Contextuality, Cohomology, and Paradox (arXiv:1502.03097) Samson A - PowerPoint PPT Presentation

0 • • 0 • • 0 Hardy model: • 1 • 00 01 10 11 • 1 a 0 b 0 1 1 1 1 • 1 a 0 b 1 0 1 1 1 b 1 a 1 b 0 0 1 1 1 • a 1 b 1 1 1 1 0 • a 1 a 0 • • b 0 7

0 • • 0 • • 0 Hardy model: • 1 • 00 01 10 11 • 1 a 0 b 0 1 1 1 1 • 1 a 0 b 1 0 1 1 1 b 1 a 1 b 0 0 1 1 1 • a 1 b 1 1 1 1 0 • a 1 a 0 • • b 0 7

0 • • 0 • • 0 Hardy model: • 1 • 00 01 10 11 • 1 a 0 b 0 1 1 1 1 • 1 a 0 b 1 0 1 1 1 b 1 a 1 b 0 0 1 1 1 • a 1 b 1 1 1 1 0 • a 1 a 0 • • b 0 7

0 • • 0 • • 0 Hardy model: • 1 • 00 01 10 11 • 1 a 0 b 0 1 1 1 1 • 1 a 0 b 1 0 1 1 1 b 1 a 1 b 0 0 1 1 1 • a 1 b 1 1 1 1 0 • a 1 a 0 • • b 0 Global section: λ ( a 0 , a 1 , b 0 , b 1 ) �→ (1 , 0 , 1 , 0) . 7

0 • • 0 • • 0 Hardy model: • 1 • 00 01 10 11 • 1 a 0 b 0 1 1 1 1 • 1 a 0 b 1 0 1 1 1 b 1 a 1 b 0 0 1 1 1 • a 1 b 1 1 1 1 0 • a 1 a 0 • • b 0 Global section: λ ( a 0 , a 1 , b 0 , b 1 ) �→ (1 , 0 , 1 , 0) . 7

0 • • 0 • • 0 Hardy model: • 1 • 00 01 10 11 • 1 a 0 b 0 1 1 1 1 • 1 a 0 b 1 0 1 1 1 b 1 a 1 b 0 0 1 1 1 • a 1 b 1 1 1 1 0 • a 1 a 0 • • b 0 Global section: λ ( a 0 , a 1 , b 0 , b 1 ) �→ (1 , 0 , 1 , 0) . 7

0 • • 0 • • 0 Hardy model: • 1 • 00 01 10 11 • 1 a 0 b 0 1 1 1 1 • 1 a 0 b 1 0 1 1 1 b 1 a 1 b 0 0 1 1 1 • a 1 b 1 1 1 1 0 • a 1 a 0 • • b 0 Global section: λ ( a 0 , a 1 , b 0 , b 1 ) �→ (1 , 0 , 1 , 0) . 7

0 • • 0 • • 0 Hardy model: • 1 • 00 01 10 11 • 1 a 0 b 0 1 1 1 1 • 1 a 0 b 1 0 1 1 1 b 1 a 1 b 0 0 1 1 1 • a 1 b 1 1 1 1 0 • a 1 a 0 • • b 0 Global section: λ ( a 0 , a 1 , b 0 , b 1 ) �→ (1 , 0 , 1 , 0) . 7

0 • • 0 • • 0 Hardy model: • 1 • 00 01 10 11 • 1 a 0 b 0 1 1 1 1 • 1 a 0 b 1 0 1 1 1 b 1 a 1 b 0 0 1 1 1 • a 1 b 1 1 1 1 0 • a 1 a 0 • • b 0 Global section: λ ( a 0 , a 1 , b 0 , b 1 ) �→ (1 , 0 , 1 , 0) . 7

0 • • 0 • • 0 Hardy model: • 1 • 00 01 10 11 • 1 a 0 b 0 1 1 1 1 • 1 a 0 b 1 0 1 1 1 b 1 a 1 b 0 0 1 1 1 • a 1 b 1 1 1 1 0 • a 1 a 0 • • b 0 Global section: λ ( a 0 , a 1 , b 0 , b 1 ) �→ (1 , 0 , 1 , 0) . 7

0 • • 0 • • 0 Hardy model: • 1 • 00 01 10 11 • 1 a 0 b 0 1 1 1 1 • 1 a 0 b 1 0 1 1 1 b 1 a 1 b 0 0 1 1 1 • a 1 b 1 1 1 1 0 • a 1 a 0 • • b 0 Global section: λ ( a 0 , a 1 , b 0 , b 1 ) �→ (1 , 0 , 1 , 0) . 7

