(Non-)Contextuality of Physical Theories as an Axiom

Transcription

1 (Non-)Contextuality of Physical Theories as an Axiom Simone Severini Department of Computer Science QIP 2011

2 Plan 1. Introduction: non-contextuality 2. Results: a general framework to study non-contextuality; non-contextuality and non-locality 3. Open problems: theoretical; applied; a complexity perspective. Plan:

3 1. Introduction: non-contextuality for 1. classical theories 2. non-signaling theories 3. quantum theory

4 measuring means to ask questions to a system What is your velocity? What is your angular What is the momentum? colour of your eyes? Is your entropy 5 bits?

5 measuring means to ask questions to a system what is your velocity? What is your angular What is the momentum? colour of your eyes? Is your entropy 5 bits?

6 measuring means to ask questions to a system what is your velocity? what is your angular What is the momentum? colour of your eyes? Is your entropy 5 bits?

7 measuring means to ask questions to a system what is your velocity? what is your angular what is the momentum? colour of your eyes? Is your entropy 5 bits?

8 measuring means to ask questions to a system what is your velocity? what is your angular what is the momentum? colour of your eyes? is your entropy 5 bits?

9 measuring means to ask questions to a system what is your velocity? what is your angular what is the momentum? colour of your eyes? is your entropy 5 bits?

10 Question 1 Question 5 Question 2 Question 4 Question 3

11 Question 1 context: a set of mutually compatible questions Question 2 Question 3

12 compatibility: Question 1: Is the colour red? Yes Question 2: What is the suit? Diamonds

13 non-contextuality: answers do not depend on contexts

14 Context 1 Question 1: Is X true? Yes Question 2: Is Y true? Yes non-contextuality: answers do not depend on contexts

15 Context 2 Question 2: Is Y true? Yes non-contextuality: answers do not depend on contexts Question 3: Is Z true? No

16 Context 1 Context 2 Question 1: Is X true? Yes Question 2: Is Y true? Yes non-contextuality: answers do not depend on contexts Question 3: Is Z true? No

17 Context 1 Context 2 Question 1: Is X true? Yes Question 2: Is Y true? Yes non-contextuality: answers do not depend on contexts Question 3: Is Z true? No

18 compatibility structure Question 1 Question 2 Question 3 Question 5 Question 4

19 compatibility structure Question 1 Question 2 Question 3 Question 5 Question 4

20 compatibility structure Question 1 Question 2 Question 3 Question 5 Question 4

21 compatibility structure Question 1 Question 2 Question 3 Question 5 Question 4

22 compatibility structure Question 1 Question 2 Question 3 Question 5 Question 4

23 compatibility structure Question 1 Question 2 Question 3 Question 5 Question 4

24 exclusiveness: answers to adjacent questions are not both 1

25 Question 1 0/1 Question 2 0/1 Question 3 0/1 Question 5 0/1 exclusiveness: answers to adjacent questions are not both 1 Question 4 0/1

26 Question 1 1 Question 2 0/1 Question 3 0/1 Question 5 0/1 exclusiveness: answers to adjacent questions are not both 1 Question 4 0/1

27 Question 1 1 Question 2 0 Question 3 0/1 Question 5 0/1 exclusiveness: answers to adjacent questions are not both 1 Question 4 0/1

28 Question 1 1 Question 2 0 Question 3 0/1 Question 5 0 exclusiveness: answers to adjacent questions are not both 1 Question 4 0/1

29 the answers 1 have expectation 2 if non-contextual and exclusive

30 classical theories give expectation 2

31 non-contextuality: probabilities of answers do not depend on contexts

32 Context 1 Question 1: Is X true? Pr[answer Yes] = a Question 2: Is Y true? Pr[answer Yes] = b non-contextuality: probabilities of answers do not depend on contexts

33 Context 2 Question 2: Is Y true? Pr[answer Yes] = b non-contextuality: probabilities of answers do not depend on contexts Question 3: Is Z true? Pr[answer Yes] = c

34 Context 1 Context 2 Question 1: Is X true? Pr[answer Yes] = a Question 2: Is Y true? Pr[answer Yes] = b non-contextuality: probabilities of answers do not depend on contexts Question 3: Is Z true? Pr[answer Yes] = c

35 Context 1 Context 2 Question 1: Is X true? Pr[answer Yes] = a Question 2: Is Y true? Pr[answer Yes] = b non-contextuality: probabilities of answers do not depend on contexts Question 3: Is Z true? Pr[answer Yes] = c

