Entanglement and Area

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Entanglement and Area
Imperial College London
Cambridge, 25th August 2004

• Martin Plenio
• Imperial College London

On work with K. Audenaert, M. Cramer, J.
Dreißig, J. Eisert, R.F. Werner
Sponsored by
Royal Society Senior Research Fellowship
QUPRODIS
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Imperial College London
Cambridge, 25th August 2004
The three basic questions of a theory of
entanglement
Provide efficient methods to

• decide which states are entangled and which are
  disentangled (Characterize)
• decide which LOCC entanglement manipulations are
  possible and provide the protocols to
  implement them (Manipulate)
• decide how much entanglement is in a state and
  how efficient entanglement manipulations can
  be (Quantify)

Mathematical characterization of all multi-party
states
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Imperial College London
Cambridge, 25th August 2004
Consider natural states of interacting quantum
systems instead.
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Imperial College London
Cambridge, 25th August 2004
Entanglement in Quantum Many-Body Systems

• Entanglement in infinite interacting harmonic
  systems

• Static Properties Entanglement and Area
• Dynamics of entanglement and long-range
  entanglement

K. Audenaert, J. Eisert, M.B. Plenio and R.F.
Werner, Phys. Rev. A 66, 042327 (2002) M.B.
Plenio, J. Eisert, J. Dreißig and M. Cramer,
quant-ph/0405142 M. Cramer, J. Dreissig, J.
Eisert and M.B. Plenio, in preparation
J. Eisert, M.B. Plenio and J. Hartley,
quant-ph/0311113, to appear in Phys. Rev. Lett.
(2004) M.B. Plenio, J. Hartley and J. Eisert,
New J. Physics. 6, 36 (2004) F. Semião and M.B.
Plenio, quant-ph/0407034

• Entanglement in infinite interacting spin systems

• Entanglement and phase transitions

J.K. Pachos and M.B. Plenio, Phys. Rev. Lett. 93,
056402 (2004) A. Key, D.K.K. Lee, J.K. Pachos,
M.B. Plenio, M. E. Reuter, and E. Rico,
quant-ph/0407121
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Imperial College London
Cambridge, 25th August 2004
Entanglement and Area

• Entanglement in infinite interacting harmonic
  systems

• Static Properties Entanglement and Area
• Dynamics of entanglement and long-range
  entanglement

K. Audenaert, J. Eisert, M.B. Plenio and R.F.
Werner, Phys. Rev. A 66, 042327 (2002) M.B.
Plenio, J. Eisert, J. Dreißig and M. Cramer,
quant-ph/0405142 M. Cramer, J. Dreissig, J.
Eisert and M.B. Plenio, in preparation
J. Eisert, M.B. Plenio and J. Hartley,
quant-ph/0311113, to appear in Phys. Rev. Lett.
(2004) M.B. Plenio, J. Hartley and J. Eisert,
New J. Physics. 6, 36 (2004) F. Semião and M.B.
Plenio, quant-ph/0407034

• Entanglement in infinite interacting spin systems

• Entanglement and phase transitions

J.K. Pachos and M.B. Plenio, Phys. Rev. Lett. 93,
056402 (2004) A. Key, D.K.K. Lee, J.K. Pachos,
M.B. Plenio, M. E. Reuter, and E. Rico,
quant-ph/0407121
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Imperial College London
Cambridge, 25th August 2004
Entanglement properties of the harmonic chain
Arrange n harmonic oscillators on a ring and let
them interact harmonically by springs.
. . .
n - 1
n
. . .
1
2
. . .
K. Audenaert, J. Eisert, M.B. Plenio and R.F.
Werner, Phys. Rev. A 66, 042327 (2002)
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Imperial College London
Cambridge, 25th August 2004
Entanglement properties of the harmonic chain
Arrange n harmonic oscillators on a ring and let
them interact harmonically by springs.
. . .
n - 1
n
. . .
1
2
. . .

V
K. Audenaert, J. Eisert, M.B. Plenio and R.F.
Werner, Phys. Rev. A 66, 042327 (2002)
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Imperial College London
Cambridge, 25th August 2004
Basic Techniques

• Characteristic function

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Imperial College London
Cambridge, 25th August 2004
Basic Techniques

• Characteristic function

• A state is called Gaussian, if and only if its
  characteristic function (or its Wigner function)
  is a Gaussian

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Imperial College London
Cambridge, 25th August 2004
Basic Techniques

• Characteristic function

• A state is called Gaussian, if and only if its
  characteristic function (or its Wigner function)
  is a Gaussian

• Ground states of Hamiltonians quadratic in X and
  P are Gaussian

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Imperial College London
Cambridge, 25th August 2004
Basic Techniques

• Characteristic function

• A state is called Gaussian, if and only if its
  characteristic function (or its Wigner function)
  is a Gaussian

• Ground states of Hamiltonians quadratic in X and
  P are Gaussian

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Imperial College London
Cambridge, 25th August 2004
Basic Techniques

