Entanglement, Scaling Laws & Coupled Arrays of Micro Cavities

Entanglement, Scaling Laws & Coupled Arrays of Micro Cavities

Imperial College London

Institute for Mathematical Sciences, 53 Princes Gate, Exhibition Road, Imperial College London, London SW7 2PG

&

Quantum Optics and Laser Science Group, Blackett Laboratory, Prince Consort Road, Imperial College London, London SW7 2BW

http://www.imperial.ac.uk/quantuminformation

Martin B Plenio

Kish Island, 9th Sep 2007

http://www.imperial.ac.uk/quantuminformation


Entanglement Theory

Part IEntanglement and Quantum-Many-Body Systems


How does entanglement scale with size of region?

Audenaert, Eisert, Plenio & Werner, PRA 2002; Plenio, Eisert, Dreissig, Cramer, PRL 2005; Cramer, Eisert & Plenio, PRA 2006; Cramer, Eisert & Plenio, PRL 2007


Block Entropies in 1-D Critical Systems

Audenaert, Eisert, Plenio & Werner, PRA (2002); Cramer, Eisert and Plenio, Cramer, Eisert, Plenio, PRL 98, 220603 (2007)

Logarithmic divergencein 1-D systems

N ~ log L

Fermions give the same divergence

Wolf, Korepin etc


Block Entropies in 2-D


Block Entropies in 2-D

Fourier T

ransformation

Bosonic field limit Few critical chains Vanishing contribution

in the field limit

Fermionic criticalsystems

Finite Fermi surfaceLogarithmic correction

to area law persists


“Fermi surfaces”


Correlations and Area in Classical SystemsCan 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 finite ranged interactions General shape of the regions

Classical harmonic oscillators in thermal state: Classical correlations obey the same area law

Field limit 1-D yields logarithmic divergence 2-D bosons yield entanglement ~ area again 2-D fermions yield log-correction

Audenaert, Eisert, Plenio & Werner, Phys. Rev. A 66, 042327 (2002)

Cramer, Eisert, Plenio, Phys. Rev. Lett. 98, 220603 (2007)

Cramer, Eisert, Plenio & Dreissig, Phys. Rev. A 73, 012309 (2006)

Cramer, Eisert, Plenio & Dreissig, Phys. Rev. A 73, 012309 (2006)

Plenio, Eisert, Dreissig, Cramer, Phys. Rev. Lett. 94, 060503 (2005)


The classical state space is small…

Hilbert space is really large …


The classical state space is small …

Hilbert space is really large …


Why is this interesting ?

Entropy of sub-system quantifies entanglement

Entropy also measures how disorder subsystem is and thusmeasures how much information is required to describe the system accurately.

Slow growth of entropy (saturation, area scaling)

Fast growth of entropy (volume scaling)

Efficient description on classical computer my be possible

Accurate description is much harder


Approximating Ground States

efficient compute reduced density matrices

improve state approximation efficiently

describe all states in principle

Variational approaches

Define a class of states such that we can


DMRG & PEPS

} } } } }

Two virtual particlesper physical site





• Entanglement of block bounded by dimension d of bond area scaling enforced• Correlations drop of exponentially

DMRG works well in 1-D non-critical systemsDynamics/Algorithms are hard

SR White, PRL 1992; Ostlund & Rommer, PRL 1995

Variational approaches


Quantum States with Long-Range Correlations

i

Arbitrary patterns in arbitrary dimensions including long range correlations can be encoded in a weighted graph state

Anders, Plenio, Verstraete, Dur & Briegel, PRL 2006

DMRG & PEPSVariational approaches

Start with product state and apply arbitrary sequence of CPhase gates


Ising Model 1-D, 2-D, 3-D

Estimate position of critical point of 30 latticeD

1D

2D 3D


Ising Model 1-D, 2-D, 3-DCombine DMRG and WGS to form RAGE

Take MPS state and apply abritrary sequence of CPhase gatesto obtain:

Plenio & Eisert, in preparation


Quantum – Classical boundary

Simulation on a classical computerrequires exponential resources

Efficient description on a classical computer is possible


Entanglement Theory

Part II

Creating Quantum-Many-Body Systems


The basic set-up


The basic set-up


Putting non-linearities: The basic set-up

+


Putting non-linearities: The basic set-up

+

Hartmann, Brandao, Plenio, Nature Phys. 2, 849 (2006); Hartmann, Plenio, PRL 99, (2007) & Hartmann, Brandao, Plenio, PRL 99, (2007)


+

Effective Dynamics & Polaritons





+


Non-linearities in the Polariton Picture

Gives rise to dispersive term


Hopping in the Polariton Picture

Photon hopping:c

for |2 Energy separation between polariton species<<

turns into polariton hopping

U > 0 repulsive Bose-Hubbard

U < 0 attractive Bose-Hubbard


Exact dynamics with losses

+ N +


Numerical simulation of phase transition

Difference with exact BH dynamics

Difference with exact BH dynamics Polariton number fluctuations

Loss of norm vs number fluctuations


Phase Diagram of our model for finite N

D. Rossini and R. Fazio, quant-ph/0705.1062


Spin models

Hartmann, Brandão & Plenio, quant-ph/0704.3056 to appear in PRL


Phase interfaces

Mott Superfluid

Start with one particle per site

Hartmann & Plenio, cond-mat/0708.2667


real pred.

Fabry-Perot: 160 5 x 103

Photonic bg: 10 5.5 x 105

MCs @ Imperial: 40 ?

Micro-toroid: 53 5 x 106

SC Cavities 1000?Spillane et al, PRA 2005 Soda et al, Nature Mater 2005Aoki et al, Nature 2006 Schuster et al, Nature 2007

2 / ΓCg


real pred.

Fabry-Perot: 10 10

Photonic bg: 5 4 x 103

MCs @ Imperial: 1 ?

Micro-toroid: 3 1.25 x 105

SC Cavities 50?Spillane et al, PRA 2005 Soda et al, Nature Mater 2005Aoki et al, Nature 2006 Schuster et al, Nature 2007

/ cg


Entanglement Theory

Part IV


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Entanglement, Scaling Laws & Coupled Arrays of Micro Cavities