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Discriminating Thomas–Wigner from Pancharatnam–Berry Rotations in Sequential Weak Qubit Measurements: A Variable-Visibility Protocol

Bharath Srivats

Published Apr 23, 2026• 0 citations
Publisher

Abstract

The Fisher–Lorentz identity I(V) = γ²(V) = 1/(1−V²) for binary quantum measurements predicts that sequential non-collinear weak measurements on a qubit accumulate a Thomas–Wigner (TW) rotation with a specific, visibility-dependent closed form. This prediction is numerically distinct from the Pancharatnam–Berry (PB) geometric phase that governs the same measurement cycle under the standard pure-state geometric-phase interpretation. This paper presents a pre-registered experimental protocol that discriminates between the two predictions on existing superconducting, trapped-ion, and neutral-atom platforms by sweeping the measurement visibility V ∈ [0.3, 0.99]. For the octant spherical triangle ẑ → x̂ → ŷ → ẑ, the PB prediction is Ω_PB = 45° (fixed by the enclosed solid angle of π/2 sr), while Ω_TW varies from 2.70° at V = 0.30 to 73.94° at V = 0.99, crossing the PB baseline at V_× ≈ 0.9102. The signed discrepancy Ω_TW − Ω_PB ranges from −42° to +29° across the sweep and provides two independent discrimination landmarks: a monotonic V-dependence and a sign-flipping crossing point. Shot counts, tomographic precision, readout fidelity, and systematic-error budgets sufficient for 3σ discrimination are specified per platform (≈45k shots total for pure initial state, ≈180k for mixed-state preparation at |r| ≈ 0.5). The protocol requires no new hardware. A successful run on any reasonable-fidelity qubit platform (IBM, Google, Rigetti, OQC, IQM transmons; Quantinuum or IonQ trapped ions; QuEra, Atom Computing, or Pasqal Rydberg arrays) either confirms a central prediction of the Fisher–Lorentz framework or falsifies it at 3σ. This is a methods paper: its purpose is to enable the experiment, not to re-establish the theory. Relation to companion preprints — This paper operationalises predictions derived in the Fisher–Lorentz framework developed in Srivats (2026), Fisher information is the squared Lorentz factor: why the qubit is special[10.5281/zenodo.19363449] and Srivats (2026), From measurement to reality: I(V) = γ²(V) and the geometry of binary distinction [10.5281/zenodo.19486140]. The SL(2,ℂ) ≅ Spin⁺(1,3) boost-composition structure on monitored qubits is established independently by Burns, Greenfield, and Dressel (Quantum Stud.: Math. Found. 13, art. 10, 2026); the present work extracts and pre-registers a specific crossing-point prediction (V_× ≈ 0.9102) as an operational TW/PB discriminator.

