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Review of “Geometric Formalism for Quantum Entanglement via B³ and S⁰ Mappings”

The manuscript proposes a geometric description of quantum entanglement in which entangled particles are represented by points in a three-dimensional ball, while measurement outcomes are represented by the two points of a zero-dimensional sphere. The author also attempts to describe quantum steering using quotient spaces, tangent bundles and fibers.

The idea of exploring quantum information through geometry is not unreasonable. There is already a substantial literature on geometric quantum mechanics and the geometry of quantum state spaces. The difficulty is that the present manuscript does not develop a consistent mathematical model. Instead, it moves between physical space, quantum states, measurement outcomes and geometric objects without clearly distinguishing them.

The first major problem is the meaning of the three-dimensional ball. At the beginning, its points appear to represent the positions of two particles. Later, the same points are treated as quantum states and as inputs to measurement and quotient maps. A particle’s spatial position is not the same thing as its quantum state. More importantly, two maximally entangled particles cannot be described merely by assigning one point to each particle. The manuscript never introduces a joint quantum state, a density operator, measurement operators or a probability rule. Therefore, the claim that the construction represents maximal entanglement is assumed rather than shown.

There is also a direct problem with the quotient-space argument in Proposition 1. The manuscript identifies two points as equivalent but assigns them different measurement outcomes. It then defines a function on the equivalence class using the value of one of its representatives. This is not well defined. Once two points belong to the same equivalence class, a function defined on that class must give the same result regardless of which representative is chosen. The manuscript assumes exactly the opposite. This is not a minor issue of notation; it invalidates the central construction and the diagram associated with the first proposition.

The role of the two-point space is also overstated. A set containing two possible values can certainly be used to label the outcomes of a binary measurement. However, this alone does not model quantum entanglement or explain state collapse. The same set could represent a classical coin toss. Its disconnectedness is a basic topological property, not an explanation of quantum jumps.

The manuscript further assumes that the outcome for one particle determines the outcome for the other through either the same sign or the opposite sign. That is not a general description of entangled measurements. The relation between outcomes depends on the particular entangled state and on the measurements chosen by both observers. Without specifying the state, measurement basis and probabilities, the proposed relation simply builds the desired correlation into the assumptions.

The use of the term “quantum steering” is similarly problematic. In quantum information theory, steering has a precise operational definition involving conditional states and different measurement choices. None of those elements is introduced here. The proof instead states that the second particle is forced to collapse because the particles are entangled. This is not a derivation of steering; it is a restatement of the claim being made.

Proposition 2 is even less convincing. Standard differential-geometric terms are used incorrectly. A tangent space is treated as if it were a cross product of two vectors. Cross products are later called cotangent spaces. A map is described as a fiber, although a fiber is normally a set associated with a point under a projection. The manuscript also shifts between base spaces, fibers and projections without defining a valid bundle structure. As written, this section does not establish a meaningful connection between differential geometry and quantum steering.

Another concern is that the paper does not produce a concrete result. It does not calculate measurement probabilities, reproduce a known entanglement correlation, derive a steering criterion or demonstrate an advantage over existing approaches. The main conclusion appears to be that a binary measurement has two possible outcomes. That follows from choosing a two-point set as the target and does not require the geometric construction developed in the paper.

The proposed extension to a complex ball does not resolve these problems. Quantum mechanics does use complex Hilbert spaces, but introducing complex coordinates alone is not enough. A valid quantum model still needs states, inner products, operators, measurements and probabilities. The conclusion mentions these ideas only in general terms and does not connect them to the two propositions.

In my view, the manuscript would need to be rewritten from the ground up. The author should first choose a concrete quantum system and define an explicit entangled state and measurement procedure. Only then should the geometric spaces be introduced, with a precise explanation of what each point, map and fiber represents. The mathematical constructions must also be checked against the standard definitions used in topology and differential geometry.

At present, the proposed formalism does not provide a valid model of entanglement or quantum steering. The main propositions are internally inconsistent, and the physical conclusions are asserted rather than derived.

Recommendation: Reject in the present form.

Competing interests

The author declares that they have no competing interests.

Use of Artificial Intelligence (AI)

The author declares that they did not use generative AI to come up with new ideas for their review.