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Published byReynard Carr Modified over 9 years ago
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Quantum info tools & toys for quantum gravity LOOPS `05 Daniel Terno Perimeter Institute
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MEASUREMENTS Outline POVM Information gain DYNAMICS Completely positive maps Non completely positive maps ENTANGLEMENT Entang’t 101 BH applications
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POVM discrete continuous Projections/von Neumann Realization ancillary system+ unitary evolution+ PVM Moments MEASUREMENTS
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Construction: covariance considerations and /or optimization Use: decision/identification unsharp properties non-commuting variables/ phase space observables Coexistence & uncertainty
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Classical geometry 6 edges 3 edges, 3 angles 3 areas, 3 dihedral angles 4 areas, 2 dihedral angles 3 edges, 3 products Volume TETRAHEDRON
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5 commuting observables Standard uncertainty relation Basis: eigenvectors of Quantum mechanics
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Question How uncertain is the shape and how this uncertainty decreases in the classical limit? Observation 1 [numeric] Naïve bound Observation 2
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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
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Priors At least two natural probability distributions or Fixing 4 areas
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Encoding & distance Condition Figure of merit POVM Spin POVM
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(1,1,1,1) tetrahedron ILLUSTRATION Optimization: Constraint: Independent variables: phases
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Unitary Completely positive Def: unital map Definition: Physics: DYNAMICS
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Unitary evolution & partial trace Non completely positive Physically acessible
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Causal sets CNOT gate Hawkins, Markopoulou, Sahlmann CQG 20, 3839 (2003)
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Causal sets Partial sets: unital CP dynamics? Lemma: physically accessible and unital => CP
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a brief history Ancient times: 1935-1993 “The sole use of entanglement was to subtly humiliate the opponents of QM” Modern age: 1993- Resource of QIT Teleportation, quantum dense coding, quantum computation…. Postmodern age: 1986 (2001)- Entanglement in physics ENTANGLEMENT
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a closer encounter Pure states 0.2 0.4 0.6 0.8 1 1 Mixed states hierarchy Direct product Separable Entangled ENTANGLEMENT
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Entanglement of formation Minimal weighted average entanglement of constituents measures ENTANGLEMENT
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“Good” measures of entanglement: satisfy three axioms Coincide on pure states with Do not increase under LOCC Zero on unentangled states Almost never known
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Entropy and entanglement on the horizon gr-qc/0508085 gr-qc/0505068 Phys. Rev. A 72 022307 (2005) Etera Livine, Tuesday I, 16:00
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Evaporation
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MEASUREMENTS Summary POVM Information gain DYNAMICS Completely positive maps Non completely positive maps ENTANGLEMENT Entang’t 101 BH applications
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Thanks to Hilary Carteret Viqar Husain Netanel Lindner Etera Livine Lee Smolin Oliver Winkler Karol Życzkowski
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