Relative State Quantum Logic

TL;DR

This paper develops a quantum logic framework based on relative states, emphasizing information transfer and the non-commutative nature of conjunctions involving conjugate variables.

Contribution

It introduces a new projective quantum logic that accounts for system-environment information transfer and addresses limitations of traditional quantum logic approaches.

Findings

Conjunctions of conjugate variables can be consistently defined but are non-commutative.

The proposed logic scheme is not distributive, with discrepancies linked to interference effects.

Probabilities can be mapped to an orthocomplemented ternary logic where the law of excluded middle holds.

Abstract

A projective quantum logic in terms of relative states is developed, emphasizing the importance of information transfer between a system under study and its environment. The need for accounting for the historical evolution of system is highlighted and it is found that the conjunction of observations involving conjugate variables can be consistently defined but is found to be non-commutative. It is shown that the Birkhoff and von Neumann approach to quantum logic is unable to deal with such conjunctions. It is found that whilst the proposed scheme is still not distributive in general, the discrepancy is directly related to interference effects that may disappear when information is transferred from the system to its environment. It is argued that the probabilities associated with projections be mapped to an orthocomplemented ternary logic, in which it is shown that the law of the excluded middle still holds.