Quantum correlations from simple assumptions

Abstract

We address the problem of deriving the set of quantum correlations for every Bell and Kochen-Specker (KS) contextuality scenario from simple assumptions. We show that the correlations that are possible according to quantum theory are equal to those possible under the assumptions that there is a nonempty set of correlations for every KS scenario and a statistically independent realization of any two KS experiments. The proof uses tools of the graph-theoretic approach to correlations and deals with Bell nonlocality and KS contextuality in a unified way.

Hypergraph Contextuality

“…Finally, we stress that the MMPH constructive generation of non-binary quantum sets from operationally chosen vectors out of all possible ones within such sets contribute to our understanding of the physical origin of quantum correlations since they represent a new MMPH scenario for getting “quantum correlations from simple assumptions” presented in Reference [ 73 ].…”

The problem of quantum correlations and the totalitarian principle

“…By definition of Bell scenario, behaviours must satisfy normalization and non-signalling (i.e., non-disturbance). It can be proven [10,23] that, for bipartite Bell scenarios, the set of behaviours satisfying the EP (applied to a single copy) is equal to the set of behaviours that satisfy normalization and the non-signalling principle. Similar arguments apply to n statistically independent experiments.…”

“…There are two properties of H(G) that are important for us. The first one is that any assignment of probabilities h[H(G)] ∈ P[H(G)] can be implemented by suitably choosing assignments p(G) ∈ P(G), x(G) ∈ P(G), y(G) ∈ P(G) and z(G) ∈ P(G) for, respectively, {e k } n k=1 , {x k } n k=1 , {y k } n k=1 and {z k } n k=1 , and assignments of probabilities a(K 2 ) ∈ P(K 2 ), b(K 2 ) ∈ P(K 2 ) and c(K 2 ) ∈ P(K 2 ) for, respectively, {a 0 , a 1 }, {b 0 , b 1 } and {c 0 , c 1 } (K 2 is the complete graph on two vertices; the graph of exclusivity of the events of tossing a coin) [10]. Therefore, the largest P(G) allowed by assumptions 3.1 and 3.2 can always be obtained from the largest P[H(G)] allowed by these assumptions by suitably tracing out its elements.…”

General Bayesian theories and the emergence of the exclusivity principle

“…The framework assumes only basic Bayesian laws, such as the validity of the rule of conditional probability, and the consistency of beliefs at different moments of time. Surprisingly, we find that these rather minimalistic assumptions imply the validity of the exclusivity principle, a feature of quantum theory that characterizes a large portion of the set of quantum correlations [12][13][14][15][16][17][18][19].…”