Contextuality A (compatible) empirical model is non-contextual if there exists a global distribution d ∈ Prob ( O X ) (on the joint assignments of out- comes to all measurements) that marginalises to all the e C : ∃ d ∈ Prob ( O X ) . ∀ C ∈M . d | C = e C . S Abramsky, R S Barbosa, & S Mansfield 8/27
Contextuality A (compatible) empirical model is non-contextual if there exists a global distribution d ∈ Prob ( O X ) (on the joint assignments of out- comes to all measurements) that marginalises to all the e C : ∃ d ∈ Prob ( O X ) . ∀ C ∈M . d | C = e C . That is, we can glue all the local information together into a global con- sistent description from which the local information can be recovered. S Abramsky, R S Barbosa, & S Mansfield 8/27
Contextuality A (compatible) empirical model is non-contextual if there exists a global distribution d ∈ Prob ( O X ) (on the joint assignments of out- comes to all measurements) that marginalises to all the e C : ∃ d ∈ Prob ( O X ) . ∀ C ∈M . d | C = e C . That is, we can glue all the local information together into a global con- sistent description from which the local information can be recovered. Contextuality : family of data which is locally consistent but globally inconsistent . S Abramsky, R S Barbosa, & S Mansfield 8/27
Contextuality A (compatible) empirical model is non-contextual if there exists a global distribution d ∈ Prob ( O X ) (on the joint assignments of out- comes to all measurements) that marginalises to all the e C : ∃ d ∈ Prob ( O X ) . ∀ C ∈M . d | C = e C . That is, we can glue all the local information together into a global con- sistent description from which the local information can be recovered. 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 con- textual. S Abramsky, R S Barbosa, & S Mansfield 8/27
Strong contextuality Strong Contextuality: no event can be extended to a global assignment. S Abramsky, R S Barbosa, & S Mansfield 9/27
Strong contextuality • 0 Strong Contextuality: no event can be extended to a • 0 • global assignment. • 0 • 1 • E.g. K–S models, GHZ, the PR • 1 box: • 1 A B ( 0 , 0 ) ( 0 , 1 ) ( 1 , 0 ) ( 1 , 1 ) • b 2 a 1 b 1 � × × � • a 2 a 1 b 2 � × × � a 2 b 1 � × × � a 1 • a 2 b 2 × � � × • b 1 S Abramsky, R S Barbosa, & S Mansfield 9/27
The contextual fraction
The contextual fraction Non-contextuality: global distribution d ∈ Prob ( O X ) such that: ∀ C ∈M . d | C = e C . S Abramsky, R S Barbosa, & S Mansfield 10/27
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? S Abramsky, R S Barbosa, & S Mansfield 10/27
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 . S Abramsky, R S Barbosa, & S Mansfield 10/27
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-contetual fraction : maximum weigth of such a subdistribution. S Abramsky, R S Barbosa, & S Mansfield 10/27
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-contetual fraction : maximum weigth of such a subdistribution. Equivalently, maximum weight λ over all convex decompositions e = λ e NC + ( 1 − λ ) e ′ where e NC is a non-contextual model. S Abramsky, R S Barbosa, & S Mansfield 10/27
