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

SLIDE 1

Contextuality, Cohomology, and Paradox

(arXiv:1502.03097) Samson Abramsky, Rui Soares Barbosa, Kohei Kishida

(speaking)

, Ray Lal, and Shane Mansfield QPL2015 17 July, 2015

SLIDE 2

Outline

1 Topological model for contextuality. 2 Cohomology: Contextuality is like “impossible figures”. 3 Relation to QM no-go theorems.

1

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Bell Non-Locality

Bell-type setup. Input-output box for (2, 2, 2) scenario:

a0 or a1 b0 or b1 0 or 1 0 or 1

Distribution p(oA, oB | ai, bj) for each context {ai, bj}. So a probability table: (0, 0) (0, 1) (1, 0) (1, 1) (a0, b0)

1/ 2 1/ 2

(a0, b1)

3/ 8 1/ 8 1/ 8 3/ 8

(a1, b0)

3/ 8 1/ 8 1/ 8 3/ 8

(a1, b1)

1/ 8 3/ 8 3/ 8 1/ 8

2

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(0, 0) (0, 1) (1, 0) (1, 1) (a0, b0)

1/ 2 1/ 2

(a0, b1)

3/ 8 1/ 8 1/ 8 3/ 8

(a1, b0)

3/ 8 1/ 8 1/ 8 3/ 8

(a1, b1)

1/ 8 3/ 8 3/ 8 1/ 8

3

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Possiblility table: non-zero → 1 (“possible”) → 0 (“impossible”). Support of a probability table is a possibility table. (0, 0) (0, 1) (1, 0) (1, 1) (a0, b0)

1/ 2 1/ 2

(a0, b1)

3/ 8 1/ 8 1/ 8 3/ 8

(a1, b0)

3/ 8 1/ 8 1/ 8 3/ 8

(a1, b1)

1/ 8 3/ 8 3/ 8 1/ 8

3

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Possiblility table: non-zero → 1 (“possible”) → 0 (“impossible”). Support of a probability table is a possibility table. (0, 0) (0, 1) (1, 0) (1, 1) (a0, b0) 1 1 (a0, b1) 1 1 1 1 (a1, b0) 1 1 1 1 (a1, b1) 1 1 1 1

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Possiblility table: non-zero → 1 (“possible”) → 0 (“impossible”). Support of a probability table is a possibility table. Marginals, convex combination, no-signalling, locality, etc. all carry over to the possibilistic, logical versions. (0, 0) (0, 1) (1, 0) (1, 1) (a0, b0) 1 1 (a0, b1) 1 1 1 1 (a1, b0) 1 1 1 1 (a1, b1) 1 1 1 1

3

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Possiblility table: non-zero → 1 (“possible”) → 0 (“impossible”). Support of a probability table is a possibility table. Marginals, convex combination, no-signalling, locality, etc. all carry over to the possibilistic, logical versions. A table may be logically non-local / contextual. (0, 0) (0, 1) (1, 0) (1, 1) (a0, b0) 1 1 (a0, b1) 1 1 1 1 (a1, b0) 1 1 1 1 (a1, b1) 1 1 1 1

3

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Possiblility table: non-zero → 1 (“possible”) → 0 (“impossible”). Support of a probability table is a possibility table. Marginals, convex combination, no-signalling, locality, etc. all carry over to the possibilistic, logical versions. A table may be logically non-local / contextual. E.g. model by Hardy 1993: (0, 0) (0, 1) (1, 0) (1, 1) (a0, b0) 1 1 1 1 (a0, b1) 1 1 1 (a1, b0) 1 1 1 (a1, b1) 1 1 1 No local probability table has this support. (Logical non-locality / contextuality implies probabilistic one.)

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Theorem (Fine 1982 / Abramsky-Brandenburger 2011). A table p(· | ai, bj)i,j∈{0,1} is local iff

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Theorem (Fine 1982 / Abramsky-Brandenburger 2011). A table p(· | ai, bj)i,j∈{0,1} is local iff

There is a distribution p(· | a0, a1, b0, b1) that gives

each p(· | ai, bj) as a marginal, e.g., p(oA, oB | a0, b0) = ∑

,o′

p(oA, o, oB, o′ | a0, a1, b0, b1);

4

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Theorem (Fine 1982 / Abramsky-Brandenburger 2011). A table p(· | ai, bj)i,j∈{0,1} is local iff

