Bridging Classical and Quantum Worlds: Maps, States, and Evolutions

TL;DR

This paper explores the relationship between classical and quantum theories, focusing on mathematical properties of maps, states, and evolutions, and proposing methods to derive quantum features from classical systems and vice versa.

Contribution

It introduces intermediate notions between positivity and complete positivity for linear maps, linking them to quantum dynamics and entanglement, and discusses constructions bridging classical and quantum states and evolutions.

Findings

Equivalence between positivity and complete positivity in the classical setting

Introduction of intermediate notions connecting classical and quantum maps

Proposed methods to derive quantum states and evolutions from classical counterparts

Abstract

In this work, we present several aspects of the interplay between classical and quantum theories. After reviewing the equivalence between positivity and complete positivity in the commutative setting, we introduce and analyze intermediate notions that interpolate between these two properties for linear maps on the space of operators on a Hilbert space, highlighting their natural connections to quantum dynamics and entanglement theory. Finally, we discuss possible constructions that allow one to derive quantum states and evolutions from classical counterparts, and vice versa.