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arxiv: 2211.05692 · v2 · pith:KR6UPEO5new · submitted 2022-11-10 · 🪐 quant-ph

Exploring weak value arguments and Bargmann invariants in N-level quantum systems through the Majorana symmetric representation

Lorena B Ferraz , Dominique L Lambert , Yves Caudano This is my paper

Pith reviewed 2026-05-24 11:00 UTC · model grok-4.3

classification 🪐 quant-ph
keywords weak valuesMajorana representationBargmann invariantssymplectic areaBloch sphereN-level systemsKirkwood-Dirac distribution
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The pith

The argument of any weak value in an N-level system equals the sum of N-1 solid angles on the Bloch sphere.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a geometric picture for the phase of weak values of arbitrary observables in N-level quantum systems. It first notes that the weak value of any observable is proportional to the weak value of an effective projector constructed from that observable acting on the initial state. It then applies the Majorana symmetric representation to convert the symplectic area that defines the projector's weak-value argument in CP^{N-1} into a sum of N-1 solid angles on the ordinary Bloch sphere. A reader would care because this supplies an explicit, visual rule for phases that otherwise remain abstract in higher-dimensional state space.

Core claim

The argument of the weak value of a general observable equals the sum of N-1 solid angles on the Bloch sphere. This follows because the weak value of any observable is proportional (by a real constant) to the weak value of an effective projector, and the argument of any projector weak value is a symplectic area in CP^{N-1} that the Majorana representation decomposes into N-1 solid angles.

What carries the argument

Majorana symmetric representation, which maps each N-level state to a symmetric multi-qubit configuration whose symplectic area is the sum of N-1 solid angles on the Bloch sphere.

If this is right

  • The argument of any third-order Bargmann invariant equals the same sum of N-1 solid angles.
  • The argument of the Kirkwood-Dirac quasi-probability distribution likewise reduces to a sum of solid angles.
  • For a general spin-1 operator whose weak-value modulus diverges, the argument is still given by the sum of two solid angles.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same solid-angle sum may give a practical way to extract geometric phases from weak measurements in qutrit systems without explicitly computing the symplectic form.
  • If the mapping holds for weak values, it should also supply a Bloch-sphere route to higher-order Bargmann invariants that appear in multi-particle interference.
  • Testing the rule on spin-1 systems near divergence points could reveal whether the phase jumps discontinuously when the effective projector passes through an orthogonal state.

Load-bearing premise

The symplectic area of the projector weak value in CP^{N-1} remains exactly the sum of those N-1 solid angles after the Majorana mapping is applied to the effective projector.

What would settle it

Compute the weak-value argument for a non-projector observable in a three-level system, map the initial and final states to their Majorana stars, and check whether the numerical phase equals the sum of the two solid angles subtended by those stars; any mismatch falsifies the claim.

Figures

Figures reproduced from arXiv: 2211.05692 by Dominique L Lambert, Lorena B Ferraz, Yves Caudano.

Figure 1
Figure 1. Figure 1: FIG. 1. Representation of the argument of the weak value of the CNOT gate. The pre-selected state is [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Representation of the maximum value of the modulus [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Color map of of the angle between the vectors rep [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. a) Argument of the weak value of the spin operator, [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
read the original abstract

This work examines the argument of weak values for general observables and develops a geometric description on the Bloch sphere. We apply the Majorana symmetric representation to reach this goal. The weak value of a general observable is proportional to the weak value of an effective projector: it is constructed from the application of the observable over the initial state, after normalization by a constant of proportionality that is real. The argument of the weak value of a projector on a pure state of an $N$-level system corresponds to a symplectic area in the complex projective space $(\text{CP}^{N-1})$. This symplectic area cannot be visualized directly but it can be represented geometrically with a sum of $N-1$ solid angles on the Bloch sphere using the Majorana stellar representation. By combining these two ideas, we show that the argument of the weak value of any observable (i.e., not just projectors) can be described with the Majorana representation, as the sum of $N-1$ solid angles on the Bloch sphere. These two approaches provide two geometrical descriptions, a first one in the complex projective space $\text{CP}^{N-1}$ and a second one on the Bloch sphere, after mapping the problem from the original $N$-dimensional quantum state space $(\text{CP}^{N-1})$ to a multi-qubit description in three-dimensional space by making use of the Majorana representation. These results can also be applied to the argument of the third-order Bargmann invariant, the most fundamental order as the argument of any higher order invariant can be expressed as a sum of the argument of third-order Bargmann invariants, as well as to the argument of the Kirkwood-Dirac quasi-probability distribution. Finally, we focus on the argument of the weak value of a general spin-1 operator when its modulus diverges towards infinity.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript claims that the argument of the weak value of any observable in an N-level system equals the sum of N-1 solid angles on the Bloch sphere. This follows from (i) the weak value of a general observable being proportional (with real constant) to the weak value of an effective projector constructed from O applied to the initial state, and (ii) the Majorana stellar representation mapping the symplectic area in CP^{N-1} that equals the argument of a projector weak value to that sum of solid angles. The same geometric description is applied to the argument of the third-order Bargmann invariant and the Kirkwood-Dirac distribution; a final section examines the spin-1 case in the limit of diverging modulus.