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-contetual fraction : maximum weigth of such a subdistribution. Equivalently, maximum weight λ over all convex decompositions e = λ e NC + ( 1 − λ ) e SC where e NC is a non-contextual model. e SC is strongly contextual! S Abramsky, R S Barbosa, & S Mansfield 10/27
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-contetual fraction : maximum weigth of such a subdistribution. Equivalently, maximum weight λ over all convex decompositions e = λ e 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 S Abramsky, R S Barbosa, & S Mansfield 10/27
(Non-)contextual fraction via linear programming Checking contextuality of e corresponds to solving d ∈ R n Find M d = v e such that d ≥ 0 . and S Abramsky, R S Barbosa, & S Mansfield 11/27
(Non-)contextual fraction via linear programming Checking contextuality of e corresponds to solving d ∈ R n Find M d = v e such that d ≥ 0 . and Computing the non-contextual fraction corresponds to solving the fol- lowing linear program: c ∈ R n Find 1 · c maximising M c ≤ v e subject to c ≥ 0 . and S Abramsky, R S Barbosa, & S Mansfield 11/27
E.g. Equatorial measurements on GHZ( n ) (a) (b) Figure: Non-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. S Abramsky, R S Barbosa, & S Mansfield 12/27
Violations of Bell inequalities
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 ∈E ( C ) ◮ a bound R S Abramsky, R S Barbosa, & S Mansfield 13/27
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 ∈E ( C ) ◮ a bound R For a model e , the inequality reads as B α ( e ) ≤ R , where � B α ( e ) := α ( C , s ) e C ( s ) . C ∈M , s ∈E ( C ) S Abramsky, R S Barbosa, & S Mansfield 13/27
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 ∈E ( C ) ◮ a bound R For a model e , the inequality reads as B α ( e ) ≤ R , where � B α ( e ) := α ( C , s ) e C ( s ) . C ∈M , s ∈E ( C ) Wlog we can take R non-negative (in fact, we can take R = 0). S Abramsky, R S Barbosa, & S Mansfield 13/27
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 ∈E ( C ) ◮ a bound R For a model e , the inequality reads as B α ( e ) ≤ R , where � B α ( e ) := α ( C , s ) e C ( s ) . C ∈M , s ∈E ( C ) 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 . S Abramsky, R S Barbosa, & S Mansfield 13/27
Violation of a Bell inequality A Bell inequality establishes a bound for the value of B α ( e ) amongst NC models. S Abramsky, R S Barbosa, & S Mansfield 14/27
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 ∈ E ( C ) } C ∈M S Abramsky, R S Barbosa, & S Mansfield 14/27
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 ∈ E ( 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 S Abramsky, R S Barbosa, & S Mansfield 14/27
Bell inequality violation and the contextual fraction Proposition Let e be an empirical model. S Abramsky, R S Barbosa, & S Mansfield 15/27
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 ) . S Abramsky, R S Barbosa, & S Mansfield 15/27
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 is attained: there exists a Bell inequality whose normalised violation by e is exactly CF ( e ) . S Abramsky, R S Barbosa, & S Mansfield 15/27
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 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 : e = NCF ( e ) e NC + CF ( e ) e SC . S Abramsky, R S Barbosa, & S Mansfield 15/27