There is a distribution p(· | a0, a1, b0, b1) that gives

each p(· | ai, bj) as a marginal, e.g., p(oA, oB | a0, b0) = ∑

,o′

p(oA, o, oB, o′ | a0, a1, b0, b1);

i.e. a distribution over

deterministic λ(a0,a1,b0,b1)→(0,0,0,0), λ(a0,a1,b0,b1)→(0,0,0,1), . . . λ(a0,a1,b0,b1)→(1,1,1,1);

λ(0,0,0,1)

0, 0 0, 1

a0 b1 1

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Theorem (Fine 1982 / Abramsky-Brandenburger 2011). A table p(· | ai, bj)i,j∈{0,1} is local iff

There is a distribution p(· | a0, a1, b0, b1) that gives

each p(· | ai, bj) as a marginal, e.g., p(oA, oB | a0, b0) = ∑

,o′

p(oA, o, oB, o′ | a0, a1, b0, b1);

i.e. a distribution over

deterministic λ(a0,a1,b0,b1)→(0,0,0,0), λ(a0,a1,b0,b1)→(0,0,0,1), . . . λ(a0,a1,b0,b1)→(1,1,1,1);

λ(0,0,0,1)

0, 0 0, 1

a0 b1 1

i.e. the table is a convex

combination of the deterministic tables for such λ’s.

4

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Theorem (Fine 1982 / Abramsky-Brandenburger 2011). A table p(· | ai, bj)i,j∈{0,1} is local iff

There is a distribution p(· | a0, a1, b0, b1) that gives

each p(· | ai, bj) as a marginal, e.g., p(oA, oB | a0, b0) = ∑

,o′

p(oA, o, oB, o′ | a0, a1, b0, b1);

Upshot. A no-signalling but non-local table is
“Locally consistent”:

able to assign probabilities / possibilities consistently to the family of measurement contexts {ai, bj};

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Theorem (Fine 1982 / Abramsky-Brandenburger 2011). A table p(· | ai, bj)i,j∈{0,1} is local iff

There is a distribution p(· | a0, a1, b0, b1) that gives

each p(· | ai, bj) as a marginal, e.g., p(oA, oB | a0, b0) = ∑

,o′

p(oA, o, oB, o′ | a0, a1, b0, b1);

Upshot. A no-signalling but non-local table is
“Locally consistent”:

able to assign probabilities / possibilities consistently to the family of measurement contexts {ai, bj};

“Globally inconsistent”:

not able to to the set {a0, a1, b0, b1} of all measurements.

4

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Theorem (Fine 1982 / Abramsky-Brandenburger 2011). A table p(· | ai, bj)i,j∈{0,1} is local iff

There is a distribution p(· | a0, a1, b0, b1) that gives

each p(· | ai, bj) as a marginal, e.g., p(oA, oB | a0, b0) = ∑

,o′

p(oA, o, oB, o′ | a0, a1, b0, b1);

Upshot. A no-signalling but non-local table is
“Locally consistent”:

able to assign probabilities / possibilities consistently to the family of measurement contexts {ai, bj};

“Globally inconsistent”:

not able to to the set {a0, a1, b0, b1} of all measurements. Topology on the set of measurements.

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SLIDE 17

Topological Model for Contextuality

Topological spaces of variables and of their values.

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Topological Model for Contextuality

Topological spaces of variables and of their values.

measurements and outcomes
sentences and truth values
questions and answers

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Topological Model for Contextuality

Topological spaces of variables and of their values.

measurements and outcomes
sentences and truth values
questions and answers

For each variable x, x

z

X

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Topological Model for Contextuality

Topological spaces of variables and of their values.

measurements and outcomes
sentences and truth values
questions and answers

For each variable x, a dependent type F(x) of values. x

z

F(x) F() F(z) X

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SLIDE 21

Topological Model for Contextuality

Topological spaces of variables and of their values.

measurements and outcomes
sentences and truth values
questions and answers

For each variable x, a dependent type F(x) of values. “Bundle” ∑

x∈X

F(x) x

z

F(x) F() F(z) X ∑

x∈X

F(x) π

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When we ask several questions, answers may obey constraints:

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When we ask several questions, answers may obey constraints:

laws of physics,

e.g., Charles’s law

laws of logic

ϕ ¬ϕ ¬¬ϕ

t t ff

t

F() F(t)

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When we ask several questions, answers may obey constraints:

laws of physics,

e.g., Charles’s law

laws of logic

ϕ ¬ϕ ¬¬ϕ

t t ff

t

F() F(t) Distinguish good and bad ways of connecting dots in bundles .. . just like “continuous sections”!

