Contextuality as a resource

Transcription

1 Contextuality as a resource Rui Soares Barbosa Department of Computer Science, University of Oxford rui.soares.barbosa@cs.ox.ac.uk Combining Viewpoints in Quantum Theory Edinburgh, 20th March 2018

2 Joint work with: Samson Abramsky (Oxford) Shane Mansfield (Paris VII) and also: Kohei Kishida (Dalhousie) Giovanni Carù (Oxford) Nadish de Silva (UCL) Octavio Zapata (UCL) R S Barbosa Contextuality as a resource 1

3 Motivation Contextuality and non-locality: fundamental non-classical phenomenona of QM R S Barbosa Contextuality as a resource 2

4 Motivation Contextuality and non-locality: fundamental non-classical phenomenona of QM Contextuality as a resource for QIP and QC: R S Barbosa Contextuality as a resource 2

5 Motivation Contextuality and non-locality: fundamental non-classical phenomenona of QM Contextuality as a resource for QIP and QC: Non-local games XOR games (CHSH; Cleve Høyer Toner Watrous) quantum graph homomorphisms (Mančinska Roberson) constraint satisfaction (Cleve Mittal) etc. (Abramsky B de Silva Zapata) R S Barbosa Contextuality as a resource 2

6 Motivation Contextuality and non-locality: fundamental non-classical phenomenona of QM Contextuality as a resource for QIP and QC: Non-local games XOR games (CHSH; Cleve Høyer Toner Watrous) quantum graph homomorphisms (Mančinska Roberson) constraint satisfaction (Cleve Mittal) etc. (Abramsky B de Silva Zapata) MBQC Raussendorf (2013) Contextuality in measurement-based quantum computation R S Barbosa Contextuality as a resource 2

7 Motivation Contextuality and non-locality: fundamental non-classical phenomenona of QM Contextuality as a resource for QIP and QC: Non-local games XOR games (CHSH; Cleve Høyer Toner Watrous) quantum graph homomorphisms (Mančinska Roberson) constraint satisfaction (Cleve Mittal) etc. (Abramsky B de Silva Zapata) MBQC Raussendorf (2013) Contextuality in measurement-based quantum computation MSD Howard Wallman Veith Emerson (2014) Contextuality supplies the magic for quantum computation R S Barbosa Contextuality as a resource 2

8 Overview Contextuality formulated in a theory-independent fashion R S Barbosa Contextuality as a resource 3

9 Overview Contextuality formulated in a theory-independent fashion Abramsky & Brandenburger: unified framework for non-locality and contextuality (cf. Cabello Severini Winter, Acín Fritz Leverrier Sainz) R S Barbosa Contextuality as a resource 3

10 Overview Contextuality formulated in a theory-independent fashion Abramsky & Brandenburger: unified framework for non-locality and contextuality (cf. Cabello Severini Winter, Acín Fritz Leverrier Sainz) Towards a resource theory of contextuality: R S Barbosa Contextuality as a resource 3

11 Overview Contextuality formulated in a theory-independent fashion Abramsky & Brandenburger: unified framework for non-locality and contextuality (cf. Cabello Severini Winter, Acín Fritz Leverrier Sainz) Towards a resource theory of contextuality: Combine and transform contextual blackboxes R S Barbosa Contextuality as a resource 3

12 Overview Contextuality formulated in a theory-independent fashion Abramsky & Brandenburger: unified framework for non-locality and contextuality (cf. Cabello Severini Winter, Acín Fritz Leverrier Sainz) Towards a resource theory of contextuality: Combine and transform contextual blackboxes Measure of contextuality R S Barbosa Contextuality as a resource 3

13 Overview Contextuality formulated in a theory-independent fashion Abramsky & Brandenburger: unified framework for non-locality and contextuality (cf. Cabello Severini Winter, Acín Fritz Leverrier Sainz) Towards a resource theory of contextuality: Combine and transform contextual blackboxes Measure of contextuality Quantifiable advantages in QC and QIP tasks R S Barbosa Contextuality as a resource 3

14 Contextuality

15 Empirical data o A {0, 1} o B {0, 1} measurement device measurement device m A {a 1, a 2 } m B {b 1, b 2 } preparation p R S Barbosa Contextuality as a resource 4

16 Empirical data A B (0, 0) (0, 1) (1, 0) (1, 1) a 1 b 1 1/ /2 a 1 b 2 3/8 1/8 1/8 3/8 a 2 b 1 3/8 1/8 1/8 3/8 a 2 b 2 1/8 3/8 3/8 1/8 o A {0, 1} o B {0, 1} measurement device measurement device m A {a 1, a 2 } m B {b 1, b 2 } preparation p R S Barbosa Contextuality as a resource 4

17 A simple observation (Abramsky Hardy) Propositional formulae φ 1,..., φ N R S Barbosa Contextuality as a resource 5

18 A simple observation (Abramsky Hardy) Propositional formulae φ 1,..., φ N p i := Prob(φ i ) R S Barbosa Contextuality as a resource 5

19 A simple observation (Abramsky Hardy) Propositional formulae φ 1,..., φ N p i := Prob(φ i ) Not simultaneously satisfiable, hence Prob( phi i ) = 0 R S Barbosa Contextuality as a resource 5

20 A simple observation (Abramsky Hardy) Propositional formulae φ 1,..., φ N p i := Prob(φ i ) Not simultaneously satisfiable, hence Prob( phi i ) = 0 Using elementary logic and probability: 1 = Prob( phi i ) = Prob( φ i ) N Prob( φ i ) = i=1 N N (1 p i ) = N p i. i=1 i=1 R S Barbosa Contextuality as a resource 5

21 A simple observation (Abramsky Hardy) Propositional formulae φ 1,..., φ N p i := Prob(φ i ) Not simultaneously satisfiable, hence Prob( phi i ) = 0 Using elementary logic and probability: 1 = Prob( phi i ) = Prob( φ i ) N Prob( φ i ) = i=1 N N (1 p i ) = N p i. i=1 i=1 Hence, N i=1 p i N 1. R S Barbosa Contextuality as a resource 5

