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Macroscopic non-contextuality as a principle for Almost Quantum Correlations Joe Henson and Ana Bel en Sainz University of Bristol Characterising correlations Nonlocality: nontrivial communication complexity 1 , no advantage for nonlocal

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Macroscopic non-contextuality as a principle for Almost Quantum Correlations

Joe Henson and Ana Bel´ en Sainz

University of Bristol

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Characterising correlations

Nonlocality: nontrivial communication complexity1, no advantage for nonlocal computation2, information causality3, macroscopic locality4, local

  • rthogonality5.

Not enough5,6 → Almost quantum correlations6

1van Dam, PhD thesis, University of Oxford (2000). 2Linden, Popescu, Short, Winter, arXiv:quant-ph/0610097. 3Pawlowski et al., Nature 461, 1101-1104 (2009). 4Navascu´ es and Wunderlich, Proc. Roy. Soc. Lond. A 466:881-890 (2009). 5Fritz et al., Nat. Comm. 4, 2263 (2013). 6Navascu´ es, Guryanova, Hoban and Ac´ ın, Nat. Comm. 6, 6288 (2015). 7 8 Joe Henson, ABS – PRA 91, 042114 (2015) Macroscopic non-contextuality

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Characterising correlations

Nonlocality: nontrivial communication complexity1, no advantage for nonlocal computation2, information causality3, macroscopic locality4, local

  • rthogonality5.

Not enough5,6 → Almost quantum correlations6 Contextuality: Consistent exclusivity7 Not enough7, extra assumptions8 → Q1

1van Dam, PhD thesis, University of Oxford (2000). 2Linden, Popescu, Short, Winter, arXiv:quant-ph/0610097. 3Pawlowski et al., Nature 461, 1101-1104 (2009). 4Navascu´ es and Wunderlich, Proc. Roy. Soc. Lond. A 466:881-890 (2009). 5Fritz et al., Nat. Comm. 4, 2263 (2013). 6Navascu´ es, Guryanova, Hoban and Ac´ ın, Nat. Comm. 6, 6288 (2015). 7Ac´ ın, Fritz, Leverrier and Sainz, Comm. Math. Phys. 334(2), 533-628 (2015). 8Amaral, Terra Cunha and Cabello, PRA 89, 030101 (2014) . Joe Henson, ABS – PRA 91, 042114 (2015) Macroscopic non-contextuality

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Contextuality scenarios

x a

  • 9A. Cabello, S. Severini and A. Winter, arXiv:1010.2163 (2010)
  • 10A. Ac´

ın, T. Fritz, A. Leverrier, ABS arXiv:1210:4084 (2012). Joe Henson, ABS – PRA 91, 042114 (2015) Macroscopic non-contextuality

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Contextuality scenarios

x a “Exclusivity” structure9,10 Set of measurements Set of outcomes Identify outcomes of different measurements: same probability

  • 9A. Cabello, S. Severini and A. Winter, arXiv:1010.2163 (2010)
  • 10A. Ac´

ın, T. Fritz, A. Leverrier, ABS arXiv:1210:4084 (2012). Joe Henson, ABS – PRA 91, 042114 (2015) Macroscopic non-contextuality

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Contextuality scenarios

x a “Exclusivity” structure9,10 Set of measurements Set of outcomes Identify outcomes of different measurements: same probability → measurements “share” outcomes

  • 9A. Cabello, S. Severini and A. Winter, arXiv:1010.2163 (2010)
  • 10A. Ac´

ın, T. Fritz, A. Leverrier, ABS arXiv:1210:4084 (2012). Joe Henson, ABS – PRA 91, 042114 (2015) Macroscopic non-contextuality

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Contextuality scenarios

Hypergraphs: Vertices → events – measurement outcome (a|x) ↔ v Hyperedges → complete measurements – set of

  • utcomes

Sets of allowed p(v)

Joe Henson, ABS – PRA 91, 042114 (2015) Macroscopic non-contextuality

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Contextuality Scenarios

1 1 Probabilistic Model: G(H) p : V → [0, 1], properly normalised Classical models: C(H) Determinism → convex combination of deterministic models Quantum models: Q(H) ∃ H , ρ , {Pv : v ∈ V },

v∈e Pv =

1H

∀ e ∈ E p (v) = tr (ρPv)