36 exclusiveness: answers to adjacent questions are not both 1

37 Question 1 0/1 Question 2 0/1 Question 3 0/1 Question 5 0/1 exclusiveness: answers to adjacent questions are not both 1 Question 4 0/1

38 Question 1 Question 2 Question 3 Pr[1] = p 1 Pr[1] = p 2 Pr[1] = p 3 Question 5 Question 4 Pr[1] = p 5 Pr[1] = p 4 exclusiveness: answers to adjacent questions are not both 1

39 Context 1 Question 1 Question 2 Pr[1] = p 1 Pr[1] = p 2 p 1 p 2 =1 exclusiveness: answers to adjacent questions are not both 1

40 Question 1 Pr[1] = 1/2 Question 2 Pr[1] = 1/2 Question 3 Pr[1] = 1/2 Question 5 Pr[1] = 1/2 exclusiveness: answers to adjacent questions are not both 1 Question 4 Pr[1] = 1/2

41 non-signaling theories* give expectation 2.5 *also called general probabilistic theories

42 axiomatically: classical theories give expectation 2 non-signaling theories give expectation 2.5

43 axiomatically: classical theories give expectation 2 quantum theory give expectation? non-signaling theories give expectation 2.5

44

45 Question 1 0/1 Question 2 0/1 Question 3 0/1 Question 5 0/1 exclusiveness: answers to adjacent questions are not both 1 Question 4 0/1

46 Question 1 Pr[1] = v 1 2 Question 2 Pr[1] = Question 3 Pr[1] = v 2 2 v 3 2 Question 5 Pr[1] = v 2 5 extra axiom: Born rule Question 4 Pr[1] = v 2 4 expectation: 5 i=1 v 2 i

47 Question i Context i Question i + mod(5) Pr[1] = Pr[1] = v i 2 v i 1mod 5 2 v i v i 1mod 5 =0 compatibility

48 R 3 v 5 v 4 v i 2 = 1 5 v 1 v 3 v 2

49 R 3 v 5 v 4 v i 2 = 1 5 v 1 v 3 5 i=1 v 2 i = 5 v 2

50 R 3 quantum theory gives expectation 5 v 5 v 4 v i 2 = 1 5 v 1 v 3 5 i=1 v 2 i = 5 v 2

51 classical theories* give expectation 2 quantum theory** gives expectation non-signaling theories* give expectation 2.5 *Wright (1978); **Klyachko et al. (2008)

52 2. Results: a general framework to study non-contextuality: 1. general compatibility structures 2. perfectness 3. non-locality

53 every graph/hypergraph is a compatibility structure Question 1 Question 2 Question 3 Question 5 Question 4 2. Results

54 every graph/hypergraph is a compatibility structure Question 1 Question 2 Question 3 Question 5 Question 4 2. Results

55 every graph/hypergraph is a compatibility structure Question 1 Question 2 Question 3 Question 5 Question 4 2. Results

56 Classification theorem Let be a compatibility structure, seen as a hypergraph. Let G be the graph obtained by connecting contextual questions. The maximum expectation values for exclusive answers are G G FP for classical, quantum, and non-signaling theories, respectively; where G is the independence number, G is the Lovász -function, and FP is the fractional packing number. 2. Results

57 Independence number G Is the maximum number of mutually non-adjacent vertices in a graph G. C 5 =2 NP-complete; hard to approximate 2. Results

58 Independence number G Is the maximum number of mutually non-adjacent vertices in a graph G. C 5 =2 NP-complete; hard to approximate classical theories give expectation 2 2. Results

59 Lovász -function G An orthogonal representation of G is a set of unit vectors associated to the vertices such that two vectors are orthogonal if if the corresponding vertices are adjacent: G := max n orth. repr. i=1 v i 2 v 5 v 1 v 2 C 5 = 5 semidefinite program* v 4 v 3 *Lovász (1978) 2. Results

60 Lovász -function G An orthogonal representation of G is a set of unit vectors associated to the vertices such that two vectors are orthogonal if if the corresponding vertices are adjacent: G := max n orth. repr. i=1 v i 2 v 5 *Lovász (1978) v 4 v 1 v 3 v 2 C 5 = 5 semidefinite program* quantum theory gives expectation 5 2. Results

61 Fractional packing number FP Let be a compatibility structure, seen as a hypergraph: FP =max w i i s.t. i 0 w i 1 and context C, i C w i 1 1/2 1/2 1/2 FP C 5 =5/2 linear program 1/2 1/2 2. Results