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SLIDE 25

Hardy model: 00 01 10 11 a0b0 1 1 1 1 a0b1 1 1 1 a1b0 1 1 1 a1b1 1 1 1

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Hardy model: 00 01 10 11 a0b0 1 1 1 1 a0b1 1 1 1 a1b0 1 1 1 a1b1 1 1 1

a0
a1
b0
b1

7

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Hardy model: 00 01 10 11 a0b0 1 1 1 1 a0b1 1 1 1 a1b0 1 1 1 a1b1 1 1 1

a0
a1
b0
b1

7

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Hardy model: 00 01 10 11 a0b0 1 1 1 1 a0b1 1 1 1 a1b0 1 1 1 a1b1 1 1 1

a0
a1
b0
b1
a0
b0
a1
b1

7

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Hardy model: 00 01 10 11 a0b0 1 1 1 1 a0b1 1 1 1 a1b0 1 1 1 a1b1 1 1 1

a0
b0
a1
b1

7

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Hardy model: 00 01 10 11 a0b0 1 1 1 1 a0b1 1 1 1 a1b0 1 1 1 a1b1 1 1 1

a0
b0
a1
b1
1
1
1
7

SLIDE 31

Hardy model: 00 01 10 11 a0b0 1 1 1 1 a0b1 1 1 1 a1b0 1 1 1 a1b1 1 1 1

a0
b0
a1
b1
1
1
1
7

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Hardy model: 00 01 10 11 a0b0 1 1 1 1 a0b1 1 1 1 a1b0 1 1 1 a1b1 1 1 1

a0
b0
a1
b1
1
1
1
7

SLIDE 33

Hardy model: 00 01 10 11 a0b0 1 1 1 1 a0b1 1 1 1 a1b0 1 1 1 a1b1 1 1 1

a0
b0
a1
b1
1
1
1
7

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Hardy model: 00 01 10 11 a0b0 1 1 1 1 a0b1 1 1 1 a1b0 1 1 1 a1b1 1 1 1

a0
b0
a1
b1
1
1
1
7

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Hardy model: 00 01 10 11 a0b0 1 1 1 1 a0b1 1 1 1 a1b0 1 1 1 a1b1 1 1 1

a0
b0
a1
b1
1
1
1
7

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Hardy model: 00 01 10 11 a0b0 1 1 1 1 a0b1 1 1 1 a1b0 1 1 1 a1b1 1 1 1

a0
b0
a1
b1
1
1
1
Global section: λ(a0,a1,b0,b1)→(1,0,1,0).

7

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Hardy model: 00 01 10 11 a0b0 1 1 1 1 a0b1 1 1 1 a1b0 1 1 1 a1b1 1 1 1

a0
b0
a1
b1
1
1
1
Global section: λ(a0,a1,b0,b1)→(1,0,1,0).

7

SLIDE 38

Hardy model: 00 01 10 11 a0b0 1 1 1 1 a0b1 1 1 1 a1b0 1 1 1 a1b1 1 1 1

a0
b0
a1
b1
1
1
1
Global section: λ(a0,a1,b0,b1)→(1,0,1,0).

7

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Hardy model: 00 01 10 11 a0b0 1 1 1 1 a0b1 1 1 1 a1b0 1 1 1 a1b1 1 1 1

a0
b0
a1
b1
1
1
1
Global section: λ(a0,a1,b0,b1)→(1,0,1,0).

7

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Hardy model: 00 01 10 11 a0b0 1 1 1 1 a0b1 1 1 1 a1b0 1 1 1 a1b1 1 1 1

a0
b0
a1
b1
1
1
1
Global section: λ(a0,a1,b0,b1)→(1,0,1,0).

7

SLIDE 41

Hardy model: 00 01 10 11 a0b0 1 1 1 1 a0b1 1 1 1 a1b0 1 1 1 a1b1 1 1 1

a0
b0
a1
b1
1
1
1
Global section: λ(a0,a1,b0,b1)→(1,0,1,0).

7

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Hardy model: 00 01 10 11 a0b0 1 1 1 1 a0b1 1 1 1 a1b0 1 1 1 a1b1 1 1 1

a0
b0
a1
b1
1
1
1
Global section: λ(a0,a1,b0,b1)→(1,0,1,0).

7

SLIDE 43

Hardy model: 00 01 10 11 a0b0 1 1 1 1 a0b1 1 1 1 a1b0 1 1 1 a1b1 1 1 1

a0
b0
a1
b1
1
1
1
Global section: λ(a0,a1,b0,b1)→(1,0,1,0).

