All quantum expectation values as classical statistical mean values

TL;DR

This paper demonstrates that all quantum expectation values can be represented as classical statistical mean values through the construction of hidden observable functions and hidden mixed states, bridging quantum and classical frameworks.

Contribution

It introduces a general method to express quantum expectation values as classical averages using hidden variables, applicable to any bounded quantum observable and mixed state.

Findings

Quantum expectation values can be represented as classical integrals.

The method applies to any bounded observable and mixed state.

Provides a unified classical representation of quantum statistics.

Abstract

Given a physical quantum system described by a Hilbert H, for any bounded quantum observable (a bounded self-adjoint operator) T it is possible to define several ''hidden observable'' functions f:H->R associated to T and for any quantum mixed state (a density matrix) D it is possible to define several ''hidden mixed states'' (probability measures) m on H associated to D in such a way that the following equality is verified: Trace[ b(T). D] =integral[b(f(psi)).dm(psi) whatever is the continuous function b:R->R. This formula gives a general way to express any expectation value computable in a quantum theory as a classical statistical mean value.