1. Quantum Non-locality: From Bell to Information Causality Alex Thompson Physics 486 March 7, 2016

2. Overview EPR Bell --> CHSH inequalities, violations Tsirelson Bound Information Causality IC experiment

3. Sources of Interest B.S. Cirel'son, Quantum generalizations of Bell's inequality. Letters in Mathematical Physics 4, 93-100. 1980. Pawlowski, M., Paterek, T., Kaszlikowski, D., Scarani, V., Winter, A., & Zdotukowski, M. Information Causality as a Physical Principle. Nature 461, 1101- 1104 (2009) | doi:10.1038/nature08400 Popescu, S. & Rohrlich, D. Quantum Nonlocality as an Axiom. Mar 1994. Foundations of Physics. 24, 3, p. 379-385. Popescu, S. Nonlocality Beyond Quantum Mechanics. Nature Physics 10, 264–270 (2014) doi:10.1038/nphys2916 Ringbauer, M., Fedrizzi, A., Berry, D. W., & White, A. G. Information Causality in the Quantum and Post-Quantum Regime. Scientific Reports 4, Article number 6955 (2014). doi:10.1038/srep06955 (as well as linked supplemental information).

4. EPR, 1935 Concerned with “Spooky action at a distance” … a form of correlation.... Example system: Pion decay SzSz

5. Setting Alice and Bob are space-like separated. They each measure one of two spin components of a spin-1/2 particle. (A,A’,B,B’) They get results a and b, respectively. Given what they measure, what are the constraints on their results? A B Image: http://www.nature.com/nphys/journal/v10/n4/box/nphys2916_BX2.html

6. Bell, 1964 Quantified EPR’s concerns: Local hidden variable theories satisfy inequality, QM does not! Correlation of spin projections

7. CHSH, 1969 Generalization of Bell’s Inequality Reduces to Bell’s Inequality when A’ = B. Quantum mechanics violates it, but only up to a point…

8. Tsirelson’s Bound, 1980 S CHSH ≤ 2 violated by QM S Q ≤ 2√2 obeyed Given A i,B i observables, A i 2 = I, B i 2 = I;...Interpretable as spin components

9. PR Boxes Is 2√2 the maximal violation? No! Relativistic constraint requires S CHSH ≤ 4, the algebraic maximum (1+1+1-(-1) = 4) (Three sets of measurements fully correlated, one fully anti- correlated)

10. PR Boxes Popescu and Rohrlich (1994) introduce shared non-signaling resource: Assigns probabilities that respect relativistic causal constraints:

11. Information Causality From: Pawlowski, et. al. (2009)

12. Information Causality Alice has some number of bits, K. Bob receives m classical bits from Alice. They can share any non-signaling resources (PR boxes allowed) Number of bits Bob can learn (mutual information between his bits and Alice’s) ≤ m When Alice and Bob respect IC, their correlations can reach, but do not violate Tsirelson’s bound! (IC is sufficient for QM correlations… but not necessary)

13. Information Causality Standard protocol for violating IC with super-quantum correlations Bob can guess either of Alice’s bits at will!

14. Non-signaling Polytope Image: http://journals.aps.org/rmp/pdf/10.1103/RevModPhys.86.419 Local Constrained by: Bell-CHSH inequalities Quantum Constrained by: Tsirelson’s Bound & Information Causality Non-Signaling Constrained by algebra of correlations / relativity

15. Experiment: Ringbauer et. al. 2014 Simulate classical, quantum, super-quantum correlations. Use polarization-dependent loss in a photonic system. Image: http://www.nature.com/articles/srep06955

16. Experiment: Ringbauer et. al. 2014 Lines correspond to number of bits transferred in Information Causality protocol/game. Tsirelson’s bound is respected: efficiency of information transfer (η) cannot be higher than 1 when correlations are lower than S Q. (note rescaling) Image: http://www.nature.com/articles/srep06955

17. Conclusion Fundamental research taking place at the boundaries of QM. Characterizing what it means for a system to obey quantum mechanics. Tsirelson’s bound provides a good window into this field.

18. Additional References Barrett, J., Linden, N., Massar, S., Pironio, S., Popescu, S., Roberts, D. Nonlocal Correlations as an Information Theoretic Resource. Phys. Rev. A 71, 022101 (2005) http://arXiv:quant-ph/0404097v1 Catalano, Jesse L., Spontaneous Parametric Down-Conversion and Quantum Entanglement (2014). University Honors Theses. Paper 52. http://pdxscholar.library.pdx.edu/cgi/ viewcontent.cgi?article=1088&context=honorstheses Jonathan Allcock, J., Brunner N., Pawlowski, M., & Scarani, V. Recovering part of the boundary between quantum and nonquantum correlations from information causality. Phys. Rev. A 80, 040103(R). (2009). http://arxiv.org/abs/0906.3464v3 https://en.wikipedia.org/wiki/Quantum_nonlocality https://en.wikipedia.org/wiki/Tsirelson%27s_bound http://journals.aps.org/rmp/pdf/10.1103/RevModPhys.86.419 -- The review I wish I had found sooner!!!