22 Analysis of the Bell table A B (0, 0) (0, 1) (1, 0) (1, 1) a 1 b 1 1/ /2 a 1 b 2 3/8 1/8 1/8 3/8 a 2 b 1 3/8 1/8 1/8 3/8 a 2 b 2 1/8 3/8 3/8 1/8 R S Barbosa Contextuality as a resource 6

23 Analysis of the Bell table A B (0, 0) (0, 1) (1, 0) (1, 1) a 1 b 1 1/ /2 a 1 b 2 3/8 1/8 1/8 3/8 a 2 b 1 3/8 1/8 1/8 3/8 a 2 b 2 1/8 3/8 3/8 1/8 φ 1 = a 1 b 1 φ 2 = a 1 b 2 φ 3 = a 2 b 1 φ 4 = a 2 b 2 R S Barbosa Contextuality as a resource 6

24 Analysis of the Bell table A B (0, 0) (0, 1) (1, 0) (1, 1) a 1 b 1 1/ /2 a 1 b 2 3/8 1/8 1/8 3/8 a 2 b 1 3/8 1/8 1/8 3/8 a 2 b 2 1/8 3/8 3/8 1/8 φ 1 = a 1 b 1 φ 2 = a 1 b 2 φ 3 = a 2 b 1 φ 4 = a 2 b 2 These formulae are contradictory. R S Barbosa Contextuality as a resource 6

25 Analysis of the Bell table A B (0, 0) (0, 1) (1, 0) (1, 1) a 1 b 1 1/ /2 a 1 b 2 3/8 1/8 1/8 3/8 a 2 b 1 3/8 1/8 1/8 3/8 a 2 b 2 1/8 3/8 3/8 1/8 φ 1 = a 1 b 1 φ 2 = a 1 b 2 φ 3 = a 2 b 1 φ 4 = a 2 b 2 These formulae are contradictory. But p 1 + p 2 + p 3 + p 4 = 3.35 R S Barbosa Contextuality as a resource 6

26 Analysis of the Bell table A B (0, 0) (0, 1) (1, 0) (1, 1) a 1 b 1 1/ /2 a 1 b 2 3/8 1/8 1/8 3/8 a 2 b 1 3/8 1/8 1/8 3/8 a 2 b 2 1/8 3/8 3/8 1/8 φ 1 = a 1 b 1 φ 2 = a 1 b 2 φ 3 = a 2 b 1 φ 4 = a 2 b 2 These formulae are contradictory. But p 1 + p 2 + p 3 + p 4 = 3.35 The inequality is violated by 1 /4. R S Barbosa Contextuality as a resource 6

27 Contextuality But the Bell table can be realised in the real world. R S Barbosa Contextuality as a resource 7

28 Contextuality But the Bell table can be realised in the real world. What was our unwarranted assumption? R S Barbosa Contextuality as a resource 7

29 Contextuality But the Bell table can be realised in the real world. What was our unwarranted assumption? That all variables could in principle be observed simultaneously. R S Barbosa Contextuality as a resource 7

30 Contextuality But the Bell table can be realised in the real world. What was our unwarranted assumption? That all variables could in principle be observed simultaneously. Local consistency vs global inconsistency. R S Barbosa Contextuality as a resource 7

31 Abramsky Brandenburger framework Measurement scenario X, M, O : X is a finite set of measurements or variables O is a finite set of outcomes or values M is a cover of X, indicating joint measurability (contexts) R S Barbosa Contextuality as a resource 8

32 Abramsky Brandenburger framework Measurement scenario X, M, O : X is a finite set of measurements or variables O is a finite set of outcomes or values M is a cover of X, indicating joint measurability (contexts) Example: (2,2,2) Bell scenario The set of variables is X = {a 1, a 2, b 1, b 2 }. The outcomes are O = {0, 1}. The measurement contexts are: { {a 1, b 1 }, {a 1, b 2 }, {a 2, b 1 }, {a 2, b 2 } }. R S Barbosa Contextuality as a resource 8

33 Measurement scenarios b 2 b 2 a 1 a 2 c 1 a 1 a 2 c 2 b 1 Examples: Bell-type scenarios, KS configurations, and more. b 1 R S Barbosa Contextuality as a resource 9

34 Another example: 18-vector Kochen Specker A set of 18 variables, X = {A,..., O} R S Barbosa Contextuality as a resource 10

35 Another example: 18-vector Kochen Specker A set of 18 variables, X = {A,..., O} A set of outcomes O = {0, 1} R S Barbosa Contextuality as a resource 10

36 Another example: 18-vector Kochen Specker A set of 18 variables, X = {A,..., O} A set of outcomes O = {0, 1} A measurement cover M = {C 1,..., C 9 }, whose contexts C i correspond to the columns in the following table: U 1 U 2 U 3 U 4 U 5 U 6 U 7 U 8 U 9 A A H H B I P P Q B E I K E K Q R R C F C G M N D F M D G J L N O J L O R S Barbosa Contextuality as a resource 10

37 Empirical Models Joint outcome or event in a context C is s O C, e.g. s = [a 1 0, b 1 1]. R S Barbosa Contextuality as a resource 11

38 Empirical Models Joint outcome or event in a context C is s O C, e.g. s = [a 1 0, b 1 1]. Empirical model: family {e C } C M where e C Prob(O C ) for C M. It specifies a probability distribution over the events in each context. Each distribution is a row of the probability table. R S Barbosa Contextuality as a resource 11

39 Empirical Models Joint outcome or event in a context C is s O C, e.g. s = [a 1 0, b 1 1]. Empirical model: family {e C } C M where e C Prob(O C ) for C M. It specifies a probability distribution over the events in each context. Each distribution is a row of the probability table. Compatibility condition: the distributions agree on overlaps C, C M. e C C C = e C C C. R S Barbosa Contextuality as a resource 11