DOI
10.5281/zenodo.19722697

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•Mar 4, 2026•1 citation
The Fisher–Lorentz identity I(V) = γ²(V) = 1/(1−V²) for binary quantum measurements predicts that sequential non-collinear weak measurements on a qubit accumulate a Thomas–Wigner (TW) rotation with a specific, visibility-dependent closed form. This prediction is numerically distinct from the Pancharatnam–Berry (PB) geometric phase that governs the same measurement cycle under the standard pure-state geometric-phase interpretation. This paper presents a pre-registered experimental protocol that discriminates between the two predictions on existing superconducting, trapped-ion, and neutral-atom platforms by sweeping the measurement visibility V ∈ [0.3, 0.99]. For the octant spherical triangle ẑ → x̂ → ŷ → ẑ, the PB prediction is Ω_PB = 45° (fixed by the enclosed solid angle of π/2 sr), while Ω_TW varies from 2.70° at V = 0.30 to 73.94° at V = 0.99, crossing the PB baseline at V_× ≈ 0.9102. The signed discrepancy Ω_TW − Ω_PB ranges from −42° to +29° across the sweep and provides two independent discrimination landmarks: a monotonic V-dependence and a sign-flipping crossing point. Shot counts, tomographic precision, readout fidelity, and systematic-error budgets sufficient for 3σ discrimination are specified per platform (≈45k shots total for pure initial state, ≈180k for mixed-state preparation at |r| ≈ 0.5). The protocol requires no new hardware. A successful run on any reasonable-fidelity qubit platform (IBM, Google, Rigetti, OQC, IQM transmons; Quantinuum or IonQ trapped ions; QuEra, Atom Computing, or Pasqal Rydberg arrays) either confirms a central prediction of the Fisher–Lorentz framework or falsifies it at 3σ. This is a methods paper: its purpose is to enable the experiment, not to re-establish the theory. Relation to companion preprints — This paper operationalises predictions derived in the Fisher–Lorentz framework developed in Srivats (2026), Fisher information is the squared Lorentz factor: why the qubit is special[10.5281/zenodo.19363449] and Srivats (2026), From measurement to reality: I(V) = γ²(V) and the geometry of binary distinction [10.5281/zenodo.19486140]. The SL(2,ℂ) ≅ Spin⁺(1,3) boost-composition structure on monitored qubits is established independently by Burns, Greenfield, and Dressel (Quantum Stud.: Math. Found. 13, art. 10, 2026); the present work extracts and pre-registers a specific crossing-point prediction (V_× ≈ 0.9102) as an operational TW/PB discriminator.
The Fisher information of a binary quantum measurement with visibility V equals the squared Lorentz factor: I(V) = 1/(1 − V²) = γ²(V). This mathematical identity, confirmed through six independent derivations, underlies a conformal equivalence between the Bures and Beltrami-Klein metrics on the qubit Bloch ball, unique to N = 2 via a Weyl tensor obstruction for N ≥ 3. Consequences are traced across quantum metrology, thermodynamics, population genetics, cooperative biology, and pure mathematics. Results are organized by epistemic confidence, with proven theorems separated from conjectural extensions. Falsifiable predictions — including a Thomas-Wigner rotation angle for sequential qubit measurements testable on existing hardware — distinguish this framework from existing alternatives. Three honest tensions constrain its scope. This is the expanded companion preprint to the mathematical proof published as DOI: 10.5281/zenodo.19363449.
This document collects fourteen theorems extending the Fisher information identity I(V) = γ²(V) = 1/(1−V²), where V is the visibility of a binary quantum measurement and γ is the Lorentz factor. The identity was established in the author's anchor preprint (Zenodo DOI: 10.5281/zenodo.19363449). Part A — Internal Structure (T1–T5): Uniqueness of the Jeffreys prior as a Diff⁺- invariant half-density on the binary Fisher-Rao manifold (T1). Closed-form Thomas-Wigner rotation angle for sequential binary qubit measurements (T2). SLD saturation theorem with explicit mixed-state tilt formula tan β = b(1−r²)/a (T3). Uniqueness of the Gudermannian as the arclength bridge between Bures and Beltrami-Klein metrics (T4). Closed-form isotropic retention ratio R(r) interpolating from 1/3 to 1/2 (T5). Part B — Observer Limits (T6–T7): Lawvere fixed-point obstruction proving no finite observer can reconstruct the full γ² field (T6). Energy-precision uncertainty bound E·Var(θ̂) ≥ kT ln 2 / γ²(V) with distinguished Planckian operating point at V_P = √(1−ln 2) (T7). Part C — Symmetry and Classification (T8–T11): Möbius form-invariance of the conformal equivalence (T8). Classification of unsharp binary qubit effects by visibility (T9). Modular flow as imaginary-time SL(2,ℂ) boost with spinor rapidity ζ = 2itη (T10). SLOCC covariance of single-qubit visibility via Einstein addition (T11). Part D — Quantum Gravity (T12–T13): Scaling correspondence between holographic subregion-CV complexity and Fisher information under boost, with dimension-specific constants κ_d (T12; identity form κ_d = 1 stated as conjecture). γ² Page curve for black hole evaporation (T13; proof sketch). Part E — Partial Result (T14): Partial result deriving 3+1D spacetime signature from SL(2,ℂ) qubit measurement symmetry via exceptional isomorphisms, with explicit gap acknowledged. Complete proofs are provided for T1 through T11. T12 and T13 are proof sketches. T14 is a partial result with an explicit gap acknowledged. Each theorem is intended for subsequent individual journal submission; four companion papers (Bayesian-geometric, Lorentz kinematics, observer thermodynamics, symmetry-holography) are in preparation. This record establishes priority on the theorems as a collection. Related records:Anchor preprint: DOI 10.5281/zenodo.19363449Cross-domain applications: DOI 10.5281/zenodo.19545786
