Topological Formalism for Quantum Entanglement via ${\mathbb{B}}^3$ and ${\mathbb{S}}^0$ Mappings

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This is a rare case, in my experience, of a superior article, which addresses an important subject from a new perspective on quantum theory. I really have no criticisms. I would publish it as is. I am not sure how great  interests of readers will be, but they should be interested in it.  I might make one comment. The author might want to check on some existing works on more general topological aspects of quantum theory. There was a conference on this subject two year ago at the Fields institute. The article will be coming out in Royal Transaction later this year. There is no need to wait for them. There is the one published, Chilchilnisky G. 2022 The topology of quantum theory and social choice. Quantum Reports 4, 201–220 (https://doi.org/10.3390/quantum4020014). There are also some commentaries on it, including in connection with complementarity (not addressed by Chilchilnisky). Complementarity  implies a topological structure because one deals with two disjoint Hilbert spaces, unlike a single phase space of classical physics. This is also linked to the EPR experiment and hence to entanglement. In fact, my sense is that the topology considered is related or even correlated with this general topological structure of the formalism of QM.  I reiterate, however, that this merely something the author might want to check/add this as a reference, but it is not required for publishing the article. I fully support  publication.

Author Response

Thanks for the tip on the publication to cite. However, I feel it would need a whole study to connect the given framework to the quantum formalism that you mention. I will read the citation and see if it can bring an idea on how to develop such a notion.

 

Best wishes

Sergio

Reviewer 2 Report

Comments and Suggestions for Authors

General Evaluation

This paper proposes an original and conceptually appealing framework for describing quantum entanglement using topological and differential-geometric structures. By introducing mappings between the 3-dimensional ball (B₃) and the 0-sphere (S₀), the author provides a minimalist representation of measurement duality and quantum steering. The manuscript is clearly written and intellectually stimulating, and it demonstrates the author’s ability to bridge mathematical formalism with quantum physical concepts.

However, at its current stage, the paper remains largely conceptual. The author is encouraged to strengthen the mathematical precision, physical interpretation, and contextual relevance to existing formulations in quantum information theory. With these clarifications, the work could provide a meaningful contribution to the geometric study of entanglement.

Specific Comments

1. Clarify the physical motivation and interpretation. The mapping (B₃→ S₀) is an elegant abstraction, but the physical meaning—especially how thisrelates to quantum measurement, collapse, and steering—should be more explicitly discussed.
2. Strengthen mathematical formulation. The propositions and theorem would benefit from clearer definitions of mappings, equivalence relations, and fiber bundle structures. Some symbols (e.g., π*, π−1) could be defined more rigorously.
3. Enhance the link to existing literature. The author may briefly compare this approach with geometric quantum mechanics or projective Hilbert space methods to better position the contribution within current research.
4. Improve logical flow between sections. The transition betweenProposition 1, Proposition 2, and Theorem 1 could be made smoother by adding short bridging explanations.

Recommendation

The paper presents an original and promising idea with a clear conceptual structure. With modest revisions to clarify the mathematics and strengthen the physical narrative, it could be suitable for publication.

Author Response

1. Regarding this comment:

Clarify the physical motivation and interpretation. The mapping (B₃→ S₀) is an elegant abstraction, but the physical meaning—especially how this relates to quantum measurement, collapse, and steering—should be more explicitly discussed.

I have added this in the Introduction:

"Physically, the mapping $\mathbb{B}^{3} \to \mathbb{S}^{0}$ encapsulates the algorithmic correspondence between entanglement and measurement processes in quantum systems, whereby continuous state spaces are projected onto discrete outcome sets."



2. Regarding this comment:

Strengthen mathematical formulation. The propositions and theorem would benefit from clearer definitions of mappings, equivalence relations, and fiber bundle structures. Some symbols (e.g., π*, π−1) could be defined more rigorously.

A) The following addition have been made after the following sentence in Proposition 1:

" For all operations \( A \), we have \( Ax = Ay \) in \( \mathbb{S}^0 \)."

the following has been added:

Here, the equivalence relation $\sim$ is defined by 
$x \sim y \Leftrightarrow f(y) = R(f(x))$, 
and the projection $\pi : \mathbb{B}^{3} \to \mathbb{B}^{3}/\!\sim$ 
is the canonical quotient map in the category of topological spaces. 
The induced map $\tilde{f} : \mathbb{B}^{3}/\!\sim \to \mathbb{S}^{0}$ 
is therefore well defined and continuous by construction.

 

B) Before Proposition 2. The following has been added:


"We now define $\pi^{*} : T_{p}M \to \mathbb{S}^{0}$ as a continuous surjection assigning to each tangent space 
its discrete measurement outcome. The total space $E = \bigsqcup_{p \in M} T_{p}M$ together with 
$\pi^{*}$ forms a trivial fiber bundle $(E, M, \pi^{*}, \mathbb{S}^{0})$ in the topological sense. We can then form the second proposition."

C) After Remark 2, the following has been added:
"Now, let $(E, B, \pi^{*}, (\mathbb{S}^{0})^{n})$ denote a discrete fiber bundle,  where $\pi^{*}: E \to B$ is a continuous surjection and each fiber $\pi^{*-1}(p)$ is homeomorphic to $(\mathbb{S}^{0})^{n}$, then we can form the following theorem."

 

3. Regarding this comment:


Enhance the link to existing literature. The author may briefly compare this approach with geometric quantum mechanics or projective Hilbert space methods to better position the contribution within current research.

 

the following has been added in the Introduction:


"In contrast to geometric quantum mechanics \cite{hughston2017geometric}, which relies on the Fubini--Study metric of projective Hilbert space $\mathbb{CP}^{n}$, 
the present framework replaces the metric structure with a purely topological one.  This substitution emphasizes the discrete and non-metric nature of quantum measurement outcomes, while retaining a geometric intuition consistent with the projective viewpoint. 
Hence, the mapping $\mathbb{B}^{3} \to \mathbb{S}^{0}$ may be regarded as a minimal topological analogue of the projection from a continuous state manifold to discrete measurement results."

4.Regardint this comment 

"Improve logical flow between sections. The transition between Proposition 1, Proposition 2, and Theorem 1 could be made smoother by adding short bridging explanations."

 

The made revisions automatically satisfy this point as well.