Bell inequality violation and the contextual fraction Quantifying Contextuality LP: c ∈ R n Find 1 · c maximising M c ≤ v e subject to c ≥ 0 . and e = λ e NC + ( 1 − λ ) e SC with λ = 1 · x ∗ . SC C Q v e NC S Abramsky, R S Barbosa, & S Mansfield 16/27
Bell inequality violation and the contextual fraction Quantifying Contextuality LP: Dual LP: y ∈ R m c ∈ R n Find Find y · v e minimising 1 · c maximising M T y ≥ 1 M c ≤ v e subject to subject to y ≥ 0 . c ≥ 0 . and and e = λ e NC + ( 1 − λ ) e SC with λ = 1 · x ∗ . SC C Q v e NC S Abramsky, R S Barbosa, & S Mansfield 16/27
Bell inequality violation and the contextual fraction Quantifying Contextuality LP: Dual LP: y ∈ R m c ∈ R n Find Find y · v e minimising 1 · c maximising M T y ≥ 1 M c ≤ v e subject to subject to y ≥ 0 . c ≥ 0 . and and e = λ e NC + ( 1 − λ ) e SC with λ = 1 · x ∗ . a := 1 − |M| y SC C Q v e NC S Abramsky, R S Barbosa, & S Mansfield 16/27
Bell inequality violation and the contextual fraction Quantifying Contextuality LP: Dual LP: y ∈ R m c ∈ R n Find Find y · v e minimising 1 · c maximising M T y ≥ 1 M c ≤ v e subject to subject to y ≥ 0 . c ≥ 0 . and and e = λ e NC + ( 1 − λ ) e SC with λ = 1 · x ∗ . a := 1 − |M| y SC a ∈ R m Find a · v e C maximising M T a ≤ 0 Q subject to v e a ≤ 1 . and NC S Abramsky, R S Barbosa, & S Mansfield 16/27
Bell inequality violation and the contextual fraction Quantifying Contextuality LP: Dual LP: y ∈ R m c ∈ R n Find Find y · v e minimising 1 · c maximising M T y ≥ 1 M c ≤ v e subject to subject to y ≥ 0 . c ≥ 0 . and and e = λ e NC + ( 1 − λ ) e SC with λ = 1 · x ∗ . a := 1 − |M| y SC a ∈ R m Find a · v e C maximising M T a ≤ 0 Q subject to v e a ≤ 1 . and NC computes tight Bell inequality (separating hyperplane) S Abramsky, R S Barbosa, & S Mansfield 16/27
Operations on empirical models
Contextuality as a resource S Abramsky, R S Barbosa, & S Mansfield 17/27
Contextuality as a resource ◮ More than one possible measure of contextuality. S Abramsky, R S Barbosa, & S Mansfield 17/27
Contextuality as a resource ◮ More than one possible measure of contextuality. ◮ What properties should a good measure satisfy? S Abramsky, R S Barbosa, & S Mansfield 17/27
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 S Abramsky, R S Barbosa, & S Mansfield 17/27
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, . . . S Abramsky, R S Barbosa, & S Mansfield 17/27
Algebra of empirical models ◮ Consider operations on empirical models. S Abramsky, R S Barbosa, & S Mansfield 18/27
Algebra of empirical models ◮ Consider operations on empirical models. ◮ These operations should not increase contextuality. S Abramsky, R S Barbosa, & S Mansfield 18/27
Algebra of empirical models ◮ Consider operations on empirical models. ◮ These operations should not increase contextuality. ◮ Write type statements e : � X , M , O � to mean that e is a (compatible) emprical model on the scenario � X , M , O � . S Abramsky, R S Barbosa, & S Mansfield 18/27
Algebra of empirical models ◮ Consider operations on empirical models. ◮ These operations should not increase contextuality. ◮ Write type statements e : � X , M , O � to mean that e is a (compatible) emprical model on the scenario � X , M , O � . ◮ The operations remind one of process algebras. S Abramsky, R S Barbosa, & S Mansfield 18/27