40 Empirical Models Joint outcome or event in a context C is s O C, e.g. s = [a 1 0, b 1 1]. Empirical model: family {e C } C M where e C Prob(O C ) for C M. It specifies a probability distribution over the events in each context. Each distribution is a row of the probability table. Compatibility condition: the distributions agree on overlaps C, C M. e C C C = e C C C. In multipartite scenarios, compatibility = the no-signalling principle. R S Barbosa Contextuality as a resource 11

41 Contextuality A (compatible) empirical model is non-contextual if there exists a global distribution d Prob(O X ) on the joint assignments of outcomes to all measurements that marginalises to all the e C : d Prob(O X ). C M. d C = e C. R S Barbosa Contextuality as a resource 12

42 Contextuality A (compatible) empirical model is non-contextual if there exists a global distribution d Prob(O X ) on the joint assignments of outcomes to all measurements that marginalises to all the e C : d Prob(O X ). C M. d C = e C. i.e. all the local information can be glued into a consistent global description. R S Barbosa Contextuality as a resource 12

43 Contextuality A (compatible) empirical model is non-contextual if there exists a global distribution d Prob(O X ) on the joint assignments of outcomes to all measurements that marginalises to all the e C : d Prob(O X ). C M. d C = e C. i.e. all the local information can be glued into a consistent global description. Contextuality: family of data which is locally consistent but globally inconsistent. R S Barbosa Contextuality as a resource 12

44 Contextuality A (compatible) empirical model is non-contextual if there exists a global distribution d Prob(O X ) on the joint assignments of outcomes to all measurements that marginalises to all the e C : d Prob(O X ). C M. d C = e C. i.e. all the local information can be glued into a consistent global description. Contextuality: family of data which is locally consistent but globally inconsistent. The import of results such as Bell s and Bell Kochen Specker s theorems is that there are empirical models arising from quantum mechanics that are contextual. R S Barbosa Contextuality as a resource 12

45 Possibilistic collapse Given an empirical model e, define possibilistic model poss(e) by taking the support of each distributions. Contains the possibilistic, or logical, information of that model. R S Barbosa Contextuality as a resource 13

46 Possibilistic collapse Given an empirical model e, define possibilistic model poss(e) by taking the support of each distributions. Contains the possibilistic, or logical, information of that model a 1 b 1 1/ /2 a 1 b 2 3/8 1/8 1/8 3/8 a 2 b 1 3/8 1/8 1/8 3/8 a 2 b 2 1/8 3/8 3/8 1/ a 1 b a 1 b a 2 b a 2 b R S Barbosa Contextuality as a resource 13

47 Logical contextuality: Hardy model a 0 b a 0 b a 1 b a 1 b R S Barbosa Contextuality as a resource 14

48 Logical contextuality: Hardy model a 0 b a 0 b a 1 b a 1 b a 0 b 1 b 0 a 1 R S Barbosa Contextuality as a resource 14

49 Logical contextuality: Hardy model a 0 b a 0 b a 1 b a 1 b b a 0 a 1 b 0 R S Barbosa Contextuality as a resource 14

50 Logical contextuality: Hardy model a 0 b a 0 b a 1 b a 1 b b a 0 a 1 b 0 R S Barbosa Contextuality as a resource 14

51 Logical contextuality: Hardy model a 0 b a 0 b a 1 b a 1 b b a 0 a 1 b 0 R S Barbosa Contextuality as a resource 14

52 Logical contextuality: Hardy model a 0 b a 0 b a 1 b a 1 b b a 0 a 1 b 0 R S Barbosa Contextuality as a resource 14

53 Logical contextuality: Hardy model a 0 b a 0 b a 1 b a 1 b b a 0 a 1 b 0 R S Barbosa Contextuality as a resource 14

54 Logical contextuality: Hardy model a 0 b a 0 b a 1 b a 1 b b a 0 a 1 b 0 R S Barbosa Contextuality as a resource 14

55 Logical contextuality: Hardy model a 0 b a 0 b a 1 b a 1 b b a 0 a 1 There are some global sections, b 0 R S Barbosa Contextuality as a resource 14

56 Logical contextuality: Hardy model a 0 b a 0 b a 1 b a 1 b b a 0 a 1 There are some global sections, but... b 0 R S Barbosa Contextuality as a resource 14

57 Logical contextuality: Hardy model a 0 b a 0 b a 1 b a 1 b b a 0 a 1 There are some global sections, but... b 0 R S Barbosa Contextuality as a resource 14

58 Logical contextuality: Hardy model a 0 b a 0 b a 1 b a 1 b b a 0 a 1 There are some global sections, but... b 0 R S Barbosa Contextuality as a resource 14

59 Logical contextuality: Hardy model a 0 b a 0 b a 1 b a 1 b b a 0 a 1 There are some global sections, but... b 0 R S Barbosa Contextuality as a resource 14

60 Logical contextuality: Hardy model a 0 b a 0 b a 1 b a 1 b b a 0 a 1 There are some global sections, but... b 0 R S Barbosa Contextuality as a resource 14

61 Logical contextuality: Hardy model a 0 b a 0 b a 1 b a 1 b b a 0 a 1 There are some global sections, but... b 0 R S Barbosa Contextuality as a resource 14

62 Logical contextuality: Hardy model a 0 b a 0 b a 1 b a 1 b b a 0 a 1 There are some global sections, but... b 0 R S Barbosa Contextuality as a resource 14

63 Logical contextuality: Hardy model a 0 b a 0 b a 1 b a 1 b b a 0 a 1 There are some global sections, but... b 0 R S Barbosa Contextuality as a resource 14

64 Logical contextuality: Hardy model a 0 b a 0 b a 1 b a 1 b There are some global sections, but... a 0 0 b 1 1 b 0 Logical contextuality: Not all sections extend to global ones. 0 1 a 1 R S Barbosa Contextuality as a resource 14

