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The Algebra of the Qubit: Fourteen Theorems on I(V) = γ²(V)

Bharath Srivats

Published Apr 17, 2026• 0 citations
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Abstract

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.

DOI
10.5281/zenodo.19639557

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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

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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.
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
On the one-dimensional Fisher--Rao manifold of binary probability distributions parameterized by the visibility coordinate V∈(−1,1)V \in (-1,1)V∈(−1,1), the Fisher information equals the squared Lorentz factor: I(V)=γ2(V)=1/(1−V2)I(V) = \gamma^2(V) = 1/(1-V^2)I(V)=γ2(V)=1/(1−V2). We derive three consequences of this identity concerning reparameterization-invariant priors, arclength bridges between canonical metrics, and coarse-graining of qubit measurements. First, the Jeffreys prior πJ(V)=γ(V)/π\pi_J(V) = \gamma(V)/\piπJ​(V)=γ(V)/π is the unique normalized smooth probability density whose associated half-density coincides with the canonical half-density of the metric, and is therefore the unique prior covariant under every smooth reparameterization. Second, the Gudermannian function is the unique smooth bijection relating the Bures arclength coordinate sB=12arcsin⁡(V)s_B = \tfrac{1}{2}\arcsin(V)sB​=21​arcsin(V) to the Beltrami--Klein arclength coordinate sBK=arctanh⁡(V)s_{BK} = \operatorname{arctanh}(V)sBK​=arctanh(V) via 2 sB=gd⁡(sBK)2\,s_B = \operatorname{gd}(s_{BK})2sB​=gd(sBK​); equivalently, arcsin⁡(V)=gd⁡(arctanh⁡(V))\arcsin(V) = \operatorname{gd}(\operatorname{arctanh}(V))arcsin(V)=gd(arctanh(V)). Third, for a mixed qubit state of Bloch radius rrr undergoing transverse rotation, the isotropically averaged ratio of classical to quantum Fisher information has the closed form R(r)=[2r+(r2−1)ln⁡((1+r)/(1−r))]/(4r3)R(r) = [2r + (r^2-1)\ln((1+r)/(1-r))]/(4r^3)R(r)=[2r+(r2−1)ln((1+r)/(1−r))]/(4r3), interpolating monotonically between R(0)=1/3R(0) = 1/3R(0)=1/3 and R(1)=1/2R(1) = 1/2R(1)=1/2. Each identity is proved from scratch. These three identities are elementary consequences of I(V)=γ2(V)I(V) = \gamma^2(V)I(V)=γ2(V); their joint geometric content positions the binary Fisher--Rao manifold at the intersection of Bayesian statistics, hyperbolic geometry, and quantum measurement theory
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.