A quantum logical and geometrical approach to the study of improper mixtures

TL;DR

This paper introduces a new quantum logical and geometrical framework that treats improper mixtures as fundamental states, addressing limitations of the standard approach and better capturing quantum correlations.

Contribution

It constructs an extended lattice of quantum states including improper mixtures, overcoming issues in the standard formalism and capturing quantum correlations algebraically.

Findings

New lattice includes all physical states, pure and mixed.

Addresses the representation of improper mixtures in quantum logic.

Studies maps between composite systems and subsystems.

Abstract

We study improper mixtures from a quantum logical and geometrical point of view. Taking into account the fact that improper mixtures do not admit an ignorance interpretation and must be considered as states in their own right, we do not follow the standard approach which considers improper mixtures as measures over the algebra of projections. Instead of it, we use the convex set of states in order to construct a new lattice whose atoms are all physical states: pure states and improper mixtures. This is done in order to overcome one of the problems which appear in the standard quantum logical formalism, namely, that for a subsystem of a larger system in an entangled state, the conjunction of all actual properties of the subsystem does not yield its actual state. In fact, its state is an improper mixture and cannot be represented in the von Neumann lattice as a minimal property which determines all other properties as is the case for pure states or classical systems. The new lattice also contains all propositions of the von Neumann lattice. We argue that this extension expresses in an algebraic form the fact that -alike the classical case- quantum interactions produce non trivial correlations between the systems. Finally, we study the maps which can be defined between the extended lattice of a compound system and the lattices of its subsystems.