Significance. If the derivations hold, the work supplies a concrete, visualizable geometric interpretation on the Bloch sphere for phases appearing in weak values, Bargmann invariants, and Kirkwood-Dirac distributions in arbitrary dimension. The construction rests on standard properties of the Majorana representation and the Kähler form on projective space, which is a strength; the result could aid intuition in quantum foundations and metrology without introducing new free parameters or ad-hoc entities.

major comments (2)
  1. [construction of effective projector] The proportionality step (abstract and main derivation): the claim that the constant relating the weak value of a general observable to that of the effective projector is always real must be accompanied by an explicit formula for the projector and a short proof that no imaginary part is introduced for arbitrary O; this step is load-bearing for preserving the argument exactly when extending beyond projectors.
  2. [diverging-modulus case] Final section on spin-1 diverging modulus: the limit |weak value| → ∞ must be treated explicitly (e.g., by examining the phase extraction or the corresponding limit of the symplectic area) to confirm that the sum of N-1 solid angles remains valid; without this check the generality of the central claim is not fully established.
minor comments (2)
  1. Notation for the effective projector and the individual solid angles should be introduced with a dedicated equation or diagram early in the text to improve readability.
  2. A brief comparison table or paragraph relating the CP^{N-1} symplectic-area description to the Bloch-sphere description would help readers track the two geometric pictures.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment of the work's significance, and constructive major comments. We address each point below and will revise the manuscript to incorporate the requested clarifications.

read point-by-point responses

  1. Referee: [construction of effective projector] The proportionality step (abstract and main derivation): the claim that the constant relating the weak value of a general observable to that of the effective projector is always real must be accompanied by an explicit formula for the projector and a short proof that no imaginary part is introduced for arbitrary O; this step is load-bearing for preserving the argument exactly when extending beyond projectors.

    Authors: We agree this step requires explicit support. Let |ψ_i⟩ be the initial state and O the observable. Define the normalized vector |φ⟩ = O|ψ_i⟩ / ‖O|ψ_i⟩‖ (with the norm taken to be real and positive). The effective projector is then the rank-1 projector |φ⟩⟨φ|. The weak value satisfies A_w = ‖O|ψ_i⟩‖ ⋅ (⟨ψ_f|φ⟩ / ⟨ψ_f|ψ_i⟩). Because the prefactor ‖O|ψ_i⟩‖ is real and positive, arg(A_w) equals the argument of the projector weak value ⟨ψ_f|φ⟩ / ⟨ψ_f|ψ_i⟩ exactly. We will insert this explicit construction together with the short argument above into the revised manuscript. revision: yes

  2. Referee: [diverging-modulus case] Final section on spin-1 diverging modulus: the limit |weak value| → ∞ must be treated explicitly (e.g., by examining the phase extraction or the corresponding limit of the symplectic area) to confirm that the sum of N-1 solid angles remains valid; without this check the generality of the central claim is not fully established.

    Authors: We thank the referee for highlighting the need for an explicit limit analysis. In the |A_w| → ∞ regime the post-selected state |ψ_f⟩ approaches a direction orthogonal to |ψ_i⟩ while the phase of A_w is fixed by the direction in which the denominator vanishes. We will augment the final section with a direct computation showing that the extracted phase coincides with the limiting symplectic area in CP^{N-1} and, via the Majorana map, with the sum of the N-1 solid angles on the Bloch sphere. This confirms the geometric description continues to hold in the divergent case. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation applies standard Majorana geometry to weak-value arguments

full rationale

The paper's central step is the observation that the argument of a general weak value equals that of an effective projector (because the proportionality factor is real) and that the projector's argument equals a symplectic area in CP^{N-1} which the Majorana stellar representation maps to a sum of N-1 Bloch solid angles. Both the proportionality and the Majorana mapping are standard, externally established facts; neither is defined in terms of the target result nor obtained by fitting data inside the paper. No self-citation is invoked as a uniqueness theorem, no ansatz is smuggled, and no known empirical pattern is merely renamed. The derivation therefore remains self-contained against external geometric benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard quantum mechanics, the definition of weak values, and the Majorana representation; no free parameters, ad-hoc axioms, or new postulated entities are indicated in the abstract.

axioms (1)
  • domain assumption Standard properties of weak values, projectors, and the Majorana symmetric representation in finite-dimensional quantum systems
    Invoked throughout the geometric construction described in the abstract.

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