7

SLIDE 44

Hardy model: 00 01 10 11 a0b0 1 1 1 1 a0b1 1 1 1 a1b0 1 1 1 a1b1 1 1 1

a0
b0
a1
b1
1
1
1
Global section: λ(a0,a1,b0,b1)→(1,0,1,0).

7

SLIDE 45

Hardy model: 00 01 10 11 a0b0 1 1 1 1 a0b1 1 1 1 a1b0 1 1 1 a1b1 1 1 1

a0
b0
a1
b1
1
1
1
Global section: λ(a0,a1,b0,b1)→(1,0,1,0).

Logical contextuality: Not all sections extend to global ones.

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Hardy model: 00 01 10 11 a0b0 1 1 1 1 a0b1 1 1 1 a1b0 1 1 1 a1b1 1 1 1

a0
b0
a1
b1
1
1
1
Global section: λ(a0,a1,b0,b1)→(1,0,1,0).

Logical contextuality: Not all sections extend to global ones. Local consistency, global inconsistency

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Hardy:

a0
b0
a1
b1
1
1
1
Logical contextuality: Not all sections extend to global.

8

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00 01 10 11 a0b0 1 1 a0b1 1 1 a1b0 1 1 a1b1 1 1 PR box: Logical contextuality: Not all sections extend to global.

8

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00 01 10 11 a0b0 1 1 a0b1 1 1 a1b0 1 1 a1b1 1 1 PR box:

a0
b0
a1
b1
1
1
1
Logical contextuality: Not all sections extend to global.

8

SLIDE 50

00 01 10 11 a0b0 1 1 a0b1 1 1 a1b0 1 1 a1b1 1 1 PR box:

a0
b0
a1
b1
1
1
1
Logical contextuality: Not all sections extend to global.

8

SLIDE 51

00 01 10 11 a0b0 1 1 a0b1 1 1 a1b0 1 1 a1b1 1 1 PR box:

a0
b0
a1
b1
1
1
1
Logical contextuality: Not all sections extend to global.

8

SLIDE 52

00 01 10 11 a0b0 1 1 a0b1 1 1 a1b0 1 1 a1b1 1 1 PR box:

a0
b0
a1
b1
1
1
1
Logical contextuality: Not all sections extend to global.

8

SLIDE 53

00 01 10 11 a0b0 1 1 a0b1 1 1 a1b0 1 1 a1b1 1 1 PR box:

a0
b0
a1
b1
1
1
1
Logical contextuality: Not all sections extend to global.

8

SLIDE 54

Hardy:

a0
b0
a1
b1
1
1
1
PR box:
a0
b0
a1
b1
1
1
1
Logical contextuality: Not all sections extend to global.

8

SLIDE 55

Hardy:

a0
b0
a1
b1
1
1
1
PR box:
a0
b0
a1
b1
1
1
1
Logical contextuality: Not all sections extend to global.

8

SLIDE 56

Hardy:

a0
b0
a1
b1
1
1
1
PR box:
a0
b0
a1
b1
1
1
1
Logical contextuality: Not all sections extend to global.

8

SLIDE 57

Hardy:

a0
b0
a1
b1
1
1
1
PR box:
a0
b0
a1
b1
1
1
1
Logical contextuality: Not all sections extend to global.

Strong contextuality: No global section at all.

8

SLIDE 58

Hardy:

a0
b0
a1
b1
1
1
1
PR box:
a0
b0
a1
b1
1
1
1
Logical contextuality: Not all sections extend to global.

Strong contextuality: No global section at all. Hieararchy of contextuality: Probabilistic ⊋ Logical ⊋ Strong contextuality

8

SLIDE 59

Hardy:

a0
b0
a1
b1
1
1
1
a
b
c
1
1
Logical contextuality: Not all sections extend to global.

Strong contextuality: No global section at all. Hieararchy of contextuality: Probabilistic ⊋ Logical ⊋ Strong contextuality

8

SLIDE 60

Contextuality in Logical Paradoxes

Read bundles π : ∑

x∈X F(x) → X in logic terms:

x ∈ X are sentences,

t t, ff ∈ F(x) are truth values.

9

SLIDE 61

Contextuality in Logical Paradoxes

“West is true” “South is true” “East is true” “North is false”

Read bundles π : ∑

x∈X F(x) → X in logic terms:

x ∈ X are sentences,

t t, ff ∈ F(x) are truth values.

