Quantum Measurements from a Logical Point of View

TL;DR

This paper develops a logical framework for modeling quantum measurement processes without referencing quantum states, revealing constraints on models and demonstrating the completeness of quantum mechanics regarding outcome possibilities.

Contribution

It introduces a new logic-based approach to quantum measurement, avoiding quantum states, and characterizes the structure and limitations of models consistent with quantum predictions.

Findings

No model can predict more than one atomic outcome in Hilbert spaces of dimension ≥3.

All models agree on the possibility of outcome sequences in finite-dimensional spaces.

Quantum mechanics is complete in predicting outcome possibilities within this logical framework.

Abstract

We introduce a logic modelling some aspects of the behaviour of the measurement process, in such a way that no direct mention of quantum states is made, thus avoiding the problems associated to this rather evasive notion. We then study some properties of the models of this logic, and deduce some characteristics that any model (and hence, any formulation of quantum mechanics compatible with its predictions and relying on a notion of measurement) should verify. The main results we obtain are that in the case of a Hilbert space of dimension at least 3, using a strengthening of the Kochen-Specker theorem, we show that no model can lead to the certain prediction of more than one atomic outcome. Moreover, if the Hilbert space is finite dimensional, then we are able to precisely describe the structure of the predictions of any model of our logic. In particular, we show that all the models of our logic do exactly make the same predictions regarding whether a given sequence of outcomes is possible or not, so that quantum mechanics can be considered complete as long as the possibility of outcomes is considered.