• Characteristic function

• A Gaussian with vanishing firstmoments

• Ground states of Hamiltonians quadratic in X and
  P are Gaussian

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Imperial College London
Cambridge, 25th August 2004
Basic Techniques

• Characteristic function

• A Gaussian with vanishing firstmoments

• Ground states of Hamiltonians quadratic in X and
  P are Gaussian

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Imperial College London
Cambridge, 25th August 2004
Basic Techniques

• Characteristic function

• A Gaussian with vanishing firstmoments

• Ground states of Hamiltonians quadratic in X and
  P are Gaussian

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Imperial College London
Cambridge, 25th August 2004
Entanglement Measures
Entropy of Entanglement
with
Logarithmic Negativity
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Imperial College London
Cambridge, 25th August 2004
Ground State Entanglement in the Harmonic Chain
. . .
n/2 2
n - 1
n/2 1
n
n/2
1
n/2 - 1
2
. . .
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Imperial College London
Cambridge, 25th August 2004
Ground State Entanglement in the Harmonic Chain
. . .
n/2 2
n - 1
n/2 1
n
n/2
1
n/2 - 1
2
. . .
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Cambridge, 25th August 2004
Imperial College London
Ground State Entanglement in the Harmonic Chain
Even versus odd oscillators.
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Cambridge, 25th August 2004
Imperial College London
Ground State Entanglement in the Harmonic Chain
Even versus odd oscillators.
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Cambridge, 25th August 2004
Imperial College London
Ground State Entanglement in the Harmonic Chain
Even versus odd oscillators.
Entanglement proportionalto number of contact
points.
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Imperial College London
Cambridge, 25th August 2004
D-dimensional lattices Entanglement and Area
Entanglement per unit length of boundary red
square and environment versus length of side of
inner square on a 30x30 lattice of oscillators.
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Imperial College London
Cambridge, 25th August 2004
D-dimensional lattices Entanglement and Area
Can prove

• Upper and lower bound on entropy of entanglement
  that are proportional to the number of
  oscillators on the surface
• For ground state for general interactions
• For thermal states for squared interaction
• General shape of the regions

• Classical harmonic oscillators in thermal state

• Valence bond-solids obey entanglement area law

M.B. Plenio, J. Eisert, J. Dreißig and M. Cramer,
quant-ph/0405142
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Imperial College London
Cambridge, 25th August 2004
Why should this be true An Intuition
Can prove this exactly Intuition from squared
interaction.
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Imperial College London
Cambridge, 25th August 2004
Why should this be true An Intuition
Can prove this exactly Intuition from squared
interaction.
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Imperial College London
Cambridge, 25th August 2004
Why should this be true An Intuition
Obtain a simple normal form
Decouple oscillators except on surface
M.B. Plenio, J. Eisert, J. Dreißig and M. Cramer,
quant-ph/0405142
26
Imperial College London
Cambridge, 25th August 2004
Why should this be true An Intuition
Obtain a simple normal form
Disentangle oscillators except on surface
Disentangle
ViaGLOCC
M.B. Plenio, J. Eisert, J. Dreißig and M. Cramer,
quant-ph/0405142
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Imperial College London
Cambridge, 25th August 2004
Why should this be true An Intuition
Can prove this exactly Intuition from amended
interaction.
V
M.B. Plenio, J. Eisert, J. Dreißig and M. Cramer,
quant-ph/0405142
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Imperial College London
Cambridge, 25th August 2004
Why should this be true An Intuition
Can prove this exactly Intuition from amended
interaction.

V

M.B. Plenio, J. Eisert, J. Dreißig and M. Cramer,
quant-ph/0405142
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Imperial College London
Cambridge, 25th August 2004
Why should this be true An Intuition
of independent columns in B proportional to
of oscillators on the surface of A.
A
B

C
B
V
t

M.B. Plenio, J. Eisert, J. Dreißig and M. Cramer,
quant-ph/0405142
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Imperial College London
Cambridge, 25th August 2004
Why should this be true An Intuition
of independent columns in B proportional to
of oscillators on the surface of A. Entropy of
entanglement from by eigenvalues of
which has at most nonzero eigenvalues
A
B

C
B
V
t

M.B. Plenio, J. Eisert, J. Dreißig and M. Cramer,
quant-ph/0405142
31
Imperial College London
Cambridge, 25th August 2004
Why should this be true An Intuition
of independent columns in B proportional to
of oscillators on the surface of A. Entropy of
entanglement from by eigenvalues of
which has at most nonzero eigenvalues
A
B

Only need to bound eigenvalues
C
B
V
t

M.B. Plenio, J. Eisert, J. Dreißig and M. Cramer,
quant-ph/0405142
32
Imperial College London
Cambridge, 25th August 2004
Disentangling also works for thermal states!
Disentangle
ViaGLOCC
Now decoupled oscillators are in mixed state, but
they are NOT entangled to any other oscillator
(only to environment). Then make eigenvalue
estimates to find bounds on entanglement.
M.B. Plenio, J. Eisert, J. Dreißig and M. Cramer,
quant-ph/0405142
33
Imperial College London
Cambridge, 25th August 2004
D-dimensional lattices Entanglement and Area
Can prove

