
First order: discontinuity in
(level crossing)
Second order:
diverges for some
At criticality the correlation length diverges
GS energy:
scaling hypothesis
Differentiating w.r.t. g
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The singular term appears in every reduced density
matrix containing the sites connected by .
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.
Warning: accidental cancellations may occur depending on
the specific functional form next to leading singularity
The best suited operator for detecting and classifying QPT’s
is V , that naturally contains Moreover, FSS at criticality
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Spin 1 l-D model l D
l = Ising-like D = single ion
Phase
Diagram
Symmetries: U(1)xZ2
In this case
Around the c=1 line:
(sine-Gordon)
Critical
exponents
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Derivative
The same for
Crossing effect
What about local measures
of entanglement?
Using symmetries:
Single-site entropy
Two-sites density matrix contains the same leading singularity
[ L.Campos Venuti, et. al., PRA 73, (R) (2006).]
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[ F.Verstraete, M.Popp, J.I.Cirac, PRL 92, (2004).]
Localizable Entanglement
LE is the maximum amount of entanglement that can
be localized on two q-bits by local measurements. j i
N+2 particle state
Maximum over all local measurement basis
= probability of getting
is a measure of entanglement
(concurrence)
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[ L. Campos Venuti, M. Roncaglia, PRL 94, (2005).]
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
Ising model
Quantum XXZ chain
MPS (AKLT)
LE = max of correlation
LE = string correlations 1 :
The lower bound is attained
The LE shows that spin 1 are
perfect quantum channels but is insensitive to phase transitions.
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A spin-1 model: AKLT
=Bell state
Optimal basis:
Infinite entanglement length but finite correlation length
Actually in S=1 case LE is related to string correlation
Typical configurations
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Conclusions
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)
Apart from accidental cancellations all the scaling properties of local entanglement come from the most relevant (RG) scaling operator.
The most natural local quantity is , where g is the driving parameter
across the QPT.
it shows a crossing effect
it is unique and generally applicable
Advantages:
Localizable Entanglement  It is related to some already known correlation functions. It promotes S=1 chains as perfect quantum channels.
Open problem: Hard to define entanglement for multipartite systems,
separating genuine quantum correlations and classical ones.
References:
L.Campos Venuti, C.Degli Esposti Boschi, M.Roncaglia, A.Scaramucci, PRA 73, (R) (2006).
L.Campos Venuti and M. Roncaglia, PRL 94, (2005).
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The End
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