# Entanglement and Area

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Entanglement and Area

Martin Plenio

Imperial College London

Imperial College London Cambridge, 25th August 2004

Sponsored by:Royal Society Senior Research Fellowship

On work with K. Audenaert, M. Cramer, J. Dreißig, J. Eisert, R.F. Werner

The three basic questions of a theory of entanglement

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)

Provide efficient methods to

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Mathematical characterization of all multi-party states

Cambridge, 25th August 2004

Imperial College London

Consider natural states of interacting quantum systems instead.

Cambridge, 25th August 2004

Entanglement in Quantum Many-Body Systems

Imperial College London

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/0405142M. 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 harmonic systems

Cambridge, 25th August 2004

Entanglement in infinite interacting spin systems

Entanglement and phase transitionsJ.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

Entanglement and Area

Imperial College London

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/0405142M. 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 harmonic systems

Cambridge, 25th August 2004

Entanglement in infinite interacting spin systems

Entanglement and phase transitionsJ.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

Entanglement properties of the harmonic chain

Arrange n harmonic oscillators on a ring and let them interact harmonically by springs. . . .

. . .

K. Audenaert, J. Eisert, M.B. Plenio and R.F. Werner, Phys. Rev. A 66, 042327 (2002)

Imperial College London Cambridge, 25th August 2004

1

2

n

n - 1

. . .

Entanglement properties of the harmonic chain

Arrange n harmonic oscillators on a ring and let them interact harmonically by springs. . . .

. . .

K. Audenaert, J. Eisert, M.B. Plenio and R.F. Werner, Phys. Rev. A 66, 042327 (2002)

Imperial College London Cambridge, 25th August 2004

V

}1

2

n

n - 1

. . .

Basic Techniques

Imperial College London Cambridge, 25th August 2004

Characteristic function

Basic Techniques

Imperial College London Cambridge, 25th August 2004

Characteristic function

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

Basic Techniques

Imperial College London Cambridge, 25th August 2004

Characteristic function

Ground states of Hamiltonians quadratic in X and P are Gaussian

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

Basic Techniques

Imperial College London Cambridge, 25th August 2004

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

Basic Techniques

Imperial College London Cambridge, 25th August 2004

Characteristic function

A Gaussian with vanishing firstmoments

Ground states of Hamiltonians quadratic in X and P are Gaussian

Basic Techniques

Imperial College London Cambridge, 25th August 2004

Characteristic function

Ground states of Hamiltonians quadratic in X and P are Gaussian

A Gaussian with vanishing firstmoments

Basic Techniques

Imperial College London Cambridge, 25th August 2004

Characteristic function

Ground states of Hamiltonians quadratic in X and P are Gaussian

A Gaussian with vanishing firstmoments

Entanglement Measures

Imperial College London Cambridge, 25th August 2004

Entropy of Entanglement:

with

Logarithmic Negativity:

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Ground State Entanglement in the Harmonic Chain

Cambridge, 25th August 2004

. . .

1

2

n/2

n/2 - 1

. . .

n/2 + 1

n/2 + 2

n

n - 1

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Ground State Entanglement in the Harmonic Chain

Cambridge, 25th August 2004

. . .

1

2

n/2

n/2 - 1

. . .

n/2 + 1

n/2 + 2

n

n - 1

Ground State Entanglement in the Harmonic Chain

Even versus odd oscillators.

Cambridge, 25th August 2004Imperial College London

Ground State Entanglement in the Harmonic Chain

Even versus odd oscillators.

Cambridge, 25th August 2004Imperial College London

Ground State Entanglement in the Harmonic Chain

Even versus odd oscillators.

Cambridge, 25th August 2004Imperial College London

Entanglement proportionalto number of contact points.

Imperial College London

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.

Cambridge, 25th August 2004

Imperial College London

D-dimensional lattices: Entanglement and Area

Cambridge, 25th August 2004

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

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

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Why should this be true: An Intuition

Cambridge, 25th August 2004

Can prove this exactly: Intuition from squared interaction.

Imperial College London

Why should this be true: An Intuition

Cambridge, 25th August 2004

Can prove this exactly: Intuition from squared interaction.

Imperial College London

Why should this be true: An Intuition

Cambridge, 25th August 2004

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

Obtain a simple normal form

Decouple oscillators except on surface

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Why should this be true: An Intuition

Cambridge, 25th August 2004

ViaGLOCC

Disentangle

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

Obtain a simple normal form

Disentangle oscillators except on surface

Imperial College London

Why should this be true: An Intuition

Cambridge, 25th August 2004

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

Can prove this exactly: Intuition from amended interaction.

V=

Imperial College London

Why should this be true: An Intuition

Cambridge, 25th August 2004

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

Can prove this exactly: Intuition from amended interaction.

V=

…

…

Imperial College London

Why should this be true: An Intuition

Cambridge, 25th August 2004

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

V =

…

…

A

C

B

Bt

# of independent columns in B proportional to # of oscillators on the surface of A.

Imperial College London

Why should this be true: An Intuition

Cambridge, 25th August 2004

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

V =

…

…

A

C

B

Bt

# 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

Imperial College London

Why should this be true: An Intuition

Cambridge, 25th August 2004

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

V =

…

…

A

C

B

Bt

Only need to bound eigenvalues

# 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

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Disentangling also works for thermal states!

Cambridge, 25th August 2004

ViaGLOCC

Disentangle

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

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.

Imperial College London

D-dimensional lattices: Entanglement and Area

Cambridge, 25th August 2004

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

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

Imperial College London

D-dimensional lattices: Entanglement and Area

Cambridge, 25th August 2004

ViaGLOCC

Disentangle

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

For general interaction: Entanglement decreases exponentially with distance, contribution bounded

Imperial College London Cambridge, 25th August 2004

Imperial College London Cambridge, 25th August 2004

Imperial College London Cambridge, 25th August 2004

k = (3,2)

s(k,l) =

Imperial College London Cambridge, 25th August 2004

k = (3,2)

l = (5,6)

s(k,l) = (5-3) + (6-2)

Imperial College London Cambridge, 25th August 2004

k = (3,2)

l = (5,6)

s(k,l) = (5-3) + (6-2)

Imperial College London

The upper bound: Outline

Cambridge, 25th August 2004

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

Imperial College London

The upper bound: Outline

Cambridge, 25th August 2004

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

Imperial College London

The upper bound: Outline

Cambridge, 25th August 2004

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

Imperial College London

The upper bound: Outline

Cambridge, 25th August 2004

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

Number of oscillators with distance r from surface is proportional to surface Area theorem

Summation gives finite result because

Imperial College London

D-dimensional lattices: Entanglement and Area

Cambridge, 25th August 2004

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

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

Imperial College London

Correlations and Area in Classical Systems

Cambridge, 25th August 2004

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

denotes phase space point

Imperial College London

Correlations and Area in Classical Systems

Cambridge, 25th August 2004

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

denotes phase space point

Entropy depends on fine-graining in phase space

but mutual information is independent of fine-graining

Imperial College London

Correlations and Area in Classical Systems

Cambridge, 25th August 2004

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

Nearest neighbour interaction

Expression equivalent to quantum system with ‘squared interaction’

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

Get correlation-area connection for free

Imperial College London

D-dimensional lattices: Entanglement and Area

Cambridge, 25th August 2004

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

Can prove:

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’

Valence bond-solids obey entanglement area law

Imperial College London

Valence bond states

Cambridge, 25th August 2004

Imperial College London

D-dimensional lattices: Entanglement and Area

Cambridge, 25th August 2004

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

Can prove:

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’

Valence bond-solids obey entanglement area law