Geometric monotones of violations of quantum realism

TL;DR

This paper introduces geometric measures of violations of quantum realism, establishing their theoretical foundations, identifying the most suitable distances, and linking them to entropic divergences, thus enriching the understanding of quantum contextuality.

Contribution

It extends the framework of quantifying quantum realism violations from entropic to geometric distances, identifying Bures and Hellinger distances as valid monotones, and connecting them to entropic divergences.

Findings

Bures and Hellinger distances satisfy all minimal criteria for VQR monotones.

Geometric distances can be expressed in terms of symmetric and Sandwiched Rényi divergences.

The geometric and entropic approaches to VQR are deeply interconnected.

Abstract

Quantum realism, as introduced by Bilobran and Angelo [EPL 112, 40005 (2015)], states that projective measurements in quantum systems establish the reality of physical properties, even in the absence of a revealed outcome. This framework provides a nuanced perspective on the distinction between classical and quantum notions of realism, emphasizing the contextuality and complementarity inherent to quantum systems. While prior works have quantified violations of quantum realism (VQR) using measures based on entropic distances, here we extend the theoretical framework to geometric distances. Building on an informational approach, we derive geometric monotones of VQR using trace distance, Hilbert-Schmidt distance, Schatten $p$-distances, Bures, and Hellinger distances. We identify Bures and Hellinger distances as uniquely satisfying all minimal criteria for a bona fide VQR monotone. Remarkably, these distances can be expressed in terms of symmetric R\'enyi and Sandwiched R\'enyi divergences, aligning geometric and entropic approaches. Our findings suggest that the realism-information relation implies a deep connection between geometric and entropic frameworks, with only those geometric distances expressible as entropic quantities qualifying as valid monotones of VQR. This work highlights the theoretical and practical advantages of geometric distances, particularly in contexts where computational simplicity or symmetry is important.