9

SLIDE 62

Contextuality in Logical Paradoxes

“West is true” “South is true” “East is true” “North is false”

Read bundles π : ∑

x∈X F(x) → X in logic terms:

x ∈ X are sentences,

t t, ff ∈ F(x) are truth values.

9

SLIDE 63

Contextuality in Logical Paradoxes

“West is true” “South is true” “East is true” “North is false”

t

t

ff
ff
t

t

ff
t

t

9

SLIDE 64

Contextuality in Logical Paradoxes

“West is true” “South is true” “East is true” “North is false”

t

t

ff
ff
t

t

ff
t

t

9

SLIDE 65

Contextuality in Logical Paradoxes

“West is true” “South is true” “East is true” “North is false”

t

t

ff
ff
t

t

ff
t

t

9

SLIDE 66

Contextuality in Logical Paradoxes

“West is true” “South is true” “East is true” “North is false”

t

t

ff
ff
t

t

ff
t

t

9

SLIDE 67

Contextuality in Logical Paradoxes

“West is true” “South is true” “East is true” “North is false”

t

t

ff
ff
t

t

ff
t

t

9

SLIDE 68

Contextuality in Logical Paradoxes

“West is true” “South is true” “East is true” “North is false”

t

t

ff
ff
t

t

ff
t

t

9

SLIDE 69

Contextuality in Logical Paradoxes

“West is true” “South is true” “East is true” “North is false”

t

t

ff
ff
t

t

ff
t

t

9

SLIDE 70

Contextuality in Logical Paradoxes

“West is true” “South is true” “East is true” “North is false”

t

t

ff
ff
t

t

ff
t

t

This type of logical paradoxes (incl. the Liar Paradox) have

the same topology as “paradoxes” of (strong) contextuality.

9

SLIDE 71

How to Formally Define ...

Bundles that correspond to no-signalling possibility tables.

10

SLIDE 72

How to Formally Define ...

Bundles that correspond to no-signalling possibility tables. Two equivalent formulations:

10

SLIDE 73

How to Formally Define ...

Bundles that correspond to no-signalling possibility tables. Two equivalent formulations:

X

Σ π

1 Map of

simplicial complexes π : ∑

x∈X

F(x) → X. (With some axioms, e.g. no-signalling.)

10

SLIDE 74

How to Formally Define ...

Bundles that correspond to no-signalling possibility tables. Two equivalent formulations:

X

Σ π

1 Map of

simplicial complexes π : ∑

x∈X

F(x) → X. (With some axioms, e.g. no-signalling.)

10

SLIDE 75

How to Formally Define ...

Bundles that correspond to no-signalling possibility tables. Two equivalent formulations:

X

Σ π

1 Map of

simplicial complexes π : ∑

x∈X

F(x) → X. (With some axioms, e.g. no-signalling.)

10

SLIDE 76

How to Formally Define ...

Bundles that correspond to no-signalling possibility tables. Two equivalent formulations:

X

Σ π

1 Map of

simplicial complexes π : ∑

x∈X

F(x) → X. (With some axioms, e.g. no-signalling.)

10

SLIDE 77

How to Formally Define ...

Bundles that correspond to no-signalling possibility tables. Two equivalent formulations:

X

Σ π

1 Map of

simplicial complexes π : ∑

x∈X

F(x) → X. (With some axioms, e.g. no-signalling.)

10

SLIDE 78

How to Formally Define ...

Bundles that correspond to no-signalling possibility tables. Two equivalent formulations:

X

Σ π C(X)op Sets F

1 Map of

simplicial complexes π : ∑

x∈X

F(x) → X.

2 Presheaf

F : C(X)op → Sets. (With some axioms, e.g. no-signalling.)

10

SLIDE 79

How to Formally Define ...

Bundles that correspond to no-signalling possibility tables. Two equivalent formulations:

X

Σ π C(X)op Sets F

1 Map of

simplicial complexes π : ∑

x∈X

F(x) → X.

2 Presheaf

F : C(X)op → Sets. (With some axioms, e.g. no-signalling.)

10

SLIDE 80

How to Formally Define ...

Bundles that correspond to no-signalling possibility tables. Two equivalent formulations:

X

Σ π C(X)op Sets F

1 Map of

simplicial complexes π : ∑

x∈X

F(x) → X.

2 Presheaf

F : C(X)op → Sets. (With some axioms, e.g. no-signalling.)

10

SLIDE 81

How to Formally Define ...