0 • • 0 • • 0 Hardy model: • 1 • 00 01 10 11 • 1 a 0 b 0 1 1 1 1 • 1 a 0 b 1 0 1 1 1 b 1 a 1 b 0 0 1 1 1 • a 1 b 1 1 1 1 0 • a 1 a 0 • • b 0 Global section: λ ( a 0 , a 1 , b 0 , b 1 ) �→ (1 , 0 , 1 , 0) . 7

0 • • 0 • • 0 Hardy model: • 1 • 00 01 10 11 • 1 a 0 b 0 1 1 1 1 • 1 a 0 b 1 0 1 1 1 b 1 a 1 b 0 0 1 1 1 • a 1 b 1 1 1 1 0 • a 1 a 0 • • b 0 Global section: λ ( a 0 , a 1 , b 0 , b 1 ) �→ (1 , 0 , 1 , 0) . Logical contextuality: Not all sections extend to global ones. 7

0 • • 0 • • 0 Hardy model: • 1 • 00 01 10 11 • 1 a 0 b 0 1 1 1 1 • 1 a 0 b 1 0 1 1 1 b 1 a 1 b 0 0 1 1 1 • a 1 b 1 1 1 1 0 • a 1 a 0 • • b 0 Global section: λ ( a 0 , a 1 , b 0 , b 1 ) �→ (1 , 0 , 1 , 0) . Logical contextuality: Not all sections extend to global ones. Local consistency, global inconsistency 7

Hardy: • • 0 • • 0 • 1 • • 1 • 1 b 1 • • a 1 a 0 • • b 0 Logical contextuality: Not all sections extend to global. 8

PR box: 00 01 10 11 a 0 b 0 1 0 0 1 a 0 b 1 1 0 0 1 a 1 b 0 1 0 0 1 a 1 b 1 0 1 1 0 Logical contextuality: Not all sections extend to global. 8

PR box: • • 0 • • 0 00 01 10 11 • 1 • • 1 a 0 b 0 1 0 0 1 • 1 a 0 b 1 1 0 0 1 a 1 b 0 1 0 0 1 b 1 • a 1 b 1 0 1 1 0 • a 1 a 0 • • b 0 Logical contextuality: Not all sections extend to global. 8

PR box: • • 0 • • 0 00 01 10 11 • 1 • • 1 a 0 b 0 1 0 0 1 • 1 a 0 b 1 1 0 0 1 a 1 b 0 1 0 0 1 b 1 • a 1 b 1 0 1 1 0 • a 1 a 0 • • b 0 Logical contextuality: Not all sections extend to global. 8

PR box: • • 0 • • 0 00 01 10 11 • 1 • • 1 a 0 b 0 1 0 0 1 • 1 a 0 b 1 1 0 0 1 a 1 b 0 1 0 0 1 b 1 • a 1 b 1 0 1 1 0 • a 1 a 0 • • b 0 Logical contextuality: Not all sections extend to global. 8

PR box: • • 0 • • 0 00 01 10 11 • 1 • • 1 a 0 b 0 1 0 0 1 • 1 a 0 b 1 1 0 0 1 a 1 b 0 1 0 0 1 b 1 • a 1 b 1 0 1 1 0 • a 1 a 0 • • b 0 Logical contextuality: Not all sections extend to global. 8

PR box: • • 0 • • 0 00 01 10 11 • 1 • • 1 a 0 b 0 1 0 0 1 • 1 a 0 b 1 1 0 0 1 a 1 b 0 1 0 0 1 b 1 • a 1 b 1 0 1 1 0 • a 1 a 0 • • b 0 Logical contextuality: Not all sections extend to global. 8

Hardy: PR box: • • • 0 • 0 • • • • 0 0 • 1 • 1 • • • • 1 1 • 1 • 1 b 1 b 1 • • • a 1 • a 1 a 0 a 0 • • • b 0 • b 0 Logical contextuality: Not all sections extend to global. 8

Hardy: PR box: • • • 0 • 0 • • • • 0 0 • 1 • 1 • • • • 1 1 • 1 • 1 b 1 b 1 • • • a 1 • a 1 a 0 a 0 • • • b 0 • b 0 Logical contextuality: Not all sections extend to global. 8

Hardy: PR box: • • • 0 • 0 • • • • 0 0 • 1 • 1 • • • • 1 1 • 1 • 1 b 1 b 1 • • • a 1 • a 1 a 0 a 0 • • • b 0 • b 0 Logical contextuality: Not all sections extend to global. 8