62 Fractional packing number FP Let be a compatibility structure, seen as a hypergraph: FP =max w i i s.t. i 0 w i 1 and context C, i C w i 1 1/2 1/2 1/2 1/2 1/2 FP C 5 =5/2 linear program non-signaling theories give expectation Results

63 Remark. C 5 is the max. violation of the Klyachko-Can-Biniciouglu-Shumovsky (KCBS) inequality*. The inequality can be used to detect genuine quantum effects and it it is the simplest non-contextual inequality violated by a qutrit (because the orthogonal representation has dimension 3). *Klyachko et al. (2008) 2. Results

64 Quantum mechanics as a sandwich theory * E C E Q E NS Let be the convex sets of the vectors realizing the expectations for classical, quantum, and non-signaling theories, respectively. *cf. Knuth (1994) 2. Results

65 Quantum mechanics as a sandwich theory * E C E Q E NS Let be the convex sets of the vectors realizing the expectations for classical, quantum, and non-signaling theories, respectively. Maxima FP G E Q E NS G E C *cf. Knuth (1994) 2. Results

66 Quantum mechanics as a sandwich theory * E C E Q E NS Let be the convex sets of the vectors realizing the expectations for classical, quantum, and non-signaling theories, respectively. E Q membership in of a vector can be tested with a semidefinite program! E Q E NS E C *cf. Knuth (1994) 2. Results

67 Remark. Standard result about the Lovász function can be then used to give the max. violation of known inequalities. For example, the max. violation for the n-cycle generalization of the KCBS inequality, recently computed in * is C n = n cos /n 1 cos /n *Liang-Spekkens-Wiseman (2010) 2. Results

68 Classicality and perfectness A graph G is perfect* if if for every induced subgraph H. So, perfect graphs are the most classical ones. For a perfect graph G G H = H = H E C =E Q =E NS Whenever we have a difference between classical theories and quantum mechanics and a state dependent proof of the Kochen-Specker theorem**. The KCBS inequality is based on C 5 which is the smallest non-perfect graph. *Berge (1961); **where effects sum to unity 2. Results

69 Two remarks 1. Many intractable problems are tractable for perfect graphs (i.e., when classical and quantum theories coincide). G =2 G = n 1/3 2. There are graphs s.t. and (i.e., classical and quantum theories can have arbitrarily large separation)*. *Koniagin (1981) 2. Results

70 non-contextuality non-locality 2. Results

71 Observation Non-local experiments give compatibility structures: compatible questions are the local measurement. 2. Results

72 Observation Non-local experiments give compatibility structures: compatible questions are the local measurement. Alice settings x X outcomes a A Bob settings y Y outcomes b B 2. Results

73 Compatibility graph for a non-local experiment G= V,E V =A B X Y {abxy,a' b' x ' y } E iff x=x ' a a ' y =y ' b b' [ is the hypergraph of all cliques* in G] *complete subgraphs 2. Results

74 Compatibility graph for a non-local experiment G= V,E V =A B X Y {abxy,a' b' x ' y } E iff x=x ' a a ' y =y ' b b' exclusiveness; compatibility: the observables of Alice and Bob commute. 2. Results

75 A classification theorem for correlations Let be the compatibility hypergraph for a non-local experiment: E 1 C E id Q E 1 Q E 1 NS are the sets of correlations obtainable by local hidden variables, local quantum measurements on a bipartite state, idem but without completeness relation for the measurement, and non-signaling theories, respectively: 1 E X=C,Q, NS :=E X { w : xy E Q id :={ w ab xy abxy : xy ab w ab xy =1} w ab xy P ab xy =id} 2. Results

76 1 E NS E Q id E Q 1 1 E C 2. Results

77 Fact 1 E NS maximization over 1 1 E NS and E C is equivalent to maximization over and E NS E C 1 E Q id E Q 1 E C 2. Results

78 Fact maxima via SDP are efficient upper bounds to maximum quantum violations E C E Q id E Q 1 2. Results

79 Fact there is no efficient algorithm, unless the polynomial hierarchy collapses* E C E Q id E Q 1 *Ito-Kobayashi-Matsumoto (2009) 2. Results

80 Fact maxima via SDP are efficient upper bounds to maximum quantum violations E C E Q id E Q 1 1 id Problem: how well E Q approximates E Q? *Ito-Kobayashi-Matsumoto (2009) 2. Results