Operations S Abramsky, R S Barbosa, & S Mansfield 19/27
Operations ◮ relabelling e : � X , M , O � , α : ( X , M ) ∼ = ( X ′ , M ′ ) � e [ α ] : � X ′ , M ′ , O � S Abramsky, R S Barbosa, & S Mansfield 19/27
Operations ◮ relabelling e : � X , M , O � , α : ( X , M ) ∼ = ( X ′ , M ′ ) � e [ α ] : � X ′ , M ′ , O � → O , e [ α ] α ( C ) ( s ) := e C ( s ◦ α − 1 ) For C ∈ M , s : α ( C ) − S Abramsky, R S Barbosa, & S Mansfield 19/27
Operations ◮ relabelling e : � X , M , O � , α : ( X , M ) ∼ = ( X ′ , M ′ ) � e [ α ] : � X ′ , M ′ , O � → O , e [ α ] α ( C ) ( s ) := e C ( s ◦ α − 1 ) For C ∈ M , s : α ( C ) − ◮ restriction e : � X , M , O � , ( X ′ , M ′ ) ≤ ( X , M ) � e ↾ M ′ : � X ′ , M ′ , O � S Abramsky, R S Barbosa, & S Mansfield 19/27
Operations ◮ relabelling e : � X , M , O � , α : ( X , M ) ∼ = ( X ′ , M ′ ) � e [ α ] : � X ′ , M ′ , O � → O , e [ α ] α ( C ) ( s ) := e C ( s ◦ α − 1 ) For C ∈ M , s : α ( C ) − ◮ 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 S Abramsky, R S Barbosa, & S Mansfield 19/27
Operations ◮ relabelling e : � X , M , O � , α : ( X , M ) ∼ = ( X ′ , M ′ ) � e [ α ] : � X ′ , M ′ , O � → O , e [ α ] α ( C ) ( s ) := e C ( s ◦ α − 1 ) For C ∈ M , s : α ( C ) − ◮ 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 → O ′ � e / f : � X , M , O ′ � e : � X , M , O � , f : O − S Abramsky, R S Barbosa, & S Mansfield 19/27
Operations ◮ relabelling e : � X , M , O � , α : ( X , M ) ∼ = ( X ′ , M ′ ) � e [ α ] : � X ′ , M ′ , O � → O , e [ α ] α ( C ) ( s ) := e C ( s ◦ α − 1 ) For C ∈ M , s : α ( C ) − ◮ 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 → O ′ � e / f : � X , M , O ′ � e : � X , M , O � , f : O − → O ′ , ( e / f ) C ( s ) := � For C ∈ M , s : C − → O , f ◦ t = s e C ( t ) t : C − S Abramsky, R S Barbosa, & S Mansfield 19/27
Operations S Abramsky, R S Barbosa, & S Mansfield 20/27
Operations ◮ mixing e : � X , M , O � , e ′ : � X , M , O � , λ ∈ [ 0 , 1 ] � e + λ e ′ : � X , M , O � S Abramsky, R S Barbosa, & S Mansfield 20/27
Operations ◮ mixing e : � X , M , O � , e ′ : � X , M , O � , λ ∈ [ 0 , 1 ] � e + λ e ′ : � X , M , O � → O ′ , For C ∈ M , s : C − ( e + λ e ′ ) C ( s ) := λ e C ( s ) + ( 1 − λ ) e ′ C ( s ) S Abramsky, R S Barbosa, & S Mansfield 20/27
Operations ◮ mixing e : � X , M , O � , e ′ : � X , M , O � , λ ∈ [ 0 , 1 ] � e + λ e ′ : � X , M , O � → O ′ , For C ∈ M , s : C − ( e + λ e ′ ) C ( s ) := λ e C ( s ) + ( 1 − λ ) e ′ C ( s ) ◮ choice e : � X , M , O � , e ′ : � X , M , O � � e & e ′ : � X ⊔ X ′ , M ⊔ M ′ , O � S Abramsky, R S Barbosa, & S Mansfield 20/27
Operations ◮ mixing e : � X , M , O � , e ′ : � X , M , O � , λ ∈ [ 0 , 1 ] � e + λ e ′ : � X , M , O � → O ′ , For C ∈ M , s : C − ( e + λ e ′ ) C ( s ) := λ e 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 S Abramsky, R S Barbosa, & S Mansfield 20/27
Operations ◮ mixing e : � X , M , O � , e ′ : � X , M , O � , λ ∈ [ 0 , 1 ] � e + λ e ′ : � X , M , O � → O ′ , For C ∈ M , s : C − ( e + λ e ′ ) C ( s ) := λ e 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 � S Abramsky, R S Barbosa, & S Mansfield 20/27