Hardy: PR box: • • • 0 • 0 • • • • 0 0 • 1 • 1 • • • • 1 1 • 1 • 1 b 1 b 1 • • • a 1 • a 1 a 0 a 0 • • • b 0 • b 0 Logical contextuality: Not all sections extend to global. Strong contextuality: No global section at all. 8

Hardy: PR box: • • • 0 • 0 • • • • 0 0 • 1 • 1 • • • • 1 1 • 1 • 1 b 1 b 1 • • • a 1 • a 1 a 0 a 0 • • • b 0 • b 0 Logical contextuality: Not all sections extend to global. Strong contextuality: No global section at all. Hieararchy of contextuality: Probabilistic ⊋ Logical ⊋ Strong contextuality 8

Hardy: • • • 0 • • • • 0 0 • • 1 0 • • • 1 1 • 1 • 1 b 1 c • • • a 1 a 0 • • b • a • b 0 Logical contextuality: Not all sections extend to global. Strong contextuality: No global section at all. Hieararchy of contextuality: Probabilistic ⊋ Logical ⊋ Strong contextuality 8

Contextuality in Logical Paradoxes x ∈ X F ( x ) → X in logic terms: Read bundles π : ∑ x ∈ X are sentences, t , ff ∈ F ( x ) are truth values. t 9

Contextuality in Logical Paradoxes x ∈ X F ( x ) → X in logic terms: Read bundles π : ∑ x ∈ X are sentences, t , ff ∈ F ( x ) are truth values. t “West is true” • • “North is false” • “South is true” • “East is true” 9

Contextuality in Logical Paradoxes x ∈ X F ( x ) → X in logic terms: Read bundles π : ∑ x ∈ X are sentences, t , ff ∈ F ( x ) are truth values. t “West is true” • • “North is false” • “South is true” • “East is true” 9

Contextuality in Logical Paradoxes t t • • t t • • t t • ff • • ff • ff “West is true” • • “North is false” • “South is true” • “East is true” 9

Contextuality in Logical Paradoxes t t • • t t • • t t • ff • • ff • ff “West is true” • • “North is false” • “South is true” • “East is true” 9

Contextuality in Logical Paradoxes t t • • t t • • t t • ff • • ff • ff “West is true” • • “North is false” • “South is true” • “East is true” 9

Contextuality in Logical Paradoxes t t • • t t • • t t • ff • • ff • ff “West is true” • • “North is false” • “South is true” • “East is true” 9

Contextuality in Logical Paradoxes t t • • t t • • t t • ff • • ff • ff “West is true” • • “North is false” • “South is true” • “East is true” 9

Contextuality in Logical Paradoxes t t • • t t • • t t • ff • • ff • ff “West is true” • • “North is false” • “South is true” • “East is true” 9

Contextuality in Logical Paradoxes t t • • t t • • t t • ff • • ff • ff “West is true” • • “North is false” • “South is true” • “East is true” 9

Contextuality in Logical Paradoxes t t • • t t • • t t • ff • • ff • ff “West is true” • • “North is false” • “South is true” • “East is true” This type of logical paradoxes (incl. the Liar Paradox) have the same topology as “paradoxes” of (strong) contextuality. 9

How to Formally Define ... Bundles that correspond to no-signalling possibility tables. • • • • • • • • • • • • 10

How to Formally Define ... Bundles that correspond to no-signalling possibility tables. Two equivalent formulations: • • • • • • • • • • • • 10

How to Formally Define ... Bundles that correspond to no-signalling possibility tables. Two equivalent formulations: 1 Map of • simplicial complexes • • • ∑ π : F ( x ) → X . Σ • • x ∈ X • • π • (With some axioms, • e.g. no-signalling.) X • • 10

How to Formally Define ... Bundles that correspond to no-signalling possibility tables. Two equivalent formulations: 1 Map of • simplicial complexes • • • ∑ π : F ( x ) → X . Σ • • x ∈ X • • π • (With some axioms, • e.g. no-signalling.) X • • 10

How to Formally Define ... Bundles that correspond to no-signalling possibility tables. Two equivalent formulations: 1 Map of • simplicial complexes • • • ∑ π : F ( x ) → X . Σ • • x ∈ X • • π • (With some axioms, • e.g. no-signalling.) X • • 10

How to Formally Define ... Bundles that correspond to no-signalling possibility tables. Two equivalent formulations: 1 Map of • simplicial complexes • • • ∑ π : F ( x ) → X . Σ • • x ∈ X • • π • (With some axioms, • e.g. no-signalling.) X • • 10