81 Clause-Horne-Shimony-Holt (CHSH) inequality* *Clause-Horne-Shimony-Holt (1969) 2. Results

82 CHSH inequality settings and outcomes: A=B=X =Y ={0,1} constraint: w ab xy : x y=a XOR b w ab xy 2. Results

83 CHSH inequality settings and outcomes: A=B=X =Y ={0,1} constraint: w ab xy : x y=a XOR b w ab xy 1. Introduction: 2. Results non-contextuality

84 CHSH inequality settings and outcomes: A=B=X =Y ={0,1} constraint: w ab xy : x y=a XOR b w ab xy Introduction: 2. Results non-contextuality

85 CHSH inequality settings and outcomes: A=B=X =Y ={0,1} constraint: w ab xy : x y=a XOR b w ab xy G =3 classical max Introduction: 2. Results non-contextuality

86 CHSH inequality settings and outcomes: A=B=X =Y ={0,1} constraint: 1/2 1/2 1/2 w ab xy : x y=a XOR b w ab xy G =3 classical max. 1/2 1/2 FP =4 non-signaling max. 1/2 1/2 1/2 1. Introduction: 2. Results non-contextuality

87 CHSH inequality settings and outcomes: A=B=X =Y ={0,1} constraint: w ab xy : x y=a XOR b w ab xy v 8 2 v 7 2 v 1 2 v 2 2 v 3 2 G =3 classical max. FP =4 non-signaling max. v 6 2 v 5 2 v 4 2 G = quantum max. 1. Introduction: 2. Results non-contextuality

88 CHSH inequality G =3 classical max. FP G =4 non-signaling max. it attains the Tsirelson bound G = quantum max. 1. Introduction: 2. Results non-contextuality

89 Collins-Gisin inequality (I3322)* max max ** E Q id E Q 1 E Q id E Q 1 E C *Collins-Gisin (2004); **Navascués-Acín-Pironio (2008) 1. Introduction: 2. Results non-contextuality

90 3. Open problems: 1. theoretical: relations to Bell inequalities 2. applied: loophole-free experiments 3. a complexity perspective: degree of perfectness 3. Open problems

91 theoretical open problem Can any violation of a non-contextual inequality be converted into a (comparably large) violation of a Bell inequality? Is your entropy 5 bits? 3. Open problems

92 applied open problem Can any violation of a non-contextual inequality be converted into a (comparably large) violation of a Bell inequality? So far, forty years after Bell paper, all Bell experiments have loopholes: are graphs with a large separation between the independence number and the Lovász function good candidates for loophole-free experiments with inefficient detectors? Is your entropy 5 bits? 3. Open problems

93 complexity open problem Can any violation of a non-contextual inequality be converted into a (comparably large) violation of a Bell inequality? So far, forty years after Bell paper, all Bell experiments have loopholes: are graphs with a large separation between the independence number and the Lovász function good candidates for loophole-free experiments with inefficient detectors? Perfect graphs have many efficient algorithms that in general are NP-hard. We have shown that compatibility structures from perfect graphs have coincident classical and quantum Is your description. entropy 5 bits? Can we define a notion of parametric complexity according to the classical-quantum gap? 3. Open problems

94 open problems Can any violation of a non-contextual inequality be converted into a (comparably large) violation of a Bell inequality? So far, forty years after Bell paper, all Bell experiments have loopholes: are graphs with a large separation between the independence number and the Lovász function good candidates for loophole-free experiments with inefficient detectors? Perfect graphs have many efficient algorithms that in general are NP-hard. We have shown that compatibility structures from perfect graphs have coincident classical and quantum description. Can we define a notion of parametric complexity according to the classical-quantum gap? The Lovász function is fundamental in zero-error classical and quantum information theory*. Can we recast Is your entropy 5 bits? the non-contextuality framework into an information theoretic one? *09:55, Fri, Matthews' talk; poster Duan-Severini-Winter 3. Open problems

95 open problems Can any violation of a non-contextual inequality be converted into a (comparably large) violation of a Bell inequality? So far, forty years after Bell paper, all Bell experiments have loopholes: are graphs with a large separation between the independence number and the Lovász function good candidates for loophole-free experiments with inefficient detectors? Perfect graphs have many efficient algorithms that in general are NP-hard. We have shown that compatibility structures from perfect graphs have coincident classical and quantum description. Can we define a notion of parametric complexity according to the classical-quantum gap? The Lovász function is fundamental in zero-error classical and quantum information theory. Can we recast the non-contextuality framework into an information theoretic one? Is your entropy 5 bits? 3. Open problems