Topological Methods for Studying Contextuality: N-Cycle Scenarios and Beyond

Round 1

Reviewer 1 Report

In their paper \lq Topological methods for studying contextuality: $N$-cycle scenarios and beyond', the authors make use the concept of a {\it $N$-cycle simplicial scenario} in the context of spaces of quantum measurements and outcomes. According to their definition 2.3, a simplicial scenario consists of a pair $(X,Y)$ of simplicial sets where $X$ represents the space of measurements and $Y$ the space of outcomes. There is a space of distributions on $Y$ represented by the simplicial set $D_R(Y)$ (where $R$ is a commutative semiring) and a simplicial distribution on $(X,Y)$ is the map $p:X \rightarrow D_R(Y)$. The simplicial distribution $p$ is contextual if it does not ly in the image of a certain canonical map called $\Theta$.

In their earlier (still unpublished) paper \lq Simplicial quantum contextuality' the case of the CHSH scenario has been investigated [7]. The later paper refines topological and group cohomological concepts introduced in [6]. The same concepts are used in their subsequent (still unpublished) paper \lq Mermin polytopes in quantum computation and foundations' [8]. In all the three aforementioned papers, there is an emphasis on new proofs of Fine's theorem employed before for characterizing non-contextuality in Bell's scenarios.

The virtue of the present paper is the possibility of going beyond cycle scenarios and study the flower scenario illustrated in Fig. 1 that generalizes the bipartite Bell scenarios.
Several theorems dealing with the identification of contextuality in such scenarios are proposed.

The paper expands over the previous papers [6]-[8] with many new results. It is well written and made concrete by relating the topological theorems to quantum contextuality.
I did not notice mistakes. I recommend the publication in Entropy in the present form.

Comments for author File: Comments.pdf

Reviewer 2 Report

The authors present a comprehensive and well-developed theorem in their work. They introduce insightful techniques, such as FM elimination and collapsing measurement spaces, which, although previously mentioned in published literature, still offer valuable contributions to research in this direction. I would say the paper is justified in its suitability for publication.

Comments for author File: Comments.pdf