How to Formally Define ... Bundles that correspond to no-signalling possibility tables. Two equivalent formulations: 1 Map of • simplicial complexes • • • ∑ π : F ( x ) → X . Σ • • x ∈ X • • π • (With some axioms, • e.g. no-signalling.) X • • 10

How to Formally Define ... Bundles that correspond to no-signalling possibility tables. Two equivalent formulations: 1 Map of • simplicial complexes • • • ∑ π : F ( x ) → X . Σ Sets • • x ∈ X • 2 Presheaf • π F F : C ( X ) op → Sets . • (With some axioms, • C ( X ) op e.g. no-signalling.) X • • 10

How to Formally Define ... Bundles that correspond to no-signalling possibility tables. Two equivalent formulations: 1 Map of • simplicial complexes • • • ∑ π : F ( x ) → X . Σ Sets • • x ∈ X • 2 Presheaf • π F F : C ( X ) op → Sets . • (With some axioms, • C ( X ) op e.g. no-signalling.) X • • 10

How to Formally Define ... Bundles that correspond to no-signalling possibility tables. Two equivalent formulations: 1 Map of • simplicial complexes • • • ∑ π : F ( x ) → X . Σ Sets • • x ∈ X • 2 Presheaf • π F F : C ( X ) op → Sets . • (With some axioms, • C ( X ) op e.g. no-signalling.) X • • 10

How to Formally Define ... Bundles that correspond to no-signalling possibility tables. Two equivalent formulations: 1 Map of • simplicial complexes • • • ∑ π : F ( x ) → X . Σ Sets • • x ∈ X • 2 Presheaf • π F F : C ( X ) op → Sets . • (With some axioms, • C ( X ) op e.g. no-signalling.) X • • 10

How to Formally Define ... Bundles that correspond to no-signalling possibility tables. Two equivalent formulations: 1 Map of • simplicial complexes • • • ∑ π : F ( x ) → X . Σ Sets • • x ∈ X • 2 Presheaf • π F F : C ( X ) op → Sets . • (With some axioms, • C ( X ) op e.g. no-signalling.) X • • 10

How to Formally Define ... Bundles that correspond to no-signalling possibility tables. Two equivalent formulations: 1 Map of • simplicial complexes • • • ∑ π : F ( x ) → X . Σ Sets • • x ∈ X • 2 Presheaf • π F F : C ( X ) op → Sets . • (With some axioms, • C ( X ) op e.g. no-signalling.) X • • 10

How to Formally Define ... Bundles that correspond to no-signalling possibility tables. Two equivalent formulations: 1 Map of • simplicial complexes • • • ∑ π : F ( x ) → X . Σ Sets • • x ∈ X • 2 Presheaf • π F F : C ( X ) op → Sets . • (With some axioms, • C ( X ) op e.g. no-signalling.) X • (Global sections can be • defined suitably.) 10

How to Formally Define ... Bundles that correspond to no-signalling possibility tables. Two equivalent formulations: 1 Map of • simplicial complexes • • • ∑ π : F ( x ) → X . Σ Sets • • x ∈ X • 2 Presheaf • π F F : C ( X ) op → Sets . • (With some axioms, • C ( X ) op e.g. no-signalling.) X • (Global sections can be • defined suitably.) 2 makes it possible to apply cohomology. 10

Cohomology of Contextuality Local consistency, global inconsistency.. . Penrose 1991, “On the Cohomology of Impossible Figures”. 11

Cohomological test for contextuality: “ ˇ Cech cohomology” gives a group homomorphism γ that assigns to each section s an “obstruction” γ s s.th. • • 0 • • 0 • 1 • • 1 • 1 b 1 • • a 1 a 0 • • b 0 12

Cohomological test for contextuality: “ ˇ Cech cohomology” gives a group homomorphism γ that assigns to each section s an “obstruction” γ s s.th. s extends to a “cocycle” ⇐⇒ γ s = 0. • • 0 • • 0 • 1 • • 1 • 1 b 1 • • a 1 a 0 • • b 0 12

Cohomological test for contextuality: “ ˇ Cech cohomology” gives a group homomorphism γ that assigns to each section s an “obstruction” γ s s.th. s extends to a “cocycle” ⇐⇒ γ s = 0. • • 0 • • 0 • 1 • • 1 • 1 b 1 • • a 1 a 0 • • b 0 12

Cohomological test for contextuality: “ ˇ Cech cohomology” gives a group homomorphism γ that assigns to each section s an “obstruction” γ s s.th. s extends to a “cocycle” ⇐⇒ γ s = 0. • • 0 • • 0 • 1 • • 1 • 1 b 1 • • a 1 a 0 • • b 0 12