Joe Henson, ABS – PRA 91, 042114 (2015) Macroscopic non-contextuality

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Macroscopic Non-Contextuality

Micro scenario D1 D2 D|e| M S s p(v)

Joe Henson, ABS – PRA 91, 042114 (2015) Macroscopic non-contextuality

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Macroscopic Non-Contextuality

Micro scenario D1 D2 D|e| M S s p(v) Macro scenario D1 D2 D|e| M S s1 sN Pe ({Iv}v∈e)

Joe Henson, ABS – PRA 91, 042114 (2015) Macroscopic non-contextuality

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Macroscopic Non-Contextuality

Micro scenario D1 D2 D|e| M S s p(v) Macro scenario D1 D2 D|e| M S s1 sN Pe ({Iv}v∈e) Macroscopic Non-Contextuality: p(v) satisfies MNC if (N → ∞) Pe ({Iv}v∈e) is noncontextual

Joe Henson, ABS – PRA 91, 042114 (2015) Macroscopic non-contextuality

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Macro and micro scenarios

d v

i e = 0, 1 random variable

→ ¯ Iv

e = 1 √ N

N

i=1 (d v i e − p(v)) Joe Henson, ABS – PRA 91, 042114 (2015) Macroscopic non-contextuality

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Macro and micro scenarios

d v

i e = 0, 1 random variable

→ ¯ Iv

e = 1 √ N

N

i=1 (d v i e − p(v))

Constraints Normalisation:

v∈e ¯

Iv

e = 0

∀ e

Joe Henson, ABS – PRA 91, 042114 (2015) Macroscopic non-contextuality

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Macro and micro scenarios

d v

i e = 0, 1 random variable

→ ¯ Iv

e = 1 √ N

N

i=1 (d v i e − p(v))

Constraints Normalisation:

v∈e ¯

Iv

e = 0

∀ e CLT: N → ∞ distribution over ¯ Iv

e is Gaussian

γe

uv = δuvp(v) − p(u)p(v) Joe Henson, ABS – PRA 91, 042114 (2015) Macroscopic non-contextuality

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Macro and micro scenarios

d v

i e = 0, 1 random variable

→ ¯ Iv

e = 1 √ N

N

i=1 (d v i e − p(v))

Constraints Normalisation:

v∈e ¯

Iv

e = 0

∀ e CLT: N → ∞ distribution over ¯ Iv

e is Gaussian

γe

uv = δuvp(v) − p(u)p(v)

MNC: N → ∞ ∃ JPD PNC over the set of intensities for ALL outcomes. Pe({Iv}v∈e) =

v∈V (H)\e dIv

PNC({Iv}v∈V (H))

Joe Henson, ABS – PRA 91, 042114 (2015) Macroscopic non-contextuality

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Macro and micro scenarios

d v

i e = 0, 1 random variable

→ ¯ Iv

e = 1 √ N

N

i=1 (d v i e − p(v))

Constraints Normalisation:

v∈e ¯

Iv

e = 0

∀ e CLT: N → ∞ distribution over ¯ Iv

e is Gaussian

γe

uv = δuvp(v) − p(u)p(v)

MNC: N → ∞ ∃ JPD PNC over the set of intensities for ALL outcomes. Pe({Iv}v∈e) =

v∈V (H)\e dIv

PNC({Iv}v∈V (H)) γuv := ¯ Iu ¯ Iv →

u∈e γuv = 0, γ ≥ 0. Joe Henson, ABS – PRA 91, 042114 (2015) Macroscopic non-contextuality

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Macroscopic non-contextuality and Q1

Macroscopic non-contextuality p ∈ G(H) is MNC if ∃ γ ≥ 0 such that:

  • u∈e γuv = 0;

(u, v ∈ e and u = v) ⇒ γuv = −p(u)p(v); γvv = p(v) − p(v)2.