Operations ◮ mixing e : � X , M , O � , e ′ : � X , M , O � , λ ∈ [ 0 , 1 ] � e + λ e ′ : � X , M , O � → O ′ , For C ∈ M , s : C − ( e + λ e ′ ) C ( s ) := λ e 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 ′ } S Abramsky, R S Barbosa, & S Mansfield 20/27
Operations ◮ mixing e : � X , M , O � , e ′ : � X , M , O � , λ ∈ [ 0 , 1 ] � e + λ e ′ : � X , M , O � → O ′ , For C ∈ M , s : C − ( e + λ e ′ ) C ( s ) := λ e 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 ) S Abramsky, R S Barbosa, & S Mansfield 20/27
Operations and the contextual fraction S Abramsky, R S Barbosa, & S Mansfield 21/27
Operations and the contextual fraction ◮ relabelling CF ( e [ α ]) = CF ( e ) S Abramsky, R S Barbosa, & S Mansfield 21/27
Operations and the contextual fraction ◮ relabelling CF ( e [ α ]) = CF ( e ) ◮ restriction CF ( e ↾ σ ′ ) ≤ CF ( e ) S Abramsky, R S Barbosa, & S Mansfield 21/27
Operations and the contextual fraction ◮ relabelling CF ( e [ α ]) = CF ( e ) ◮ restriction CF ( e ↾ σ ′ ) ≤ CF ( e ) ◮ coarse-graining CF ( e / f ) ≤ CF ( e ) S Abramsky, R S Barbosa, & S Mansfield 21/27
Operations and the contextual fraction ◮ relabelling CF ( e [ α ]) = CF ( e ) ◮ restriction CF ( e ↾ σ ′ ) ≤ CF ( e ) ◮ coarse-graining CF ( e / f ) ≤ CF ( e ) ◮ mixing CF ( e + λ e ′ ) ≤ λ CF ( e ) + ( 1 − λ ) CF ( e ′ ) S Abramsky, R S Barbosa, & S Mansfield 21/27
Operations and the contextual fraction ◮ relabelling CF ( e [ α ]) = CF ( e ) ◮ restriction CF ( e ↾ σ ′ ) ≤ CF ( e ) ◮ coarse-graining CF ( e / f ) ≤ CF ( e ) ◮ mixing CF ( e + λ e ′ ) ≤ λ CF ( e ) + ( 1 − λ ) CF ( e ′ ) ◮ choice CF ( e & e ′ ) = max { CF ( e ) , CF ( e ′ ) } NCF ( e & e ′ ) = min { NCF ( e ) , NCF ( e ′ ) } S Abramsky, R S Barbosa, & S Mansfield 21/27
Operations and the contextual fraction ◮ relabelling CF ( e [ α ]) = CF ( e ) ◮ restriction CF ( e ↾ σ ′ ) ≤ CF ( e ) ◮ coarse-graining CF ( e / f ) ≤ CF ( e ) ◮ mixing CF ( e + λ e ′ ) ≤ λ CF ( e ) + ( 1 − λ ) CF ( e ′ ) ◮ choice CF ( e & e ′ ) = max { CF ( e ) , CF ( e ′ ) } NCF ( e & e ′ ) = min { NCF ( e ) , NCF ( 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 ) S Abramsky, R S Barbosa, & S Mansfield 21/27
Contextual fraction and quantum advantages
Contextual fraction and advantages ◮ Contextuality has been associated with quantum advantage in information-processing and computational tasks. S Abramsky, R S Barbosa, & S Mansfield 22/27
Contextual fraction and advantages ◮ Contextuality has been associated with quantum advantage in information-processing and computational tasks. ◮ Measure of contextuality � to quantify such advantages. S Abramsky, R S Barbosa, & S Mansfield 22/27
Contextual fraction and MBQC E.g. Raussendorf (2013) ℓ 2-MBQC S Abramsky, R S Barbosa, & S Mansfield 23/27
Contextual fraction and MBQC E.g. Raussendorf (2013) ℓ 2-MBQC ◮ measurement-based quantum computing scheme ( m input bits, l output bits, n parties) S Abramsky, R S Barbosa, & S Mansfield 23/27
Contextual fraction and MBQC E.g. Raussendorf (2013) ℓ 2-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. S Abramsky, R S Barbosa, & S Mansfield 23/27
Contextual fraction and MBQC E.g. Raussendorf (2013) ℓ 2-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. S Abramsky, R S Barbosa, & S Mansfield 23/27
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