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

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

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

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

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

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

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

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

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

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

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

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

16. 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

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

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

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

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

21. Evaporation

22. MEASUREMENTS Summary 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