Bundles that correspond to no-signalling possibility tables. Two equivalent formulations:

X

Σ π C(X)op Sets F

1 Map of

simplicial complexes π : ∑

x∈X

F(x) → X.

2 Presheaf

F : C(X)op → Sets. (With some axioms, e.g. no-signalling.)

10

SLIDE 82

How to Formally Define ...

Bundles that correspond to no-signalling possibility tables. Two equivalent formulations:

X

Σ π C(X)op Sets F

1 Map of

simplicial complexes π : ∑

x∈X

F(x) → X.

2 Presheaf

F : C(X)op → Sets. (With some axioms, e.g. no-signalling.)

10

SLIDE 83

How to Formally Define ...

Bundles that correspond to no-signalling possibility tables. Two equivalent formulations:

X

Σ π C(X)op Sets F

1 Map of

simplicial complexes π : ∑

x∈X

F(x) → X.

2 Presheaf

F : C(X)op → Sets. (With some axioms, e.g. no-signalling.)

10

SLIDE 84

How to Formally Define ...

Bundles that correspond to no-signalling possibility tables. Two equivalent formulations:

X

Σ π C(X)op Sets F

1 Map of

simplicial complexes π : ∑

x∈X

F(x) → X.

2 Presheaf

F : C(X)op → Sets. (With some axioms, e.g. no-signalling.) (Global sections can be defined suitably.)

10

SLIDE 85

How to Formally Define ...

Bundles that correspond to no-signalling possibility tables. Two equivalent formulations:

X

Σ π C(X)op Sets F

1 Map of

simplicial complexes π : ∑

x∈X

F(x) → X.

2 Presheaf

F : C(X)op → Sets. (With some axioms, e.g. no-signalling.) (Global sections can be defined suitably.)

2 makes it possible to apply cohomology.

10

SLIDE 86

Cohomology of Contextuality

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

11

SLIDE 87

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

a0
b0
a1
b1
1
1
1
12

SLIDE 88

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

a0
b0
a1
b1
1
1
1
12

SLIDE 89

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

a0
b0
a1
b1
1
1
1
12

SLIDE 90

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

a0
b0
a1
b1
1
1
1
12

SLIDE 91

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

a0
b0
a1
b1
1
1
1
12

SLIDE 92

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

a0
b0
a1
b1
1
1
1
12

SLIDE 93

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

False positives,

e.g. in Hardy model:

a0
b0
a1
b1
1
1
1
12

SLIDE 94

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

False positives,

e.g. in Hardy model.

Works for many cases;

e.g. PR box:

a0
b0
a1
b1
1
1
1
12

SLIDE 95

“All vs Nothing” Argument

13

SLIDE 96

“All vs Nothing” Argument

Joint outcomes may / may not satisfy parity equations: (0, 0) x ⊕ = 0 (0, 1) x ⊕ = 1 (1, 0) x ⊕ = 1 (1, 1) x ⊕ = 0

a0
b0
a1
b1
1
1
1
13

SLIDE 97

“All vs Nothing” Argument

Joint outcomes may / may not satisfy parity equations: (0, 0) x ⊕ = 0 (0, 1) x ⊕ = 1 (1, 0) x ⊕ = 1 (1, 1) x ⊕ = 0 a0 ⊕ b0 = 0 a0 ⊕ b1 = 0 a1 ⊕ b0 = 0 a1 ⊕ b1 = 1

a0
b0
a1
b1
1
1
1
13

SLIDE 98

“All vs Nothing” Argument

Joint outcomes may / may not satisfy parity equations: (0, 0) x ⊕ = 0 (0, 1) x ⊕ = 1 (1, 0) x ⊕ = 1 (1, 1) x ⊕ = 0 a0 ⊕ b0 = 0 a0 ⊕ b1 = 0 a1 ⊕ b0 = 0 a1 ⊕ b1 = 1 ⊕ LHS’s = ⊕ RHS’s

a0
b0
a1
b1
1
1
1
13

SLIDE 99

“All vs Nothing” Argument

Joint outcomes may / may not satisfy parity equations: (0, 0) x ⊕ = 0 (0, 1) x ⊕ = 1 (1, 0) x ⊕ = 1 (1, 1) x ⊕ = 0 a0 ⊕ b0 = 0 a0 ⊕ b1 = 0 a1 ⊕ b0 = 0 a1 ⊕ b1 = 1 ⊕ LHS’s ⊕ RHS’s

a0
b0
a1
b1
1
1
1
The equations are inconsistent,

13

SLIDE 100

“All vs Nothing” Argument

Joint outcomes may / may not satisfy parity equations: (0, 0) x ⊕ = 0 (0, 1) x ⊕ = 1 (1, 0) x ⊕ = 1 (1, 1) x ⊕ = 0 a0 ⊕ b0 = 0 a0 ⊕ b1 = 0 a1 ⊕ b0 = 0 a1 ⊕ b1 = 1 ⊕ LHS’s ⊕ RHS’s

a0
b0
a1
b1
1
1
1
The equations are inconsistent,

i.e. no global assignment to a0, a1, b0, b1, i.e. strongly contextual!