65 Strong contextuality Strong Contextuality: no event can be extended to a global assignment. R S Barbosa Contextuality as a resource 15

66 Strong contextuality Strong Contextuality: no event can be extended to a global assignment. E.g. K S, GHZ, the PR box: b 1 A B (0, 0) (0, 1) (1, 0) (1, 1) a 1 b b 2 a 1 b a a 2 b a 2 b a R S Barbosa Contextuality as a resource 15

67 Strong contextuality Strong Contextuality: no event can be extended to a global assignment. E.g. K S, GHZ, the PR box: b 1 A B (0, 0) (0, 1) (1, 0) (1, 1) a 1 b b 2 a 1 b a a 2 b a 2 b a Cohomological witnesses of contextuality (Abramsky B Mansfield, ABM Kisheda Lal, Carù, Raussendorf et al.) R S Barbosa Contextuality as a resource 15

68 Measuring Contextuality

69 The contextual fraction Non-contextuality: global distribution d Prob(O X ) such that: C M. d C = e C. R S Barbosa Contextuality as a resource 16

70 The contextual fraction Non-contextuality: global distribution d Prob(O X ) such that: C M. d C = e C. Which fraction of a model admits a non-contextual explanation? R S Barbosa Contextuality as a resource 16

71 The contextual fraction Non-contextuality: global distribution d Prob(O X ) such that: C M. d C = e C. Which fraction of a model admits a non-contextual explanation? Consider subdistributions c SubProb(O X ) such that: C M. c C e C. Non-contextual fraction: maximum weight of such a subdistribution. R S Barbosa Contextuality as a resource 16

72 The contextual fraction Non-contextuality: global distribution d Prob(O X ) such that: C M. d C = e C. Which fraction of a model admits a non-contextual explanation? Consider subdistributions c SubProb(O X ) such that: C M. c C e C. Non-contextual fraction: maximum weight of such a subdistribution. Equivalently, maximum weight λ over all convex decompositions e = λe NC + (1 λ)e where e NC is a non-contextual model. R S Barbosa Contextuality as a resource 16

73 The contextual fraction Non-contextuality: global distribution d Prob(O X ) such that: C M. d C = e C. Which fraction of a model admits a non-contextual explanation? Consider subdistributions c SubProb(O X ) such that: C M. c C e C. Non-contextual fraction: maximum weight of such a subdistribution. Equivalently, maximum weight λ over all convex decompositions e = λe NC + (1 λ)e where e NC is a non-contextual model. R S Barbosa Contextuality as a resource 16

74 The contextual fraction Non-contextuality: global distribution d Prob(O X ) such that: C M. d C = e C. Which fraction of a model admits a non-contextual explanation? Consider subdistributions c SubProb(O X ) such that: C M. c C e C. Non-contextual fraction: maximum weight of such a subdistribution. Equivalently, maximum weight λ over all convex decompositions e = λe NC + (1 λ)e SC where e NC is a non-contextual model. e SC is strongly contextual! NCF(e) = λ CF(e) = 1 λ R S Barbosa Contextuality as a resource 16

75 (Non-)contextual fraction via linear programming Checking contextuality of e corresponds to solving Find d R n such that M d = v e and d 0. R S Barbosa Contextuality as a resource 17

76 (Non-)contextual fraction via linear programming Checking contextuality of e corresponds to solving Find d R n such that M d = v e and d 0. Computing the non-contextual fraction corresponds to solving the following linear program: Find maximising subject to c R n 1 c M c v e and c 0. R S Barbosa Contextuality as a resource 17

77 E.g. Equatorial measurements on GHZ(n) (a) (b) Figure: Contextual fraction of empirical models obtained with equatorial measurements at φ 1 and φ 2 on each qubit of ψ GHZ(n) with: (a) n = 3; (b) n = 4. R S Barbosa Contextuality as a resource 18

78 Violations of Bell inequalities

79 Generalised Bell inequalities An inequality for a scenario X, M, O is given by: a set of coefficients α = {α(c, s)} C M,s O C a bound R R S Barbosa Contextuality as a resource 19

80 Generalised Bell inequalities An inequality for a scenario X, M, O is given by: a set of coefficients α = {α(c, s)} C M,s O C a bound R For a model e, the inequality reads as B α (e) R, where B α (e) := C M,s O C α(c, s)e C (s). R S Barbosa Contextuality as a resource 19

81 Generalised Bell inequalities An inequality for a scenario X, M, O is given by: a set of coefficients α = {α(c, s)} C M,s O C a bound R For a model e, the inequality reads as B α (e) R, where B α (e) := C M,s O C α(c, s)e C (s). Wlog we can take R non-negative (in fact, we can take R = 0). R S Barbosa Contextuality as a resource 19

82 Generalised Bell inequalities An inequality for a scenario X, M, O is given by: a set of coefficients α = {α(c, s)} C M,s O C a bound R For a model e, the inequality reads as B α (e) R, where B α (e) := C M,s O C α(c, s)e C (s). Wlog we can take R non-negative (in fact, we can take R = 0). It is called a Bell inequality if it is satisfied by every NC model. If it is saturated by some NC model, the Bell inequality is said to be tight. R S Barbosa Contextuality as a resource 19

83 Generalised Bell inequalities An inequality for a scenario X, M, O is given by: a set of coefficients α = {α(c, s)} C M,s O C a bound R For a model e, the inequality reads as B α (e) R, where B α (e) := C M,s O C α(c, s)e C (s). Wlog we can take R non-negative (in fact, we can take R = 0). It is called a Bell inequality if it is satisfied by every NC model. If it is saturated by some NC model, the Bell inequality is said to be tight. NB: A complete set of inequalities can be derived from the logical approach. R S Barbosa Contextuality as a resource 19

84 Violation of a Bell inequality A Bell inequality establishes a bound for the value of B α (e) amongst NC models. R S Barbosa Contextuality as a resource 20

