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 computation 2 , information causality 3 , macroscopic locality 4 , local orthogonality 5 . Not enough 5 , 6 → Almost quantum correlations 6 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
Characterising correlations Nonlocality: nontrivial communication complexity 1 , no advantage for nonlocal computation 2 , information causality 3 , macroscopic locality 4 , local orthogonality 5 . Not enough 5 , 6 → Almost quantum correlations 6 Contextuality: Consistent exclusivity 7 Not enough 7 , extra assumptions 8 → Q 1 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
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
Contextuality scenarios x a “Exclusivity” structure 9 , 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
Contextuality scenarios x a “Exclusivity” structure 9 , 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
Contextuality scenarios Hypergraphs: Vertices → events – measurement outcome ( a | x ) ↔ v Hyperedges → complete measurements – set of outcomes Sets of allowed p ( v ) Joe Henson, ABS – PRA 91, 042114 (2015) Macroscopic non-contextuality
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 , { P v : v ∈ V } , � v ∈ e P v = ∀ e ∈ E 1 H p ( v ) = tr ( ρP v ) Joe Henson, ABS – PRA 91, 042114 (2015) Macroscopic non-contextuality
Macroscopic Non-Contextuality Micro scenario D 1 s D 2 S D | e | M p ( v ) Joe Henson, ABS – PRA 91, 042114 (2015) Macroscopic non-contextuality
Macroscopic Non-Contextuality Micro scenario Macro scenario D 1 D 1 s 1 s D 2 D 2 s N S S D | e | D | e | M M P e ( { I v } v ∈ e ) p ( v ) Joe Henson, ABS – PRA 91, 042114 (2015) Macroscopic non-contextuality
Macroscopic Non-Contextuality Micro scenario Macro scenario D 1 D 1 s 1 s D 2 D 2 s N S S D | e | D | e | M M P e ( { I v } v ∈ e ) p ( v ) Macroscopic Non-Contextuality: p ( v ) satisfies MNC if ( N → ∞ ) P e ( { I v } v ∈ e ) is noncontextual Joe Henson, ABS – PRA 91, 042114 (2015) Macroscopic non-contextuality
Macro and micro scenarios ¯ d v I v 1 � N i =1 ( d v i e = 0 , 1 random variable → e = i e − p ( v )) √ N Joe Henson, ABS – PRA 91, 042114 (2015) Macroscopic non-contextuality
Macro and micro scenarios ¯ d v I v 1 � N i =1 ( d v i e = 0 , 1 random variable → e = i e − p ( v )) √ N Constraints v ∈ e ¯ I v Normalisation: � e = 0 ∀ e Joe Henson, ABS – PRA 91, 042114 (2015) Macroscopic non-contextuality
Macro and micro scenarios ¯ d v I v 1 � N i =1 ( d v i e = 0 , 1 random variable → e = i e − p ( v )) √ N Constraints v ∈ e ¯ I v Normalisation: � e = 0 ∀ e CLT: N → ∞ distribution over ¯ I v e is Gaussian γ e uv = δ uv p ( v ) − p ( u ) p ( v ) Joe Henson, ABS – PRA 91, 042114 (2015) Macroscopic non-contextuality
Macro and micro scenarios ¯ d v I v 1 � N i =1 ( d v i e = 0 , 1 random variable → e = i e − p ( v )) √ N Constraints v ∈ e ¯ I v Normalisation: � e = 0 ∀ e CLT: N → ∞ distribution over ¯ I v e is Gaussian γ e uv = δ uv p ( v ) − p ( u ) p ( v ) MNC: N → ∞ ∃ JPD P NC over the set of intensities for ALL outcomes. � �� v ∈ V ( H ) \ e dI v � P e ( { I v } v ∈ e ) = P NC ( { I v } v ∈ V ( H ) ) Joe Henson, ABS – PRA 91, 042114 (2015) Macroscopic non-contextuality