13

SLIDE 101

“All vs nothing” arguments in QM can be formulated the same way.

GHZ state:

a0 ⊕ b0 ⊕ c0 = 0 a0 ⊕ b1 ⊕ c1 = 1 a1 ⊕ b0 ⊕ c1 = 1 a1 ⊕ b1 ⊕ c0 = 1 ⊕ LHS’s = 0 1 = ⊕ RHS’s

Kochen-Specker-type:

18 variables, each occurs twice, so ⊕ LHS’s = 0; 9 equations, all of parity 1, so ⊕ RHS’s = 1.

14

SLIDE 102

Beyond QM, some NS tables suggest generalization.

15

SLIDE 103

Beyond QM, some NS tables suggest generalization.

“Box 25” of Pironio-Bancal-Scarani 2011

admits no parity argument, but satisfies a0 + 2b0 ≡ 0 mod 3 a1 + 2c0 ≡ 0 mod 3 a0 + b1 + c0 ≡ 2 mod 3 a0 + b1 + c1 ≡ 2 mod 3 a1 + b0 + c1 ≡ 2 mod 3 a1 + b1 + c1 ≡ 2 mod 3

15

SLIDE 104

Beyond QM, some NS tables suggest generalization.

“Box 25” of Pironio-Bancal-Scarani 2011

admits no parity argument, but satisfies a0 + 2b0 ≡ 0 mod 3 a1 + 2c0 ≡ 0 mod 3 a0 + b1 + c0 ≡ 2 mod 3 a0 + b1 + c1 ≡ 2 mod 3 a1 + b0 + c1 ≡ 2 mod 3 a1 + b1 + c1 ≡ 2 mod 3 ∑ LHS’s ≡ 0 mod 3 ∑ RHS’s ≡ 2 mod 3

15

SLIDE 105

Beyond QM, some NS tables suggest generalization.

“Box 25” of Pironio-Bancal-Scarani 2011

admits no parity argument, but satisfies a0 + 2b0 ≡ 0 mod 3 a1 + 2c0 ≡ 0 mod 3 a0 + b1 + c0 ≡ 2 mod 3 a0 + b1 + c1 ≡ 2 mod 3 a1 + b0 + c1 ≡ 2 mod 3 a1 + b1 + c1 ≡ 2 mod 3 ∑ LHS’s ≡ 0 mod 3 ∑ RHS’s ≡ 2 mod 3 Generalized all-vs-nothing argument uses any commutative ring R (e.g. Zn) instead of Z2:

15

SLIDE 106

Beyond QM, some NS tables suggest generalization.

“Box 25” of Pironio-Bancal-Scarani 2011

admits no parity argument, but satisfies a0 + 2b0 ≡ 0 mod 3 a1 + 2c0 ≡ 0 mod 3 a0 + b1 + c0 ≡ 2 mod 3 a0 + b1 + c1 ≡ 2 mod 3 a1 + b0 + c1 ≡ 2 mod 3 a1 + b1 + c1 ≡ 2 mod 3 ∑ LHS’s ≡ 0 mod 3 ∑ RHS’s ≡ 2 mod 3 Generalized all-vs-nothing argument uses any commutative ring R (e.g. Zn) instead of Z2:

Linear equations k0x0 + · · · + kmxm = p

(k0, . . . , km, p ∈ R).

15

SLIDE 107

Beyond QM, some NS tables suggest generalization.

“Box 25” of Pironio-Bancal-Scarani 2011

admits no parity argument, but satisfies a0 + 2b0 ≡ 0 mod 3 a1 + 2c0 ≡ 0 mod 3 a0 + b1 + c0 ≡ 2 mod 3 a0 + b1 + c1 ≡ 2 mod 3 a1 + b0 + c1 ≡ 2 mod 3 a1 + b1 + c1 ≡ 2 mod 3 ∑ LHS’s ≡ 0 mod 3 ∑ RHS’s ≡ 2 mod 3 Generalized all-vs-nothing argument uses any commutative ring R (e.g. Zn) instead of Z2:

Linear equations k0x0 + · · · + kmxm = p

(k0, . . . , km, p ∈ R).

