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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
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Kish Island, 9th Sep 2007
http://www.imperial.ac.uk/quantuminformation
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Entanglement Theory
Part IEntanglement and Quantum-Many-Body Systems
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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
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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
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Block Entropies in 2-D
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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
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“Fermi surfaces”
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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)
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The classical state space is small…
Hilbert space is really large …
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The classical state space is small …
Hilbert space is really large …
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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
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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
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DMRG & PEPS
} } } } }
Two virtual particlesper physical site
Two virtual particlesper physical site
Two virtual particlesper physical site
Two virtual particlesper physical site
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
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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
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Ising Model 1-D, 2-D, 3-D
Estimate position of critical point of 30 latticeD
1D
2D 3D
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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
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Quantum – Classical boundary
Simulation on a classical computerrequires exponential resources
Efficient description on a classical computer is possible
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Entanglement Theory
Part II
Creating Quantum-Many-Body Systems
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The basic set-up
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The basic set-up
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Putting non-linearities: The basic set-up
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Putting non-linearities: The basic set-up
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Hartmann, Brandao, Plenio, Nature Phys. 2, 849 (2006); Hartmann, Plenio, PRL 99, (2007) & Hartmann, Brandao, Plenio, PRL 99, (2007)
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Effective Dynamics & Polaritons
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Effective Dynamics & Polaritons
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Effective Dynamics & Polaritons
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Non-linearities in the Polariton Picture
Gives rise to dispersive term
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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
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Exact dynamics with losses
+ N +
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Numerical simulation of phase transition
Difference with exact BH dynamics
Difference with exact BH dynamics Polariton number fluctuations
Loss of norm vs number fluctuations
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Phase Diagram of our model for finite N
D. Rossini and R. Fazio, quant-ph/0705.1062
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Spin models
Hartmann, Brandão & Plenio, quant-ph/0704.3056 to appear in PRL
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Phase interfaces
Mott Superfluid
Start with one particle per site
Hartmann & Plenio, cond-mat/0708.2667
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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
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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
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Entanglement Theory
Part IV
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Quantum Information LIVE
Live broadcasts of all QI seminars at Imperial.
Launched 8 Jan 2007 with talk by G. Milburn.
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Live questions possible via Skype.
Recordings also available:
Sample of recording of S. Popescu
Sample of recording of S. Popescu
Want to expand to form web-TV channel
for QI community.
For more, see www.imperial.ac.uk/quantuminformation
Quantum Information LIVE
quantuminformationLIVE: 17:57:45
questions?
Keith Schwab 17:58:00 Question: is it possible to cool a low frequency cantilever coupled to a microwave cavity...similar to the cooling of a flexible mirror in cavity as has been recently demonstrated by a number of groups