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

TL;DR

This paper constructs a class of real-deterministic classical variables from quantum weak values and a global random variable, providing a hidden variable model that reproduces quantum uncertainty relations and offers an epistemic interpretation.

Contribution

It introduces a novel hidden variable model based on weak values and a global random variable that captures quantum uncertainty and complementarity within a classical framework.

Findings

Reproduces quantum uncertainty relations using classical variables

Shows the absence of a basis where error terms vanish for incompatible observables

Provides an epistemic interpretation of quantum uncertainty and measurement errors

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-Schr\"odinger 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 Schr\"odinger 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.