85 Violation of a Bell inequality A Bell inequality establishes a bound for the value of B α (e) amongst NC models. For a general (no-signalling) model e, the quantity is limited only by α := max {α(c, s) s O C} C M R S Barbosa Contextuality as a resource 20

86 Violation of a Bell inequality A Bell inequality establishes a bound for the value of B α (e) amongst NC models. For a general (no-signalling) model e, the quantity is limited only by α := max {α(c, s) s O C} C M The normalised violation of a Bell inequality α, R by an empirical model e is the value max{0, B α (e) R} α R. R S Barbosa Contextuality as a resource 20

87 Bell inequality violation and the contextual fraction Proposition Let e be an empirical model. R S Barbosa Contextuality as a resource 21

88 Bell inequality violation and the contextual fraction Proposition Let e be an empirical model. The normalised violation by e of any Bell inequality is at most CF(e). R S Barbosa Contextuality as a resource 21

89 Bell inequality violation and the contextual fraction Proposition Let e be an empirical model. The normalised violation by e of any Bell inequality is at most CF(e). This bound is attained: there exists a Bell inequality whose normalised violation by e is exactly CF(e). R S Barbosa Contextuality as a resource 21

90 Bell inequality violation and the contextual fraction Proposition Let e be an empirical model. The normalised violation by e of any Bell inequality is at most CF(e). This bound is attained: there exists a Bell inequality whose normalised violation by e is exactly CF(e). Moreover, this Bell inequality is tight at the non-contextual model e NC and maximally violated by the strongly contextual model e SC for any decomposition: e = NCF(e)e NC + CF(e)e SC. R S Barbosa Contextuality as a resource 21

91 Bell inequality violation and the contextual fraction Quantifying Contextuality LP: Find c R n maximising 1 c subject to M c v e and c 0. e = λe NC + (1 λ)e SC with λ = 1 x. SC C v e Q NC R S Barbosa Contextuality as a resource 22

92 Bell inequality violation and the contextual fraction Quantifying Contextuality LP: Find c R n maximising 1 c subject to M c v e and c 0. Dual LP: Find y R m minimising y v e subject to M T y 1 and y 0. e = λe NC + (1 λ)e SC with λ = 1 x. SC C v e Q NC R S Barbosa Contextuality as a resource 22

93 Bell inequality violation and the contextual fraction Quantifying Contextuality LP: Find c R n maximising 1 c subject to M c v e and c 0. e = λe NC + (1 λ)e SC with λ = 1 x. SC Dual LP: Find y R m minimising y v e subject to M T y 1 and y 0. a := 1 M y C v e Q NC R S Barbosa Contextuality as a resource 22

94 Bell inequality violation and the contextual fraction Quantifying Contextuality LP: Find c R n maximising 1 c subject to M c v e and c 0. e = λe NC + (1 λ)e SC with λ = 1 x. SC C Q v e NC Dual LP: Find y R m minimising y v e subject to M T y 1 and y 0. a := 1 M y Find a R m maximising a v e subject to M T a 0 and a 1. R S Barbosa Contextuality as a resource 22

95 Bell inequality violation and the contextual fraction Quantifying Contextuality LP: Find c R n maximising 1 c subject to M c v e and c 0. e = λe NC + (1 λ)e SC with λ = 1 x. SC C Q v e NC Dual LP: Find minimising y R m y v e subject to M T y 1 and y 0. Find a := 1 M y maximising a R m a v e subject to M T a 0 and a 1. computes tight Bell inequality (separating hyperplane) R S Barbosa Contextuality as a resource 22

96 Operations on empirical models

97 Contextuality as a resource R S Barbosa Contextuality as a resource 23

98 Contextuality as a resource More than one possible measure of contextuality. What properties should a good measure satisfy? R S Barbosa Contextuality as a resource 23

99 Contextuality as a resource More than one possible measure of contextuality. What properties should a good measure satisfy? Monotonicity wrt operations that do not introduce contextuality Towards a resource theory as for entanglement (e.g. LOCC), non-locality,... R S Barbosa Contextuality as a resource 23

100 Algebra of empirical models Think of empirical models as black boxes R S Barbosa Contextuality as a resource 24

101 Algebra of empirical models Think of empirical models as black boxes What operations can we perform (non-contextually) on them? R S Barbosa Contextuality as a resource 24

102 Algebra of empirical models Think of empirical models as black boxes What operations can we perform (non-contextually) on them? We write type statements e : X, M, O to mean that e is a (compatible) emprical model on X, M, O. R S Barbosa Contextuality as a resource 24

103 Algebra of empirical models Think of empirical models as black boxes What operations can we perform (non-contextually) on them? We write type statements e : X, M, O to mean that e is a (compatible) emprical model on X, M, O. The operations remind one of process algebras. R S Barbosa Contextuality as a resource 24

104 Operations R S Barbosa Contextuality as a resource 25

105 Operations Relabelling e : X, M, O α : (X, M) = (X, M ) e[α] : X, M, O R S Barbosa Contextuality as a resource 25

106 Operations Relabelling e : X, M, O α : (X, M) = (X, M ) e[α] : X, M, O For C M, s : α(c) O, e[α] α(c) (s) := e C (s α 1 ) R S Barbosa Contextuality as a resource 25

107 Operations Relabelling e : X, M, O α : (X, M) = (X, M ) e[α] : X, M, O For C M, s : α(c) O, e[α] α(c) (s) := e C (s α 1 ) Restriction e : X, M, O (X, M ) (X, M) e M : X, M, O R S Barbosa Contextuality as a resource 25

108 Operations Relabelling e : X, M, O α : (X, M) = (X, M ) e[α] : X, M, O For C M, s : α(c) O, e[α] α(c) (s) := e C (s α 1 ) Restriction e : X, M, O (X, M ) (X, M) e M : X, M, O For C M, s : C O, (e M ) C (s) := e C C (s) with any C M s.t. C C R S Barbosa Contextuality as a resource 25