Equations are inconsistent if a subset of them is s.th.
coefficients k of each variable x add up to 0,
parities p do not.

15

SLIDE 108

“Strongly contextual by AvN argument” is explained by “strongly contextual by cohomology”: Theorem. Let M be a no-signalling bundle model. Then

M admits a generalized AvN argument in a ring R

implies

Cohomology (using R) has γs = 0 for no section s in M.

16

SLIDE 109

“Strongly contextual by AvN argument” is explained by “strongly contextual by cohomology”: Theorem. Let M be a no-signalling bundle model. Then

M admits a generalized AvN argument in a ring R

implies

Cohomology (using R) has γs = 0 for no section s in M.

Hieararchy of strong contextuality: AvN

gen. AvN
cohom. SC

SC ⊊ ⊊ ⊆

16

SLIDE 110

“Strongly contextual by AvN argument” is explained by “strongly contextual by cohomology”: Theorem. Let M be a no-signalling bundle model. Then

M admits a generalized AvN argument in a ring R

implies

Cohomology (using R) has γs = 0 for no section s in M.

Hieararchy of strong contextuality: AvN

gen. AvN
cohom. SC

SC ⊊ ⊊ ⊆ SC ∩ Q ⊆ ?

16

SLIDE 111

Conclusion

General, structural formalism independent of QM formalism. Uniform methods of detecting / showing contextuality.

17

SLIDE 112

Conclusion

General, structural formalism independent of QM formalism. Uniform methods of detecting / showing contextuality.

Contextuality—local consistency, global inconsistency—

is topological in nature, expressed nicely with bundles.

17

SLIDE 113

Conclusion

General, structural formalism independent of QM formalism. Uniform methods of detecting / showing contextuality.

Contextuality—local consistency, global inconsistency—

is topological in nature, expressed nicely with bundles.

They capture contextuality as a phenomenon found in

various fields, e.g. logical paradoxes.

17

SLIDE 114

Conclusion

General, structural formalism independent of QM formalism. Uniform methods of detecting / showing contextuality.

Contextuality—local consistency, global inconsistency—

is topological in nature, expressed nicely with bundles.

They capture contextuality as a phenomenon found in

various fields, e.g. logical paradoxes.

Applying cohomology shows that contextuality is a

topological invariant of our bundles.

17

SLIDE 115

Conclusion

General, structural formalism independent of QM formalism. Uniform methods of detecting / showing contextuality.

Contextuality—local consistency, global inconsistency—

is topological in nature, expressed nicely with bundles.

They capture contextuality as a phenomenon found in

various fields, e.g. logical paradoxes.

Applying cohomology shows that contextuality is a

topological invariant of our bundles.

We have the all-vs-nothing argument in QM precisely

formulated and generalized. It shows strong contextuality

f a large class of models.

17

SLIDE 116

Conclusion

General, structural formalism independent of QM formalism. Uniform methods of detecting / showing contextuality.

Contextuality—local consistency, global inconsistency—

is topological in nature, expressed nicely with bundles.

They capture contextuality as a phenomenon found in

various fields, e.g. logical paradoxes.

Applying cohomology shows that contextuality is a

topological invariant of our bundles.

We have the all-vs-nothing argument in QM precisely

formulated and generalized. It shows strong contextuality

f a large class of models.
Their contextuality is captured by cohomology.

17

SLIDE 117

References

[1] Abramsky, Barbosa, Kishida, Lal, and Mansfield (2015), “Contextuality, cohomology and paradox”, arXiv:1502.03097 [2] Abramsky and Brandenburger (2011), “The sheaf-theoretic structure of non-locality and contextuality”, NJP [3] Abramsky, Mansfield, and Barbosa (2011), “The cohomology of non- locality and contextuality”, QPL2011 [4] Hardy (1993), “Nonlocality for two particles without inequalities for almost all entangled states”, PRL [5] Fine (1982), “Hidden variables, joint probability, and the Bell inequalities”, PRL [6] Penrose (1991), “On the cohomology of impossible figures”, Structural Topology [7] Mermin (1990), “Extreme quantum entanglement in a superposition of macroscopically distinct states”, PRL [8] Pironio, Bancal, and Scarani (2011), “Extremal correlations of the tripartite no-signaling polytope”, J. Phys. A: Math. Theor.

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