Quantum Mechanics with Contextually Labeled Observables

TL;DR

This paper proposes a contextual labeling scheme for quantum observables within the Contextuality-by-Default framework, emphasizing the role of measurement context and illustrating the approach by deriving the Tsirelson bound.

Contribution

It introduces a novel way to label quantum observables contextually, extending the theory to better understand measurement outcomes and their joint distributions.

Findings

Observables in different contexts can be labeled distinctly, even if they measure the same property.

Consistently connected systems have identical operators for the same property across contexts.

Derived the Tsirelson bound for cyclic systems of rank 3 under this framework.

Abstract

In the Contextuality-by-Default theory random variables representing measurement outcomes are labeled contextually, i.e., not only by what they measure but also under what conditions (in what contexts) the measurements are made, including but not reducible to what other measurements are made "together" with the given one. We propose in this paper that the quantum observables generating these random variables be labeled contextually as well, making the sets of the observables in different contexts disjoint. A quantum observable is defined as a pair consisting of the observable's label and a self-adjoint operator in a Hilbert space. If a system is consistently connected (i.e., obeys "non-disturbance," "non-invasivenes," or "non-signaling" conditions), the observables measuring the same property in different contexts have the same operator. A set of random variables possessing a joint distribution is represented by commuting observables. The reverse, however, is not true: random variables from different contexts do not have a joint distribution irrespective of whether the corresponding observables commute. We illustrate this view of observables by deriving the Tsirelson bound for consistently connected cyclic systems of rank 3.