Macro and micro scenarios ¯ d v I v 1 � N i =1 ( d v i e = 0 , 1 random variable → e = i e − p ( v )) √ N Constraints v ∈ e ¯ I v Normalisation: � e = 0 ∀ e CLT: N → ∞ distribution over ¯ I v e is Gaussian γ e uv = δ uv p ( v ) − p ( u ) p ( v ) MNC: N → ∞ ∃ JPD P NC over the set of intensities for ALL outcomes. � �� v ∈ V ( H ) \ e dI v � P e ( { I v } v ∈ e ) = P NC ( { I v } v ∈ V ( H ) ) I u ¯ γ uv := � ¯ I v � → � u ∈ e γ uv = 0 , γ ≥ 0 . Joe Henson, ABS – PRA 91, 042114 (2015) Macroscopic non-contextuality
Macroscopic non-contextuality and Q 1 Macroscopic non-contextuality p ∈ G ( H ) is MNC if ∃ γ ≥ 0 such that: � u ∈ e γ uv = 0 ; ( u, v ∈ e and u � = v ) ⇒ γ uv = − p ( u ) p ( v ) ; γ vv = p ( v ) − p ( v ) 2 . Joe Henson, ABS – PRA 91, 042114 (2015) Macroscopic non-contextuality
Macroscopic non-contextuality and Q 1 Macroscopic non-contextuality p ∈ G ( H ) is MNC if ∃ γ ≥ 0 such that: � u ∈ e γ uv = 0 ; ( u, v ∈ e and u � = v ) ⇒ γ uv = − p ( u ) p ( v ) ; γ vv = p ( v ) − p ( v ) 2 . Q 1 models p ∈ G ( H ) is a Q 1 model if ∃ M ≥ 0 such that: � u ∈ e M uv = p ( v ) for all u ∈ V ( H ) ; ( u, v ∈ e and u � = v ) ⇒ M uv = 0 ; M vv = P ( v ) ; M 1 v = P ( v ) and M 11 = 1 . Joe Henson, ABS – PRA 91, 042114 (2015) Macroscopic non-contextuality
Macroscopic non-contextuality and Q 1 Macroscopic non-contextuality p ∈ G ( H ) is MNC if ∃ γ ≥ 0 such that: � u ∈ e γ uv = 0 ; ( u, v ∈ e and u � = v ) ⇒ γ uv = − p ( u ) p ( v ) ; γ vv = p ( v ) − p ( v ) 2 . Q 1 models p ∈ G ( H ) is a Q 1 model if ∃ M ≥ 0 such that: � u ∈ e M uv = p ( v ) for all u ∈ V ( H ) ; ( u, v ∈ e and u � = v ) ⇒ M uv = 0 ; M vv = P ( v ) ; M 1 v = P ( v ) and M 11 = 1 . γ uv = M uv − p ( u ) p ( v ) − → p is MNC iff p ∈ Q 1 ( H ) Joe Henson, ABS – PRA 91, 042114 (2015) Macroscopic non-contextuality
Bell scenarios as contextuality scenarios x 1 x k x n · · · · · · a 1 a k a n P ( a 1 . . . a n | x 1 . . . x n ) Joe Henson, ABS – PRA 91, 042114 (2015) Macroscopic non-contextuality
Bell scenarios as contextuality scenarios x = ( x 1 , . . . , x n ) · · · · · · a = ( a 1 , . . . , a n ) P ( a 1 . . . a n | x 1 . . . x n ) → P ( a | x ) Joe Henson, ABS – PRA 91, 042114 (2015) Macroscopic non-contextuality
Bell Scenarios as contextuality scenarios H = B n,m,d : 00 | 00 01 | 00 00 | 01 01 | 01 Vertices – events: 10 | 00 11 | 00 10 | 01 11 | 01 { ( a 1 . . . a n | x 1 . . . x n ) } a 1 ... a n ,x 1 ... x n 00 | 10 01 | 10 00 | 11 01 | 11 Hyperedges: correlated measurements 10 | 10 11 | 10 10 | 11 11 | 11 G ( B n,m,d ) = NS ( n, m, d ) B 2 , 2 , 2 Joe Henson, ABS – PRA 91, 042114 (2015) Macroscopic non-contextuality
Bell Scenarios p is almost quantum 12 in ( n, m, d ) iff p ∈ Q 1 ( B n,m,d ) AFLS 11 : ( n, m, d ) : p is Almost quantum iff p is MNC in B n,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
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
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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