• Upper and lower bound on entropy of entanglement
  that are proportional to the number of
  oscillators on the surface
• For ground state for general interactions
• For thermal states for squared interaction
• General shape of the regions

a

• Classical harmonic oscillators in thermal state

• Valence bond-solids obey entanglement area law

M.B. Plenio, J. Eisert, J. Dreißig and M. Cramer,
quant-ph/0405142
34
Imperial College London
Cambridge, 25th August 2004
D-dimensional lattices Entanglement and Area
For general interaction Entanglement decreases
exponentially
with distance, contribution bounded
Disentangle
ViaGLOCC
M.B. Plenio, J. Eisert, J. Dreißig and M. Cramer,
quant-ph/0405142
35
Imperial College London
Cambridge, 25th August 2004
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Imperial College London
Cambridge, 25th August 2004
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Imperial College London
Cambridge, 25th August 2004
k (3,2)
s(k,l)
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Imperial College London
Cambridge, 25th August 2004
k (3,2)
l (5,6)
s(k,l) (5-3) (6-2)
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Imperial College London
Cambridge, 25th August 2004
k (3,2)
l (5,6)
s(k,l) (5-3) (6-2)
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Imperial College London
Cambridge, 25th August 2004
The upper bound Outline
M.B. Plenio, J. Eisert, J. Dreißig and M. Cramer,
quant-ph/0405142
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Imperial College London
Cambridge, 25th August 2004
The upper bound Outline
M.B. Plenio, J. Eisert, J. Dreißig and M. Cramer,
quant-ph/0405142
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Imperial College London
Cambridge, 25th August 2004
The upper bound Outline
M.B. Plenio, J. Eisert, J. Dreißig and M. Cramer,
quant-ph/0405142
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Imperial College London
Cambridge, 25th August 2004
The upper bound Outline
Number of oscillators with distance r from
surface is proportional to surface ? Area theorem
Summation gives finite result because
M.B. Plenio, J. Eisert, J. Dreißig and M. Cramer,
quant-ph/0405142
44
Imperial College London
Cambridge, 25th August 2004
D-dimensional lattices Entanglement and Area
Can prove

• Upper and lower bound on entropy of entanglement
  that are proportional to the number of
  oscillators on the surface
• For ground state for general interactions
• For thermal states for squared interaction
• General shape of the regions

a
a
a

• Classical harmonic oscillators in thermal state

• Valence bond-solids obey entanglement area law

M.B. Plenio, J. Eisert, J. Dreißig and M. Cramer,
quant-ph/0405142
45
Imperial College London
Cambridge, 25th August 2004
Correlations and Area in Classical Systems
g denotes phase space point
M.B. Plenio, J. Eisert, J. Dreißig and M. Cramer,
quant-ph/0405142
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Imperial College London
Cambridge, 25th August 2004
Correlations and Area in Classical Systems
g denotes phase space point
Entropy depends on fine-graining in phase space
but mutual information is independent of
fine-graining
M.B. Plenio, J. Eisert, J. Dreißig and M. Cramer,
quant-ph/0405142
47
Imperial College London
Cambridge, 25th August 2004
Correlations and Area in Classical Systems
Nearest neighbour interaction

• Expression equivalent to quantum system with
  squared interaction

• Get correlation-area connection for free

• Connection between correlation and area is
  independent of quantum mechanics and
  relativity

M.B. Plenio, J. Eisert, J. Dreißig and M. Cramer,
quant-ph/0405142
48
Imperial College London
Cambridge, 25th August 2004
D-dimensional lattices Entanglement and Area
Can prove

• Upper and lower bound on entropy of entanglement
  that are proportional to the number of
  oscillators on the surface
• For ground state for general interactions
• For thermal states for squared interaction
• General shape of the regions

a
a
a

• Classical harmonic oscillators in thermal state
  Classical correlations are bounded from
  above and below by expressions proportional
  to number of oscillators on surface.
• Proof via quantum systems with squared
  interactions

a

• Valence bond-solids obey entanglement area law

M.B. Plenio, J. Eisert, J. Dreißig and M. Cramer,
quant-ph/0405142
49
Imperial College London
Cambridge, 25th August 2004
Valence bond states
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Imperial College London
Cambridge, 25th August 2004
D-dimensional lattices Entanglement and Area
Can prove

• Upper and lower bound on entropy of entanglement
  that are proportional to the number of
  oscillators on the surface
• For ground state for general interactions
• For thermal states for squared interaction
• General shape of the regions

a
a
a

• Classical harmonic oscillators in thermal state
  Classical correlations are bounded from
  above and below by expressions proportional
  to number of oscillators on surface.
• Proof via quantum systems with squared
  interactions

a
a

• Valence bond-solids obey entanglement area law

M.B. Plenio, J. Eisert, J. Dreißig and M. Cramer,
quant-ph/0405142