# Quantum info tools toys for quantum gravity Daniel

- Slides: 23

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 continuous Projections/von Neumann Realization Moments ancillary system+ unitary evolution+ PVM

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

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

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

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

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

Priors At least two natural probability distributions or Fixing 4 areas

Encoding & distance Condition Figure of merit POVM Spin POVM

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

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

Non completely positive Unitary evolution & partial trace Physically acessible

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

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

ENTANGLEMENT 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 a closer encounter Pure states 1 0. 8 0. 6 0. 4 0. 2 0. 4 0. 6 Mixed states hierarchy Direct product Separable Entangled 0. 8 1

Entanglement of formation ENTANGLEMENT measures Minimal weighted average entanglement of constituents

Coincide on pure states with n w o n k Zero on unentangled states “Good” measures of entanglement: satisfy three axioms r e v e n t s o m Al Do not increase under LOCC

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

Evaporation

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

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