Entanglement, Scaling Laws & Coupled Arrays of Micro Cavities

1. Imperial College London Entanglement, Scaling Laws & Coupled Arrays of Micro Cavities 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

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

3. Entanglement Theory Part I Entanglement and Quantum-Many-Body Systems

4. 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

5. Block Entropies in 1-D Critical Systems N ~ log L Logarithmic divergencein 1-D systems Fermions give the same divergence Wolf, Korepin etc Audenaert, Eisert, Plenio & Werner, PRA (2002); Cramer, Eisert and Plenio, Cramer, Eisert, Plenio, PRL 98, 220603 (2007)

6. Block Entropies in 2-D

7. Block Entropies in 2-D Fourier Transformation Vanishing contributionin the field limit Bosonic field limit Few critical chains Logarithmic correctionto area law persists Fermionic criticalsystems Finite Fermi surface

8. “Fermi surfaces”

9. Correlations and Area in Classical Systems 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 finite ranged interactions • General shape of the regions • Plenio, Eisert, Dreissig, Cramer, Phys. Rev. Lett. 94, 060503 (2005) • Cramer, Eisert, Plenio & Dreissig, Phys. Rev. A 73, 012309 (2006) • Classical harmonic oscillators in thermal state: Classical correlations obey the same area law • Cramer, Eisert, Plenio & Dreissig, Phys. Rev. A 73, 012309 (2006) • 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)

10. The classical state space is small… Hilbert space is really large …

11. The classical state space is small … Hilbert space is really large …

12. 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) Efficient description on classical computer my be possible Fast growth of entropy (volume scaling) Accurate description is much harder

13. Approximating Ground States Variational approaches Define a class of states such that we can • efficient compute reduced density matrices • improve state approximation efficiently • describe all states in principle

14. DMRG & PEPS Variational approaches 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 systems Dynamics/Algorithms are hard SR White, PRL 1992; Ostlund & Rommer, PRL 1995

15. DMRG & PEPS Quantum States with Long-Range Correlations Variational approaches Start with product state and apply arbitrary sequence of CPhase gates Arbitrary patterns in arbitrary dimensions including long range correlations can be encoded in a weighted graph state Anders, Plenio, Verstraete, Dur & Briegel, PRL 2006

16. Ising Model 1-D, 2-D, 3-D 1D 2D 3D D Estimate position of critical point of 30 lattice

17. Ising Model 1-D, 2-D, 3-D Combine DMRG and WGS to form RAGE Take MPS state and apply abritrary sequence of CPhase gatesto obtain: Plenio & Eisert, in preparation

18. Quantum – Classical boundary Efficient description on a classical computer is possible Simulation on a classical computerrequires exponential resources

19. Entanglement Theory Part II Creating Quantum-Many-Body Systems

20. The basic set-up

21. The basic set-up

22. Putting non-linearities: The basic set-up +

23. 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)

24. Effective Dynamics & Polaritons +

25. Effective Dynamics & Polaritons

26. Effective Dynamics & Polaritons +

27. Non-linearities in the Polariton Picture Gives rise to dispersive term

28. Hopping in the Polariton Picture Photon hopping:c for |2wa| Energy separation between polariton species < < turns into polariton hopping U > 0 repulsive Bose-Hubbard U < 0 attractive Bose-Hubbard

29. Exact dynamics with losses + N +

30. Numerical simulation of phase transition Difference with exact BH dynamics Polariton number fluctuations Difference with exact BH dynamics Loss of norm vs number fluctuations

31. Phase Diagram of our model for finite N D. Rossini and R. Fazio, quant-ph/0705.1062

32. Spin models Hartmann, Brandão & Plenio, quant-ph/0704.3056 to appear in PRL

33. Phase interfaces Start with one particle per site Mott Superfluid Hartmann & Plenio, cond-mat/0708.2667

34. 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

35. 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

36. Entanglement Theory Part IV

37. Live broadcasts of all QI seminars at Imperial. Launched 8 Jan 2007 with talk by G. Milburn. Quantum Information LIVE

38. 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