Fourteen theorems on the algebraic structure of binary quantum measurement, built on the identity I(V) = γ²(V) = 1/(1 − V²), where I is the classical Fisher information of a binary measurement with visibility V = 2p − 1 and γ is the Lorentz factor of special relativity. Part A (T1–T5): internal structure. Uniqueness of the Jeffreys prior π_J(V) = γ(V)/π under Diff₊ invariance (T1). Closed-form Thomas–Wigner rotation for sequential binary measurements (T2). SLD saturation of the quantum Cramér–Rao bound (T3). Uniqueness of the Gudermannian bridge between hyperbolic and spherical Fisher–Rao geometries (T4). Closed-form mixed-state retention R(r) interpolating from 1/3 to 1/2 (T5). Part B (T6–T7): observer–manifold theorems. Futility theorem via Lawvere's fixed-point theorem: no embedded observer can surjectively reconstruct the γ² field (T6). Energy-precision uncertainty E·Var(θ̂) ≥ kT ln 2 / γ²(V), saturating Landauer at V = 0 (T7). Part C (T8, held): dimensional selection of 3+1D spacetime from exceptional isomorphisms, with explicit gap acknowledged. Part D (T9–T12): symmetry and classification. Form-invariance of the conformal equivalence under Möbius action (T9). Two-outcome binary qubit effect classification via 4γ²(V) (T10). Modular flow as SL(2,ℂ) boost at imaginary rapidity (T11). SLOCC covariance γ²(V') = γ²(V)·γ²(ξ)·(1+Vξ)² (T12). Part E (T13–T14): quantum gravity connections. Holographic subregion complexity equals Fisher information under boost (T13, proof sketch). γ² Page curve for black hole evaporation (T14, proof sketch). Complete proofs for T1–T7 and T9–T12. T8 presented with scope limitations. T13 and T14 presented as proof sketches. Mathematical companion to: B.G. Srivats, "Fisher Information Is the Squared Lorentz Factor," Zenodo DOI 10.5281/zenodo.19363449 (2026) B.G. Srivats, "From Measurement to Reality," Zenodo DOI 10.5281/zenodo.19486140 (2026) Builds on independently established work by Burns–Greenfield–Dressel (QSMF 13:10, 2026) and Fullwood–Vedral–Guzmán-González (arXiv:2604.07471, 2026). Each theorem is intended for subsequent individual journal publication; this record establishes priority on all fourteen as a collection.
Three results. (1) We prove the tensor-level conformal identity dsBK2=4γ2(r) dsBures2 between the Bures metric and the Beltrami–Klein metric on the open qubit Bloch ball, where γ2(r)=1/(1−r2) is the squared Lorentz factor. (2) For the visibility coordinate V=2p−1 of a binary quantum measurement, the classical Bernoulli Fisher information takes the closed form I(V)=1/(1−V2)=γ2(V). (3) The conformal structure is special to qubits; for N≥3, the Bures metric on full-rank density matrices has a non-constant sectional curvature at the maximally mixed state and hence a non-vanishing Weyl tensor and no conformal equivalence to any constant-curvature hyperbolic model (Theorem 2). We outline an experimentally testable operational consequence where sequential non-collinear weak measurements on a qubit predict a Thomas–Wigner rotation with a closed-form purity dependence that deviates from the Pancharatnam baseline at intermediate visibility.
The Fisher information of a binary quantum measurement with visibility V is I(V) = 1/(1-V²) = γ²(V), the squared Lorentz factor from special relativity. This identity, proven in the companion anchor preprint, holds only for qubits: for N≥3 quantum systems, the Weyl curvature tensor obstructs the conformal equivalence between the Bures metric of quantum state space and the Beltrami-Klein metric of hyperbolic/relativistic geometry. Five independent research programs—in quantum optics, condensed matter, stochastic thermodynamics, holography, and quantum measurement theory—have converged on this same γ² structure over the past decade. This document is the interpretive companion to the anchor preprint. It traces the implications of the identity from fundamental physics (a deductive chain through five published theorems connecting γ² to Einstein's equations) through cosmology (dark energy, dark matter, fine-tuning as Fisher optimality), consciousness (entropy production as the measurement process experienced from inside), biology (evolution, aging, cancer as Fisher information dynamics), and computation (transformer attention as Fisher-Rao geometry). It introduces the concepts of "beautiful change" (structured, self-organized entropy production as the distinguishing feature of rich consciousness) and the "downprojection hypothesis" (experienced spacetime as a conformal restriction of a higher-dimensional information-geometric manifold). All claims are explicitly labeled by epistemic status across four tiers: proven mathematical identity, published theorems by independent groups, strong circumstantial evidence, and conjecture. Falsifiable predictions are provided, two already experimentally confirmed. The proven mathematics stands independently of the interpretive framework built upon it.
This preprint proves that on the open qubit Bloch ball the Beltrami-Klein hyperbolic metric is conformal to the Bures (quantum Fisher information) metric with conformal factor 4γ²(r), where γ²(r) = 1/(1−r²) is simultaneously the squared Lorentz factor and the Fisher information of a binary measurement with visibility V = r. A Weyl-tensor obstruction shows this conformal equivalence fails for all N ≥ 3 quantum systems. We derive a closed-form Thomas-Wigner rotation prediction for sequential non-collinear qubit measurements, testable on current platforms.