Abstract
The set of subsystems Σ(m) of a finite quantum system Σ(n) with variables in ℤ(n), together with logical connectives, is a Heyting algebra. The probabilities τ(m|ρn)=Tr[(m)ρn] (where (m) is the projector to Σ(m)) are compatible with associativity of the join in the Heyting algebra, only if the variables belong to the same chain. Consequently, contextuality in the present formalism, has the chains as contexts. Various Bell-like inequalities are discussed. They are violated, and this proves that quantum mechanics is a contextual theory.
Highlights
In recent work [1] we have studied the mathematical structure of the set of subsystems of a finite quantum system Σ(n) with variables in Z(n)
We have shown that the set of subsystems of Σ(n) with logical connectives, is a distributive lattice Λ(Σn)
In this paper we present a methodology for deriving novel Bell-like inequalities, starting from an equality in Boolean algebra, and using Boole’s inequality for probabilities
Summary
In recent work [1] we have studied the mathematical structure of the set of subsystems of a finite quantum system Σ(n) with variables in Z(n) (the integers modulo n). In this paper we present a methodology for deriving novel Bell-like inequalities, starting from an equality in Boolean algebra, and using Boole’s inequality for probabilities (which is valid in non-contextual quantum mechanics). In similar way we define the logical operations in the set Hn of the Hilbert spaces of the subsystems of Σ(n) This is a Heyting algebra isomorphic to Λ[D(n)] and Λ(Σn), which we denote as Λ(Hn). For variables in a chain, σ(m1, m2|ρn) = 0 for all density matrices, and the τ (m|ρn) obey the equality of Eq(1) For this reason, contexts in the present formalism are the chains in the Heyting algebra.
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