From the Attempt of Certain Classical Reformulations of Quantum Mechanics to Quasi-Probability Representations

TL;DR

This paper examines classical reformulations of quantum mechanics, highlighting limitations of classical embeddings of quantum states and exploring quasi-probability representations as an alternative approach.

Contribution

It demonstrates the limitations of classical state embeddings for quantum effects and discusses quasi-probability representations as a viable alternative.

Findings

Classical embeddings of quantum states are limited by non-denseness of effects.

Injective classical embeddings cannot fully describe quantum effects.

Quasi-probability representations offer an alternative to classical reformulations.

Abstract

The concept of an injective affine embedding of the quantum states into a set of classical states, i.e., into the set of the probability measures on some measurable space, as well as its relation to statistically complete observables is revisited, and its limitation in view of a classical reformulation of the statistical scheme of quantum mechanics is discussed. In particular, on the basis of a theorem concerning a non-denseness property of a set of coexistent effects, it is shown that an injective classical embedding of the quantum states cannot be supplemented by an at least approximate classical description of the quantum mechanical effects. As an alternative approach, the concept of quasi-probability representations of quantum mechanics is considered.