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PRESENTS

DANNY TERNO & ETERA LIVINE. PRODUCTION. With contributions from Asher Peres, Viqar Hussain and Oliver Winkler. PRESENTS. B. A. L. K. H. C. E. O. L. when. T. M. E. N. E. N. T. '. L. E. m et. Outline. Volume 1: new properties. Noncovariance of reduced density matrices

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PRESENTS

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  1. DANNY TERNO & ETERA LIVINE PRODUCTION With contributions from Asher Peres, Viqar Hussain and Oliver Winkler PRESENTS

  2. B A L K H C E O L when T M E N E N T ' L E met

  3. Outline Volume1:newproperties • Noncovariance of reduced density matrices • Noninvariance of entropy • Implications to holography and thermodynamics Volume2:old applications • Entanglement and black hole entropy • Entanglement and Hawking radiation

  4. Volume1: new properties

  5. spin momentum classical info Noncovariance (1) Example: Lorentz transform of a single massive particle transform:v along thez-axis important parameter:

  6. Partial trace is not Lorentz covariant Spin entropy is not scalar Distinguishability depends on motion Peres and Terno,Rev. Mod. Phys. 76, 93 (2004) Entropy

  7. Noncovariance (2) Geometric entropy here there there trace out “there” Bombelli et al, Phys. Rev. D34, 373 (1986) Holzhey, Larsen and Wilczek, Nucl. Phys. B424, 443 (1994) Callan and Wilczek, Phys. Lett. B333, 55 (1994).

  8. Decomposition of Lorentz transformations ? = trivial 1D rep irrep of 1-particle states not irreducible no correlations no Bell-type violations Terno, Phys. Rev. Lett. 93, 051303 (2004) Transformations do not split into here and there spaces

  9. Noninvariance(1) Boundary conditions & cut-offs Bekenstein, Lett. Nuovo Cim. 4, 737 (1972) …. Busso, Rev. Mod. Phys. 74,825 (2002) Model Yurtsever, Phys. Rev. Lett. 91, 041302 (2003) Number of degrees of freedom N is frame-dependent

  10. Spacelike holographic bound Lorentz boost: factors 1/γ both area and entropy change Saved ? Terno, Phys. Rev. Lett. 93, 051303 (2004) v

  11. - Black holes: invariance Two observers with a relative boost Hawking’s area theorem Model 1+1 calculations: the same crossing point, relative boost Fiola, Preskill, Strominger, Trivedi, Phys. Rev. D 59, 3987 (1994)

  12. Noninvariance(2) Accelerated cavity Moore, J. Math. Phys. 11, 2679 (1970) Levin, Peleg, Peres, J.Phys.A 25, 6471 (1992) Accelerated observers & matter beyond the horizon Terno, Phys. Rev. Lett. 93, 051303 (2004)

  13. Volume2: old applications

  14. Entanglement on the horizon Object: static black hole States: spin network that crosses the horizon Requirement: SU(2) invariance of the horizon states Qubit BH

  15. Standard counting story area 2nspins constraint number of states entropy Fancy counting story density matrix entropy

  16. Entanglement Measure: entanglement of formation 2 vs 2n-2 States of the minimal decomposition degeneracy indices Alternative decomposition: linear combinations Its reduced density matrices: mixtures Entropy: concavity

  17. unentangled fraction entanglement n vs n Entropy of the whole vs. sum of its parts reduced density matrices BH is not made from independent qubits, but… Livine and Terno, gr-qc/0412xxx

  18. Entanglement and Hawking radiation Hussein,Terno and Winkler, in preparation

  19. T M E N E N T ' L E when B A L K H C E O L met Summary • Reduced density matrices are not covariant • Entropy (and the # of degrees of freedom) are observer-dependent • Entanglement is responsible for the logarithmic corrections of BH entropy • Entropy of the BH radiation = entanglement entropy between gravity and matter

  20. Thanks to Jacob Bekenstein Ivette Fuentes-Schuller Florian Girelli Netanel Lindner Rob Myers Johnathan Oppenheim David Poulin Terry Rudolph Frederic Schuller Lee Smolin Rafael Sorkin Rowan Thomson

  21. renormalization of entropy Technique: Entropy usually diverges Unruh effect General: cut-off

  22. Unruh+ Matter outside the horizon n particles in the mode (k,m) Audretsch and Müller, Phys. Rev. D 49, 4056 (1994) Splitting: usual + super

  23. Two observers: the same acceleration, different velocities Special case renormalized quantities temperature Of what? two subsystems General case: temperature is undefined

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