Operational Meanings of Orders of Observables Defined through Quantum Set Theories with Different Conditionals

TL;DR

This paper explores how different quantum logical conditionals influence the order relations of quantum reals within quantum set theory, linking these relations to operational joint probabilities from measurements.

Contribution

It develops quantum set theory models based on three different conditionals and characterizes their order relations through operational measurement probabilities.

Findings

The three models satisfy the transfer principle of ZFC set theory.

The quantum reals' equality truth values are consistent across models.

Order relations between quantum reals depend on the underlying conditional.

Abstract

In quantum logic there is well-known arbitrariness in choosing a binary operation for conditional. Currently, we have at least three candidates, called the Sasaki conditional, the contrapositive Sasaki conditional, and the relevance conditional. A fundamental problem is to show how the form of the conditional follows from an analysis of operational concepts in quantum theory. Here, we attempt such an analysis through quantum set theory (QST). In this paper, we develop quantum set theory based on quantum logics with those three conditionals, each of which defines different quantum logical truth value assignment. We show that those three models satisfy the transfer principle of the same form to determine the quantum logical truth values of theorems of the ZFC set theory. We also show that the reals in the model and the truth values of their equality are the same for those models. Interestingly, however, the order relation between quantum reals significantly depends on the underlying conditionals. We characterize the operational meanings of those order relations in terms of joint probability obtained by the successive projective measurements of arbitrary two observables. Those characterizations clearly show their individual features and will play a fundamental role in future applications to quantum physics.