Cohomological test for contextuality: “ ˇ Cech cohomology” gives a group homomorphism γ that assigns to each section s an “obstruction” γ s s.th. s extends to a “cocycle” ⇐⇒ γ s = 0. ⇒ • s extends to global • 0 • • 0 • 1 • • 1 • 1 b 1 • • a 1 a 0 • • b 0 12

Cohomological test for contextuality: “ ˇ Cech cohomology” gives a group homomorphism γ that assigns to each section s an “obstruction” γ s s.th. s extends to a “cocycle” ⇐⇒ γ s = 0. ⇒ ⇍ • s extends to global • 0 • • 0 • 1 • • 1 • 1 b 1 • • a 1 a 0 • • b 0 12

Cohomological test for contextuality: “ ˇ Cech cohomology” gives a group homomorphism γ that assigns to each section s an “obstruction” γ s s.th. s extends to a “cocycle” ⇐⇒ γ s = 0. ⇒ ⇍ • s extends to global • 0 • • 0 • False positives, • 1 • e.g. in Hardy model: • 1 • 1 b 1 • • a 1 a 0 • • b 0 12

Cohomological test for contextuality: “ ˇ Cech cohomology” gives a group homomorphism γ that assigns to each section s an “obstruction” γ s s.th. s extends to a “cocycle” ⇐⇒ γ s = 0. ⇒ ⇍ • s extends to global • 0 • • 0 • False positives, • 1 • e.g. in Hardy model. • 1 • Works for many cases; • 1 e.g. PR box: b 1 • • a 1 a 0 • • b 0 12

“All vs Nothing” Argument 13

“All vs Nothing” Argument Joint outcomes may / may not • satisfy parity equations: • 0 • • (0 , 0) � x ⊕ � = 0 0 • 1 (0 , 1) � x ⊕ � = 1 • 0 • 1 (1 , 0) � x ⊕ � = 1 • 1 (1 , 1) � x ⊕ � = 0 b 1 • • a 1 a 0 • • b 0 13

“All vs Nothing” Argument Joint outcomes may / may not • satisfy parity equations: • 0 • • (0 , 0) � x ⊕ � = 0 0 • 1 (0 , 1) � x ⊕ � = 1 • • 1 (1 , 0) � x ⊕ � = 1 • 1 (1 , 1) � x ⊕ � = 0 b 1 a 0 ⊕ b 0 = 0 • a 0 ⊕ b 1 = 0 • a 1 a 1 ⊕ b 0 = 0 a 0 • a 1 ⊕ b 1 = 1 • b 0 13

“All vs Nothing” Argument Joint outcomes may / may not • satisfy parity equations: • 0 • • (0 , 0) � x ⊕ � = 0 0 • 1 (0 , 1) � x ⊕ � = 1 • • 1 (1 , 0) � x ⊕ � = 1 • 1 (1 , 1) � x ⊕ � = 0 b 1 a 0 ⊕ b 0 = 0 • a 0 ⊕ b 1 = 0 • a 1 a 1 ⊕ b 0 = 0 a 0 • a 1 ⊕ b 1 = 1 • b 0 ⊕ ⊕ LHS’s = RHS’s 13

“All vs Nothing” Argument Joint outcomes may / may not • satisfy parity equations: • 0 • • (0 , 0) � x ⊕ � = 0 0 • 1 (0 , 1) � x ⊕ � = 1 • • 1 (1 , 0) � x ⊕ � = 1 • 1 (1 , 1) � x ⊕ � = 0 b 1 a 0 ⊕ b 0 = 0 • a 0 ⊕ b 1 = 0 • a 1 a 1 ⊕ b 0 = 0 a 0 • a 1 ⊕ b 1 = 1 • b 0 ⊕ ⊕ LHS’s � RHS’s The equations are inconsistent, 13

“All vs Nothing” Argument Joint outcomes may / may not • satisfy parity equations: • 0 • • (0 , 0) � x ⊕ � = 0 0 • 1 (0 , 1) � x ⊕ � = 1 • • 1 (1 , 0) � x ⊕ � = 1 • 1 (1 , 1) � x ⊕ � = 0 b 1 a 0 ⊕ b 0 = 0 • a 0 ⊕ b 1 = 0 • a 1 a 1 ⊕ b 0 = 0 a 0 • a 1 ⊕ b 1 = 1 • b 0 ⊕ ⊕ LHS’s � RHS’s The equations are inconsistent, i.e. no global assignment to a 0 , a 1 , b 0 , b 1 , i.e. strongly contextual! 13