109 Operations Relabelling e : X, M, O α : (X, M) = (X, M ) e[α] : X, M, O For C M, s : α(c) O, e[α] α(c) (s) := e C (s α 1 ) Restriction e : X, M, O (X, M ) (X, M) e M : X, M, O For C M, s : C O, (e M ) C (s) := e C C (s) with any C M s.t. C C Coarse-graining e : X, M, O f : O O e/f : X, M, O R S Barbosa Contextuality as a resource 25

110 Operations Relabelling e : X, M, O α : (X, M) = (X, M ) e[α] : X, M, O For C M, s : α(c) O, e[α] α(c) (s) := e C (s α 1 ) Restriction e : X, M, O (X, M ) (X, M) e M : X, M, O For C M, s : C O, (e M ) C (s) := e C C (s) with any C M s.t. C C Coarse-graining e : X, M, O f : O O e/f : X, M, O For C M, s : C O, (e/f ) C (s) := t:c O,f t=s e C(t) R S Barbosa Contextuality as a resource 25

111 Operations Mixing e, e : X, M, O λ [0, 1] e + λ e : X, M, O R S Barbosa Contextuality as a resource 26

112 Operations Mixing e, e : X, M, O λ [0, 1] e + λ e : X, M, O For C M, s : C O, (e + λ e ) C (s) := λe C (s) + (1 λ)e C(s) R S Barbosa Contextuality as a resource 26

113 Operations Mixing e, e : X, M, O λ [0, 1] e + λ e : X, M, O For C M, s : C O, (e + λ e ) C (s) := λe C (s) + (1 λ)e C(s) Choice e : X, M, O e : X, M, O e & e : X X, M M, O R S Barbosa Contextuality as a resource 26

114 Operations Mixing e, e : X, M, O λ [0, 1] e + λ e : X, M, O For C M, s : C O, (e + λ e ) C (s) := λe C (s) + (1 λ)e C(s) Choice e : X, M, O e : X, M, O e & e : X X, M M, O For C M, (e & e ) C := e C For D M, (e&e ) D := e D R S Barbosa Contextuality as a resource 26

115 Operations Mixing e, e : X, M, O λ [0, 1] e + λ e : X, M, O For C M, s : C O, (e + λ e ) C (s) := λe C (s) + (1 λ)e C(s) Choice e : X, M, O e : X, M, O e & e : X X, M M, O For C M, (e & e ) C := e C For D M, (e&e ) D := e D Tensor e : X, M, O e : X, M, O e e : X X, M M, O R S Barbosa Contextuality as a resource 26

116 Operations Mixing e, e : X, M, O λ [0, 1] e + λ e : X, M, O For C M, s : C O, (e + λ e ) C (s) := λe C (s) + (1 λ)e C(s) Choice e : X, M, O e : X, M, O e & e : X X, M M, O For C M, (e & e ) C := e C For D M, (e&e ) D := e D Tensor e : X, M, O e : X, M, O e e : X X, M M, O M M := {C D C M, D M } R S Barbosa Contextuality as a resource 26

117 Operations Mixing e, e : X, M, O λ [0, 1] e + λ e : X, M, O For C M, s : C O, (e + λ e ) C (s) := λe C (s) + (1 λ)e C(s) Choice e : X, M, O e : X, M, O e & e : X X, M M, O For C M, (e & e ) C := e C For D M, (e&e ) D := e D Tensor e : X, M, O e : X, M, O e e : X X, M M, O M M := {C D C M, D M } For C M, D M, s = s 1, s 2 : C D O, (e e ) C D s 1, s 2 := e C (s 1 ) e D(s 2 ) R S Barbosa Contextuality as a resource 26

118 Operations and the contextual fraction R S Barbosa Contextuality as a resource 27

119 Operations and the contextual fraction Relabelling e[α] R S Barbosa Contextuality as a resource 27

120 Operations and the contextual fraction Relabelling e[α] Restriction e M R S Barbosa Contextuality as a resource 27

121 Operations and the contextual fraction Relabelling e[α] Restriction e M Coarse-graining e/f R S Barbosa Contextuality as a resource 27

122 Operations and the contextual fraction Relabelling e[α] Restriction e M Coarse-graining e/f Mixing λe + (1 λ)e R S Barbosa Contextuality as a resource 27

123 Operations and the contextual fraction Relabelling e[α] Restriction e M Coarse-graining e/f Mixing λe + (1 λ)e Choice e & e R S Barbosa Contextuality as a resource 27

124 Operations and the contextual fraction Relabelling e[α] Restriction e M Coarse-graining e/f Mixing λe + (1 λ)e Choice e & e Tensor e 1 e 2 R S Barbosa Contextuality as a resource 27

125 Operations and the contextual fraction Relabelling e[α] Restriction e M Coarse-graining e/f Mixing λe + (1 λ)e Choice e & e Tensor e 1 e 2 Sequencing e 1 ; e 2 R S Barbosa Contextuality as a resource 27

126 Operations and the contextual fraction Relabelling CF(e[α]) = CF(e) Restriction e M Coarse-graining e/f Mixing λe + (1 λ)e Choice e & e Tensor e 1 e 2 Sequencing e 1 ; e 2 R S Barbosa Contextuality as a resource 27

127 Operations and the contextual fraction Relabelling Restriction CF(e[α]) = CF(e) CF(e M ) CF(e) Coarse-graining e/f Mixing λe + (1 λ)e Choice e & e Tensor e 1 e 2 Sequencing e 1 ; e 2 R S Barbosa Contextuality as a resource 27

128 Operations and the contextual fraction Relabelling Restriction CF(e[α]) = CF(e) CF(e M ) CF(e) Coarse-graining CF(e/f ) CF (e) Mixing λe + (1 λ)e Choice e & e Tensor e 1 e 2 Sequencing e 1 ; e 2 R S Barbosa Contextuality as a resource 27

