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Quantum info tools & toys for quantum gravity

Quantum info tools & toys for quantum gravity. Daniel Terno. Perimeter Institute. LOOPS `05. Outline. MEASUREMENTS. POVM Information gain. DYNAMICS. Completely positive maps Non completely positive maps. ENTANGLEMENT. Entang’t 101 BH applications. MEASUREMENTS. POVM. discrete.

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Quantum info tools & toys for quantum gravity

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  1. Quantum info tools & toysfor quantum gravity Daniel Terno Perimeter Institute LOOPS `05

  2. Outline MEASUREMENTS POVM Information gain DYNAMICS Completely positive maps Non completely positive maps ENTANGLEMENT Entang’t 101 BH applications

  3. MEASUREMENTS POVM discrete continuous Projections/von Neumann ancillary system+ unitary evolution+ PVM Realization Moments

  4. Construction: covariance considerations and /or optimization Use: decision/identification unsharp properties non-commuting variables/ phase space observables Coexistence & uncertainty

  5. TETRAHEDRON Classical geometry • 6 edges • 3 edges, 3 angles • 3 edges, 3 products • 3 areas, 3 dihedral angles • 4 areas, 2 dihedral angles Volume

  6. Quantum mechanics 5 commuting observables Basis: eigenvectors of Standard uncertainty relation

  7. Question How uncertain is the shape and how this uncertainty decreases in the classical limit? Observation 1 [numeric] Observation 2 Naïve bound

  8. More precise formulation: quantum communication problem 1. Fix the areas 2. Encode the angles 3. Decode 4. Calculate the figure of merit 5. Average over all angles 6. Take the limit

  9. Priors At least two natural probability distributions or Fixing 4 areas

  10. Encoding & distance Condition Figure of merit POVM Spin POVM

  11. ILLUSTRATION (1,1,1,1) tetrahedron Optimization: Constraint: Independent variables: phases

  12. DYNAMICS Unitary Completely positive Definition: Physics: Def: unital map

  13. Non completely positive Unitary evolution & partial trace Physically acessible

  14. Causal sets Hawkins, Markopoulou, Sahlmann CQG 20, 3839 (2003) CNOT gate

  15. Causal sets Partial sets: unital CP dynamics? Lemma: physically accessible and unital => CP

  16. ENTANGLEMENT a brief history Ancient times: 1935-1993 “The sole use of entanglement was to subtly humiliate the opponents of QM” Modernage: 1993- Resource of QIT Teleportation, quantum dense coding, quantum computation…. Postmodern age: 1986 (2001)- Entanglement in physics

  17. 1 0.8 0.6 0.4 0.2 1 0.4 0.6 0.2 0.8 ENTANGLEMENT a closer encounter Purestates Mixed states hierarchy Direct product Separable Entangled

  18. ENTANGLEMENT Entanglement of formation measures Minimal weighted average entanglement of constituents

  19. Coincide on pure states with Zero on unentangled states “Good” measures of entanglement: satisfy three axioms Almost never known Do not increase under LOCC

  20. Entropy and entanglementon the horizon gr-qc/0508085 gr-qc/0505068 Phys. Rev. A 72 022307 (2005) Etera Livine, Tuesday I, 16:00

  21. Evaporation

  22. Summary MEASUREMENTS POVM Information gain DYNAMICS Completely positive maps Non completely positive maps ENTANGLEMENT Entang’t 101 BH applications

  23. Thanks to Hilary Carteret Viqar Husain Netanel Lindner Etera Livine Lee Smolin Oliver Winkler Karol Życzkowski

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