[Paper Review] Comment on "Gleason-Type Theorem for Projective Measurements, Including Qubits" by F. De Zela

Michael J. W. Hall|arXiv (Cornell University)|Nov 2, 2016

Quantum Mechanics and Applications3 references3 citations

TL;DR

This paper critiques F. De Zela's claimed extension of Gleason's theorem to qubits, demonstrating that his assumptions—continuity of probability measures and existence of eigenstates—are insufficient to derive Born's rule. A counterexample with a nonlinear probability functional shows the derivation fails, as Gudder's theorem does not apply to the restricted domain of Bloch vectors used by De Zela.

ABSTRACT

It has recently been claimed by De Zela that Gleason's theorem, for probability measures on the lattice of projection operators, can be extended to qubits by adding assumptions related to continuity and the existence of 'eigenstates'. This amounts to a claim of the derivation of Born's rule for Hermitian qubit observables. I point out a simple counterexample, and the flaw in De Zela's derivation (these are equally applicable to the repetition of the derivation given in a recent Reply). I also briefly discuss a valid extension to qubits given by Busch.

Motivation & Objective

• To challenge the validity of De Zela's recent claim that Gleason’s theorem can be extended to qubits using only continuity and eigenstate assumptions.
• To identify the logical flaw in De Zela’s derivation, particularly the incorrect application of Gudder’s theorem on orthogonal additivity.
• To provide a concrete counterexample showing that nonlinear probability functionals on qubit projections satisfy De Zela’s assumptions yet violate the linearity required by density operators.
• To clarify the distinction between quantum logic-based derivations and the stronger assumptions needed for qubit extensions, contrasting De Zela’s approach with Busch’s valid extension.
• To caution against uncritical citation of flawed derivations in foundational quantum mechanics research.

Proposed method

• Construct a nonlinear probability functional $ p_{ ho}(P_{ ho}) = \frac{1}{2}\left[1 + (n^{\psi} \cdot n^{\phi})^3\right] $ on qubit projections, parameterized by Bloch vectors.
• Verify that this functional satisfies De Zela’s two additional assumptions: continuity in the Bloch vector and $ p_{\phi}(P_{\phi}) = 1 $.
• Demonstrate that the functional violates the linearity required by Gleason’s theorem and thus cannot be represented by a density operator.
• Expose the error in De Zela’s application of Gudder’s theorem, which requires continuity over full vectors, not just the restricted 3-sphere of Bloch vectors.
• Compare De Zela’s assumptions with Busch’s stronger assumption involving convex combinations of probability operators, which correctly supports Gleason-type extensions.
• Use the counterexample to show that continuity over the Bloch sphere alone does not enforce linear structure, undermining De Zela’s derivation.

Experimental results

Research questions

• RQ1 Can Gleason’s theorem be extended to qubits using only continuity of probability measures and the existence of eigenstates?
• RQ2 What is the flaw in De Zela’s application of Gudder’s theorem to the qubit projection lattice?
• RQ3 Does a nonlinear probability functional on qubit projections that satisfies De Zela’s assumptions still violate the linearity required by Born’s rule?
• RQ4 Why does the assumption of continuity over the Bloch sphere fail to enforce linear structure in the probability measure?
• RQ5 How does Busch’s valid extension of Gleason’s theorem to qubits differ in its assumptions from De Zela’s approach?

Key findings

• A counterexample is constructed where $ p_{\phi}(P_{\psi}) = \frac{1}{2}\left[1 + (n^{\phi} \cdot n^{\psi})^3\right] $ satisfies De Zela’s continuity and eigenstate assumptions but is nonlinear in $ P_{\psi} $, thus violating the density operator form required by Gleason’s theorem.
• The functional $ p_{\phi}(P_{\psi}) $ is continuous in the Bloch vector $ n^{\psi} $, satisfies $ p_{\phi}(P_{\phi}) = 1 $, and preserves normalization and additivity over orthogonal projections, yet cannot be represented by a density operator.
• De Zela’s derivation fails because Gudder’s theorem—on which it relies—requires continuity over full 4-vectors, not just the restricted $ (1, n^{\psi}) $ submanifold of the Bloch sphere.
• The counterexample proves that De Zela’s assumptions are insufficient to derive Born’s rule for qubits, invalidating his claim of a Gleason-type theorem for qubits.
• Busch’s extension remains valid because it assumes convex combinations of probability operators, a stronger physical assumption that correctly enforces linearity.
• The paper concludes that De Zela’s assumptions, while physically intuitive, are too weak to recover the linear structure of quantum probabilities in qubit systems.