Characterization of the non-classical relation between measurement outcomes represented by non-orthogonal quantum states

TL;DR

This paper explores how quantum mechanics' non-orthogonal states lead to non-classical relations between measurement outcomes, revealing that quantum theory inherently violates non-contextual assumptions through the structure of Hilbert space.

Contribution

It demonstrates that the Hilbert space formalism imposes a lower bound on certain measurement outcome probabilities, highlighting the fundamental contextuality in quantum measurement relations.

Findings

Quantum states' non-orthogonality affects outcome relations.

Hilbert space modifies classical outcome probability bounds.

Quantum theory inherently violates non-contextuality assumptions.

Abstract

Quantum mechanics describes seemingly paradoxical relations between the outcomes of measurements that cannot be performed jointly. In Hilbert space, the outcomes of such incompatible measurements are represented by non-orthogonal states. In this paper, we investigate how the relation between outcomes represented by non-orthogonal quantum states differs from the relations suggested by a joint assignment of measurement outcomes that do not depend on the actual measurement context. The analysis is based on a well-known scenario where three statements about the impossibilities of certain outcomes would seem to make a specific fourth outcome impossible as well, yet quantum theory allows the observation of that outcome with a non-vanishing probability. We show that the Hilbert space formalism modifies the relation between the four measurement outcomes by defining a lower bound of the fourth probability that increases as the total probability of the first three outcomes drops to zero. Quantum theory thus makes the violation of non-contextual consistency between the measurement outcomes not only possible, but actually requires it as a necessary consequence of the Hilbert space inner products that describe the contextual relation between the outcomes of different measurements.