Joe Henson, ABS – PRA 91, 042114 (2015) Macroscopic non-contextuality

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Macroscopic non-contextuality and Q1

Macroscopic non-contextuality p ∈ G(H) is MNC if ∃ γ ≥ 0 such that:

  • u∈e γuv = 0;

(u, v ∈ e and u = v) ⇒ γuv = −p(u)p(v); γvv = p(v) − p(v)2. Q1 models p ∈ G(H) is a Q1 model if ∃ M ≥ 0 such that:

  • u∈e Muv = p(v) for all u ∈ V (H) ;

(u, v ∈ e and u = v) ⇒ Muv = 0; Mvv = P(v); M1v = P(v) and M11 = 1.

Joe Henson, ABS – PRA 91, 042114 (2015) Macroscopic non-contextuality

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Macroscopic non-contextuality and Q1

Macroscopic non-contextuality p ∈ G(H) is MNC if ∃ γ ≥ 0 such that:

  • u∈e γuv = 0;

(u, v ∈ e and u = v) ⇒ γuv = −p(u)p(v); γvv = p(v) − p(v)2. Q1 models p ∈ G(H) is a Q1 model if ∃ M ≥ 0 such that:

  • u∈e Muv = p(v) for all u ∈ V (H) ;

(u, v ∈ e and u = v) ⇒ Muv = 0; Mvv = P(v); M1v = P(v) and M11 = 1. γuv = Muv − p(u)p(v) − → p is MNC iff p ∈ Q1(H)

Joe Henson, ABS – PRA 91, 042114 (2015) Macroscopic non-contextuality

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Bell scenarios as contextuality scenarios

· · · · · · xk ak x1 a1 xn an P (a1 . . . an|x1 . . . xn)

Joe Henson, ABS – PRA 91, 042114 (2015) Macroscopic non-contextuality

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Bell scenarios as contextuality scenarios

· · · · · · x = (x1, . . . , xn) a = (a1, . . . , an) P (a1 . . . an|x1 . . . xn) → P (a|x)

Joe Henson, ABS – PRA 91, 042114 (2015) Macroscopic non-contextuality

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Bell Scenarios as contextuality scenarios

00|00 01|00 10|00 11|00 00|01 01|01 10|01 11|01 00|10 01|10 10|10 11|10 00|11 01|11 10|11 11|11

B2,2,2 H = Bn,m,d: Vertices – events: {(a1 . . . an|x1 . . . xn)}a1 ... an,x1 ... xn Hyperedges: correlated measurements G(Bn,m,d) = NS(n, m, d)

Joe Henson, ABS – PRA 91, 042114 (2015) Macroscopic non-contextuality

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Bell Scenarios

AFLS11: p is almost quantum12 in (n, m, d) iff p ∈ Q1(Bn,m,d) (n, m, d): p is Almost quantum iff p is MNC in Bn,m,d

11Ac´ ın, Fritz, Leverrier and Sainz, Comm. Math. Phys. 334(2), 533-628 (2015). 12Navascu´ es, Guryanova, Hoban and Ac´ ın, Nat. Comm. 6, 6288 (2015). Joe Henson, ABS – PRA 91, 042114 (2015) Macroscopic non-contextuality

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Conclusions and open problems

Generalise ML to contextuality scenarios Strengthen ML in Bell scenarios → correlated measurements MNC fully characterises almost quantum models without extra assumptions (as opposed to CE) MNC directly applies to multipartite Bell scenarios (as opposed to CE) First characterisation of almost quantum correlations from a physical principle.

Joe Henson, ABS – PRA 91, 042114 (2015) Macroscopic non-contextuality

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Conclusions and open problems

Generalise ML to contextuality scenarios Strengthen ML in Bell scenarios → correlated measurements MNC fully characterises almost quantum models without extra assumptions (as opposed to CE) MNC directly applies to multipartite Bell scenarios (as opposed to CE) First characterisation of almost quantum correlations from a physical principle. From Almost quantum to quantum → sequences of measurements?

Joe Henson, ABS – PRA 91, 042114 (2015) Macroscopic non-contextuality

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Thanks !!!

Joe Henson and ABS – PRA 91, 042114 (2015). (arXiv:1501.06052) Joe Henson, ABS – PRA 91, 042114 (2015) Macroscopic non-contextuality