Games and Monogamy in the relativistically causal correlations Micha� l Kamo´ n Gda´ nsk University of Technology National Quantum Information Center, University of Gda´ nsk 49 Symposium on Mathematical Physics, Toru´ n 17-18 June 2017 Micha� l Kamo´ n Games and Monogamy in the relativistically causal correlations
Games and Monogamy in the relativistically causal correlations ∗ ∗ Paper in preparation In cooperation with: Roberto Salazar 1,2 , Dardo Goyeneche 4,5 , Karol Horodecki 3 , Debashis Saha 1,2 Ravishankar Ramanathan 6 , Pawel Horodecki 2,4 , 1) Institute of Theoretical Physics and Astrophysics, 2) National Quantum Information Centre, 3) Institute of Informatics Faculty of Mathematics, Physics and Informatics, University of Gda´ nsk, 80-308 Gda´ nsk, Poland 4) Faculty of Applied Physics and Mathematics, Gda´ nsk University of Technology, 80-233 Gda´ nsk, Poland 5) Institute of Physics, Jagiellonian University, 30-059 Krak´ ow 6) Laboratoire d’Information Quantique, Universit´ e Libre de Bruxelles, Belgium Micha� l Kamo´ n Games and Monogamy in the relativistically causal correlations
Games and Monogamy in the relativistically causal correlations Presentation Plan: 1 Local deterministic, quantum and no-signaling measurements 2 Correlations of measurements outcomes 3 Polytope of RC correlations 4 Nonlocal games in RC 5 Security against RC eavesdropper 6 Conclusions Micha� l Kamo´ n Games and Monogamy in the relativistically causal correlations
Local deterministic, quantum and no-signaling measurements General setting 1 [ n ] = { 1 , . . . , n } space-like separated parties 2 x = { x 1 , . . . , x n } inputs strings 3 a = { a 1 , . . . , a n } outputs strings 4 P ( a | x ) joint probability of a given x Micha� l Kamo´ n Games and Monogamy in the relativistically causal correlations
Local deterministic, quantum and no-signaling measurements No-signaling General setting • None measurements x S = { x i } i ∈ S 1 [ n ] = { 1 , . . . , n } space-like separated parties performed locally by any subset of parties 2 x = { x 1 , . . . , x n } inputs strings S ⊆ [ n ] can influence measurement statistics of other parties S c , i.e.: 3 a = { a 1 , . . . , a n } outputs strings P ( aS c | xS c ) = � P ( a ′| x ′ ) = � 4 P ( a | x ) joint probability of a given x P ( a ′′| x ′′ ) a ′ S a ′′ S for all a ′ , a ′′ with a ′ S c = a ′′ S c = a S c and for Local determinism all x ′ , x ′′ with x ′ S c = x ′′ S c = x S c . 1 Each observable has predefined outcome 2 Space-like separation of subsystems ⇓ measurements independence P ( a | x )= � π ( λ ) P ( a 1 | x 1 ,λ ) P ( a 2 | x 2 λ ) ... P ( an | xn ,λ ) λ Quantum measurements 1 Joint state represented by density matrix ρ 2 Measurement is given by Hermitian operator A P ( a | x )= Tr ( A x ρ a ) Micha� l Kamo´ n Games and Monogamy in the relativistically causal correlations
Local deterministic, quantum and no-signaling measurements No-signaling General setting • None measurements x S = { x i } i ∈ S 1 [ n ] = { 1 , . . . , n } space-like separated parties performed locally by any subset of parties 2 x = { x 1 , . . . , x n } inputs strings S ⊆ [ n ] can influence measurement statistics of other parties S c , i.e.: 3 a = { a 1 , . . . , a n } outputs strings P ( aS c | xS c ) = � P ( a ′| x ′ ) = � 4 P ( a | x ) joint probability of a given x P ( a ′′| x ′′ ) a ′ S a ′′ S for all a ′ , a ′′ with a ′ S c = a ′′ S c = a S c and for Local determinism all x ′ , x ′′ with x ′ S c = x ′′ S c = x S c . 1 Each observable has predefined outcome 2 Space-like separation of subsystems ⇓ measurements independence P ( a | x )= � π ( λ ) P ( a 1 | x 1 ,λ ) P ( a 2 | x 2 λ ) ... P ( an | xn ,λ ) λ Quantum measurements 1 Joint state represented by density matrix ρ 2 Measurement is given by Hermitian operator A Figure: Schematic representation of space of local deterministic L , P ( a | x )= Tr ( A x ρ a ) quantum Q and no-signaling N S sets. J. Barrett et al. Phys. Rev. A 71, 022101 (2005). Micha� l Kamo´ n Games and Monogamy in the relativistically causal correlations
