Quantum uncertainty as classical uncertainty of real-deterministic variables constructed from complex weak values and a global random variable

Abstract

What does it take for real-deterministic c-valued (i.e., classical, commuting) variables to comply with the Heisenberg uncertainty principle? Here, we construct a class of real-deterministic c-valued variables out of the weak values obtained via a non-perturbing weak measurement of quantum operators with a post-selection over a complete set of state vectors basis, which always satisfies the Kennard-Robertson-Schrödinger uncertainty relation. First, we introduce an auxiliary global random variable and couple it to the imaginary part of the weak value to transform the incompatibility between the quantum operator and the basis into the fluctuation of an 'error term', and then superimpose it onto the real-part of the weak value. We show that this class of "c-valued physical quantities" provides a real-deterministic contextual hidden variable model for the quantum expectation value of a certain class of operators. We then show that the Schrödinger and the Kennard-Robertson lower bounds can be obtained separately by imposing the classical uncertainty relation to the c-valued physical quantities associated with a pair of Hermitian operators.Within the representation, the complementarity between two incompatible quantum observables manifests the absence of a basis wherein the error terms of the associated two c-valued physical quantities simultaneously vanish. Furthermore, quantum uncertainty relation is captured by a specific irreducible epistemic restriction, foreign in classical mechanics, constraining the allowed form of the joint distribution of the two c-valued physical quantities. We then suggest an epistemic interpretation of the two terms decomposing the c-valued physical quantity as the optimal estimate under the epistemic restriction and the associated estimation error, and discuss the classical limit.

Conservation of correlation in measurement underlying the violation of Bell inequalities and a game of joint mapping

“…This theorem is a special case of a theorem presented in the previous work [28]. Moreover, taking σν = Îν , and noting that Ĩν (η is also equal to the local quantum expectation value.…”

“…The above definition of c-valued spin variables can be extended to general quantum observables acting on general quantum states [28]. It was initially conceived for phase space variables to study a specific epistemic (i.e., statistical) restriction underlying the incompatibility between quantum observables for position and momentum [29,30].…”

“…Consider two spin operators σ nµ and σ n ′ µ associated with particle µ, so that [σ nµ , σ n ′ µ ] = 0, µ = 1, 2. Then, for any preparation |ψ 12 , the associated c-valued spin variables, i.e., s nµ (η 12 , ξ|ψ 12 ) and s n ′ µ (η 12 , ξ|ψ 12 ), cannot simultaneously equal to ±1 independent of ξ [28]. For example, if for a given |ψ 12 we have s nµ (η 12 , ξ|ψ 12 ) = ±1 independent of ξ which is the case when |ψ 12 is the eigenstate of σ nµ , then we must have s n ′ µ (η 12 , ξ|ψ 12 ) = ±1 fluctuating randomly with ξ, and vice versa.…”

“…Indeed, like the standard quantum spin values, the variances of the c-valued spin variables defined in Eq. ( 1) satisfy the Heisenberg-Kennard-Robertson-Schrödinger uncertainty relation [28].…”

Operational characterization of general quantum correlation via complex weak value measurement

“…Noting this fact, it is argued in Ref. [55] that the imaginary part of the weak value can be seen as the strength of the error in a single shot estimation via the introduction of an unbiased global random variable. See also Refs.…”