129 Operations and the contextual fraction Relabelling Restriction CF(e[α]) = CF(e) CF(e M ) CF(e) Coarse-graining CF(e/f ) CF (e) Mixing CF (λe + (1 λ)e ) λcf(e) + (1 λ)cf(e ) Choice e & e Tensor e 1 e 2 Sequencing e 1 ; e 2 R S Barbosa Contextuality as a resource 27

130 Operations and the contextual fraction Relabelling Restriction CF(e[α]) = CF(e) CF(e M ) CF(e) Coarse-graining CF(e/f ) CF (e) Mixing CF (λe + (1 λ)e ) λcf(e) + (1 λ)cf(e ) Choice CF(e & e ) = max{cf(e), CF(e )} Tensor e 1 e 2 NCF(e 1 e 2 ) = NCF(e 1 ) NCF(e 2 ) Sequencing e 1 ; e 2 R S Barbosa Contextuality as a resource 27

131 Operations and the contextual fraction Relabelling Restriction CF(e[α]) = CF(e) CF(e M ) CF(e) Coarse-graining CF(e/f ) CF (e) Mixing CF (λe + (1 λ)e ) λcf(e) + (1 λ)cf(e ) Choice CF(e & e ) = max{cf(e), CF(e )} Tensor CF(e 1 e 2 ) = CF(e 1 ) + CF(e 2 ) CF(e 1 )CF(e 2 ) NCF(e 1 e 2 ) = NCF(e 1 ) NCF(e 2 ) Sequencing e 1 ; e 2 R S Barbosa Contextuality as a resource 27

132 Operations and the contextual fraction Relabelling Restriction CF(e[α]) = CF(e) CF(e M ) CF(e) Coarse-graining CF(e/f ) CF (e) Mixing CF (λe + (1 λ)e ) λcf(e) + (1 λ)cf(e ) Choice CF(e & e ) = max{cf(e), CF(e )} Tensor CF(e 1 e 2 ) = CF(e 1 ) + CF(e 2 ) CF(e 1 )CF(e 2 ) NCF(e 1 e 2 ) = NCF(e 1 ) NCF(e 2 ) Sequencing CF(e 1 e 2 ) CF(e 1 ) + CF(e 2 ) CF(e 1 )CF(e 2 ) NCF(e 1 ; e 2 ) NCF(e 1 ) NCF(e 2 ) R S Barbosa Contextuality as a resource 27

133 Resource theory of contextuality (some work in progress) R S Barbosa Contextuality as a resource 28

134 Resource theory of contextuality (some work in progress) Resource theory a la Coecke Fritz Spekkens. (resource theory of combinable processes) R S Barbosa Contextuality as a resource 28

135 Resource theory of contextuality (some work in progress) Resource theory a la Coecke Fritz Spekkens. (resource theory of combinable processes) Device-independent processes R S Barbosa Contextuality as a resource 28

136 Resource theory of contextuality (some work in progress) Resource theory a la Coecke Fritz Spekkens. (resource theory of combinable processes) Device-independent processes Operations remind one of process algebra R S Barbosa Contextuality as a resource 28

137 Resource theory of contextuality (some work in progress) Resource theory a la Coecke Fritz Spekkens. (resource theory of combinable processes) Device-independent processes Operations remind one of process algebra Process calculus: operational semantics by (probabilistic) transitions R S Barbosa Contextuality as a resource 28

138 Resource theory of contextuality (some work in progress) Resource theory a la Coecke Fritz Spekkens. (resource theory of combinable processes) Device-independent processes Operations remind one of process algebra Process calculus: operational semantics by (probabilistic) transitions bissimulation, metric / approximation R S Barbosa Contextuality as a resource 28

139 Resource theory of contextuality (some work in progress) Resource theory a la Coecke Fritz Spekkens. (resource theory of combinable processes) Device-independent processes Operations remind one of process algebra Process calculus: operational semantics by (probabilistic) transitions bissimulation, metric / approximation (modal) logic for device-independent processes R S Barbosa Contextuality as a resource 28

140 Resource theory of contextuality (some work in progress) Resource theory a la Coecke Fritz Spekkens. (resource theory of combinable processes) Device-independent processes Operations remind one of process algebra Process calculus: operational semantics by (probabilistic) transitions bissimulation, metric / approximation (modal) logic for device-independent processes Sequencing: R S Barbosa Contextuality as a resource 28

141 Resource theory of contextuality (some work in progress) Resource theory a la Coecke Fritz Spekkens. (resource theory of combinable processes) Device-independent processes Operations remind one of process algebra Process calculus: operational semantics by (probabilistic) transitions bissimulation, metric / approximation (modal) logic for device-independent processes Sequencing: so far, it hides middle steps R S Barbosa Contextuality as a resource 28

142 Resource theory of contextuality (some work in progress) Resource theory a la Coecke Fritz Spekkens. (resource theory of combinable processes) Device-independent processes Operations remind one of process algebra Process calculus: operational semantics by (probabilistic) transitions bissimulation, metric / approximation (modal) logic for device-independent processes Sequencing: so far, it hides middle steps not doing so leads to notion of causal empirical models. R S Barbosa Contextuality as a resource 28

143 Resource theory of contextuality (some work in progress) Resource theory a la Coecke Fritz Spekkens. (resource theory of combinable processes) Device-independent processes Operations remind one of process algebra Process calculus: operational semantics by (probabilistic) transitions bissimulation, metric / approximation (modal) logic for device-independent processes Sequencing: so far, it hides middle steps not doing so leads to notion of causal empirical models. Allow natural expression of measurement-based computation with feed-forward, in a device-independent form: R S Barbosa Contextuality as a resource 28

144 Resource theory of contextuality (some work in progress) Resource theory a la Coecke Fritz Spekkens. (resource theory of combinable processes) Device-independent processes Operations remind one of process algebra Process calculus: operational semantics by (probabilistic) transitions bissimulation, metric / approximation (modal) logic for device-independent processes Sequencing: so far, it hides middle steps not doing so leads to notion of causal empirical models. Allow natural expression of measurement-based computation with feed-forward, in a device-independent form: One can measure a non-maximal context (face σ of complex) R S Barbosa Contextuality as a resource 28

