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Entanglement and Area

Imperial College London. Cambridge, 25th August 2004. Entanglement and Area. On work with K. Audenaert, M. Cramer, J. Drei ßig, J. Eisert, R.F. Werner. Martin Plenio Imperial College London. Sponsored by:. Royal Society Senior Research Fellowship. QUPRODIS. Imperial College London.

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Entanglement and Area

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  1. Imperial College London Cambridge, 25th August 2004 Entanglement and Area On work with K. Audenaert, M. Cramer, J. Dreißig, J. Eisert, R.F. Werner Martin Plenio Imperial College London Sponsored by: Royal Society Senior Research Fellowship QUPRODIS

  2. 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

  3. Imperial College London Cambridge, 25th August 2004 Consider natural states of interacting quantum systems instead.

  4. 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

  5. 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

  6. 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)

  7. 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)

  8. Imperial College London Cambridge, 25th August 2004 Basic Techniques • Characteristic function

  9. 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

  10. 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

  11. 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

  12. 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

  13. 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

  14. 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

  15. Imperial College London Cambridge, 25th August 2004 Entanglement Measures Entropy of Entanglement: with Logarithmic Negativity:

  16. Imperial College London Cambridge, 25th August 2004 Ground State Entanglement in the Harmonic Chain . . . n/2 + 2 n - 1 n n/2 + 1 n/2 1 n/2 - 1 2 . . .

  17. Imperial College London Cambridge, 25th August 2004 Ground State Entanglement in the Harmonic Chain . . . n/2 + 2 n - 1 n n/2 + 1 n/2 1 n/2 - 1 2 . . .

  18. Imperial College London Cambridge, 25th August 2004 Ground State Entanglement in the Harmonic Chain Even versus odd oscillators.

  19. Imperial College London Cambridge, 25th August 2004 Ground State Entanglement in the Harmonic Chain Even versus odd oscillators.

  20. Imperial College London Cambridge, 25th August 2004 Ground State Entanglement in the Harmonic Chain Even versus odd oscillators. Entanglement proportionalto number of contact points.

  21. 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.

  22. 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

  23. Imperial College London Cambridge, 25th August 2004 Why should this be true: An Intuition Can prove this exactly: Intuition from squared interaction.

  24. Imperial College London Cambridge, 25th August 2004 Why should this be true: An Intuition Can prove this exactly: Intuition from squared interaction.

  25. 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

  27. 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

  28. 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

  29. 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 … B C V = t … M.B. Plenio, J. Eisert, J. Dreißig and M. Cramer, quant-ph/0405142

  30. 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 … B C 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 B C 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

  36. Imperial College London Cambridge, 25th August 2004

  37. Imperial College London Cambridge, 25th August 2004 k = (3,2) s(k,l) =

  38. Imperial College London Cambridge, 25th August 2004 k = (3,2) l = (5,6) s(k,l) = (5-3) + (6-2)

  39. Imperial College London Cambridge, 25th August 2004 k = (3,2) l = (5,6) s(k,l) = (5-3) + (6-2)

  40. Imperial College London Cambridge, 25th August 2004 The upper bound: Outline M.B. Plenio, J. Eisert, J. Dreißig and M. Cramer, quant-ph/0405142

  41. Imperial College London Cambridge, 25th August 2004 The upper bound: Outline M.B. Plenio, J. Eisert, J. Dreißig and M. Cramer, quant-ph/0405142

  42. Imperial College London Cambridge, 25th August 2004 The upper bound: Outline M.B. Plenio, J. Eisert, J. Dreißig and M. Cramer, quant-ph/0405142

  43. 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

  46. 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

  50. 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

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