Correlations of measurements outcomes “Boxes” - families of joint probability distributions { P ( a | x ) } a , x Two parties, two inputs, two outputs (2,2,2)-scenario � CHSH � := � A 0 B 0 � + � A 0 B 1 � + � A 1 B 0 � − � A 1 B 1 � Figure: Schematic presentation of values of CHSH-like game within local deterministic, quantum and no-signaling theory. R. Gill, Epidemiology Meets Quantum: Statistics, Causality, and Bell’s Theorem, www.slideshare.net/ 1 P. Horodecki , R. Ramanathan , arXiv: 1611.06781 (2016) . Micha� l Kamo´ n Games and Monogamy in the relativistically causal correlations
Correlations of measurements outcomes Relativistic causality 1 “Boxes” - families of joint probability Main assumption: No causal loops! distributions { P ( a | x ) } a , x Two parties, two inputs, two outputs (2,2,2)-scenario � CHSH � := � A 0 B 0 � + � A 0 B 1 � + � A 1 B 0 � − � A 1 B 1 � Figure: Schematic presentation of values of CHSH-like game within local deterministic, quantum and no-signaling theory. R. Gill, Epidemiology Meets Quantum: Statistics, Causality, and Bell’s Theorem, www.slideshare.net/ 1 P. Horodecki , R. Ramanathan , arXiv: 1611.06781 (2016) . Micha� l Kamo´ n Games and Monogamy in the relativistically causal correlations
Correlations of measurements outcomes Relativistic causality 1 “Boxes” - families of joint probability Main assumption: No causal loops! distributions { P ( a | x ) } a , x Two parties, two inputs, two outputs (2,2,2)-scenario � CHSH � := � A 0 B 0 � + � A 0 B 1 � + � A 1 B 0 � − � A 1 B 1 � Figure: Violation of causality by “point to point” signaling in two-party scenario1. Figure: Schematic presentation of values of CHSH-like game within local deterministic, quantum and no-signaling theory. R. Gill, Epidemiology Meets Quantum: Statistics, Causality, and Bell’s Theorem, www.slideshare.net/ 1 P. Horodecki , R. Ramanathan , arXiv: 1611.06781 (2016) . Micha� l Kamo´ n Games and Monogamy in the relativistically causal correlations
Correlations of measurements outcomes Relativistic causality 1 “Boxes” - families of joint probability Main assumption: No causal loops! distributions { P ( a | x ) } a , x Two parties, two inputs, two outputs (2,2,2)-scenario � CHSH � := � A 0 B 0 � + � A 0 B 1 � + � A 1 B 0 � − � A 1 B 1 � Figure: Violation of causality by “point to point” signaling in two-party scenario1. In (2,2,2)-scenario no-signaling conditions are necessary and sufficient conditions for relativistic causality Figure: Schematic presentation of values of CHSH-like game within local deterministic, quantum and no-signaling theory. R. Gill, NS (2 , 2 , 2) ≡ RC (2 , 2 , 2) Epidemiology Meets Quantum: Statistics, Causality, and Bell’s Theorem, www.slideshare.net/ 1 P. Horodecki , R. Ramanathan , arXiv: 1611.06781 (2016) . Micha� l Kamo´ n Games and Monogamy in the relativistically causal correlations
Three party, two input, two output (3,2,2)-scenario General necessary condition for random variable X M to signaling to a correlation of random variables X P and X Q J + [ X P ] ∩ J + [ X Q ] ⊆ J + [ X M ] Figure: A particular spacetime configuration of measurement events in the three-party case, where “point to region” signaling is allowed by RC1. 1 P. Horodecki, R. Ramanathan, arXiv: 1611.06781 (2016) Micha� l Kamo´ n Games and Monogamy in the relativistically causal correlations
Three party, two input, two output (3,2,2)-scenario General necessary condition for random variable X M to signaling to a correlation of random variables X P and X Q J + [ X P ] ∩ J + [ X Q ] ⊆ J + [ X M ] Figure: A particular spacetime configuration of measurement events in the three-party case, where “point to region” signaling is allowed by RC1. Necessary and sufficient conditions for relativistic causality 1 � � a P ( a , b , c | x ′ , y , z ) P ( b , c | y , z ) = a P ( a , b , c | x , y , z ) = ∀ x , x ′ , y , z , b , c � � c P ( a , b , c | x , y , z ′ ) P ( a , b | x , y ) = c P ( a , b , c | x , y , z ) = ∀ z , z ′ , x , y , a , b � � b P ( a , b , c | x , y ′ , z ) P ( a , c | x , z ) = b P ( a , b , c | x , y , z ) = ∀ y , y ′ , x , z , a , c = � � b , c P ( a , b , c | x , y ′ , z ′ ) P ( a | x ) b , c P ( a , b , c | x , y , z ) = ∀ y , y ′ , z , z ′ , x , a = � � a , b P ( a , b , c | x ′ , y ′ , z ) P ( c | z ) a , b P ( a , b , c | x , y , z ) = ∀ x , x ′ , y , y ′ , z , c 1 P. Horodecki, R. Ramanathan, arXiv: 1611.06781 (2016) Micha� l Kamo´ n Games and Monogamy in the relativistically causal correlations
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