145 Resource theory of contextuality (some work in progress) Resource theory a la Coecke Fritz Spekkens. (resource theory of combinable processes) Device-independent processes Operations remind one of process algebra Process calculus: operational semantics by (probabilistic) transitions bissimulation, metric / approximation (modal) logic for device-independent processes Sequencing: so far, it hides middle steps not doing so leads to notion of causal empirical models. Allow natural expression of measurement-based computation with feed-forward, in a device-independent form: One can measure a non-maximal context (face σ of complex) leaving a model on scenario lkσm R S Barbosa Contextuality as a resource 28

146 Contextual fraction and quantum advantages

147 Contextual fraction and advantages Contextuality has been associated with quantum advantage in information-processing and computational tasks. R S Barbosa Contextuality as a resource 29

148 Contextual fraction and advantages Contextuality has been associated with quantum advantage in information-processing and computational tasks. Measure of contextuality quantify such advantages. R S Barbosa Contextuality as a resource 29

149 Contextual fraction and cooperative games Game described by n formulae (one for each allowed input). These describe the winning condition that the corresponding outputs must satisfy. R S Barbosa Contextuality as a resource 30

150 Contextual fraction and cooperative games Game described by n formulae (one for each allowed input). These describe the winning condition that the corresponding outputs must satisfy. If the formulae are k-consistent (at most k are jointly satisfiable), hardness of the task is n k n. (cf. Abramsky & Hardy, Logical Bell inequalities ) R S Barbosa Contextuality as a resource 30

151 Contextual fraction and cooperative games Game described by n formulae (one for each allowed input). These describe the winning condition that the corresponding outputs must satisfy. If the formulae are k-consistent (at most k are jointly satisfiable), hardness of the task is n k n. (cf. Abramsky & Hardy, Logical Bell inequalities ) We have 1 p S NCF n k n R S Barbosa Contextuality as a resource 30

152 Contextuality and MBQC E.g. Raussendorf (2013) l2-mbqc R S Barbosa Contextuality as a resource 31

153 Contextuality and MBQC E.g. Raussendorf (2013) l2-mbqc measurement-based quantum computing scheme (m input bits, l output bits, n parties) R S Barbosa Contextuality as a resource 31

154 Contextuality and MBQC E.g. Raussendorf (2013) l2-mbqc measurement-based quantum computing scheme (m input bits, l output bits, n parties) classical control: pre-processes input determines the flow of measurements post-processes to produce the output only Z 2 -linear computations. R S Barbosa Contextuality as a resource 31

155 Contextuality and MBQC E.g. Raussendorf (2013) l2-mbqc measurement-based quantum computing scheme (m input bits, l output bits, n parties) classical control: pre-processes input determines the flow of measurements post-processes to produce the output only Z 2 -linear computations. additional power to compute non-linear functions resides in certain resource empirical models. R S Barbosa Contextuality as a resource 31

156 Contextuality and MBQC E.g. Raussendorf (2013) l2-mbqc measurement-based quantum computing scheme (m input bits, l output bits, n parties) classical control: pre-processes input determines the flow of measurements post-processes to produce the output only Z 2 -linear computations. additional power to compute non-linear functions resides in certain resource empirical models. Raussendorf (2013): If an l2-mbqc deterministically computes a non-linear Boolean function f : 2 m 2 l then the resource must be strongly contextual. R S Barbosa Contextuality as a resource 31

157 Contextuality and MBQC E.g. Raussendorf (2013) l2-mbqc measurement-based quantum computing scheme (m input bits, l output bits, n parties) classical control: pre-processes input determines the flow of measurements post-processes to produce the output only Z 2 -linear computations. additional power to compute non-linear functions resides in certain resource empirical models. Raussendorf (2013): If an l2-mbqc deterministically computes a non-linear Boolean function f : 2 m 2 l then the resource must be strongly contextual. Probabilistic version: non-linear function computed with sufficently large probability of success implies contextuality. R S Barbosa Contextuality as a resource 31

158 Contextual fraction and MBQC Goal: Compute Boolean function f : 2 m 2 l using l2-mbqc R S Barbosa Contextuality as a resource 32

159 Contextual fraction and MBQC Goal: Compute Boolean function f : 2 m 2 l using l2-mbqc Hardness of the problem ν(f ) := min {d(f, g) g is Z 2 -linear} (average distance between f and closest Z 2 -linear function) where for Boolean functions f and g, d(f, g) := 2 m {i 2 m f (i) g(i)}. R S Barbosa Contextuality as a resource 32

160 Contextual fraction and MBQC Goal: Compute Boolean function f : 2 m 2 l using l2-mbqc Hardness of the problem ν(f ) := min {d(f, g) g is Z 2 -linear} (average distance between f and closest Z 2 -linear function) where for Boolean functions f and g, d(f, g) := 2 m {i 2 m f (i) g(i)}. Average probability of success computing f (over all 2 m possible inputs): p S. R S Barbosa Contextuality as a resource 32

161 Contextual fraction and MBQC Goal: Compute Boolean function f : 2 m 2 l using l2-mbqc Hardness of the problem ν(f ) := min {d(f, g) g is Z 2 -linear} (average distance between f and closest Z 2 -linear function) where for Boolean functions f and g, d(f, g) := 2 m {i 2 m f (i) g(i)}. Average probability of success computing f (over all 2 m possible inputs): p S. Then, 1 p S NCF(e) ν(f ) R S Barbosa Contextuality as a resource 32

162 Questions...? The contextual fraction as a measure of contextuality Samson Abramsky, RSB, Shane Mansfield PRL 119: (2017), arxiv: [quant-ph] R S Barbosa Contextuality as a resource 33