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A Modular Speed Limit on the Qubit: Bloch Precession, Saturation, Channel Data-Processing, and the Vanishing of Modular Flow at Pure States

Bharath Srivats

Published May 16, 2026• 0 citations
Publisher

Abstract

We establish a complete operator-theoretic and operational picture of Tomita-Takesaki modular flow on the qubit. For any pair of qubit states ρ_M (modular) and ρ_E (evaluation), with Bloch radii V_M, r, axes n̂, m̂, and rapidity η_M = arctanh(V_M), the variance of the modular Hamiltonian K_M = -log ρ_M admits the cross-state closed form Var_E(K_M) = η_M²(1 - r² cos²θ), yielding a Mandelstam-Tamm bound |d⟨O⟩/dt| ≤ 2η_M √(1 - r²cos²θ) · √Var_E(O) whose maximizing-observable saturation ratio is r sin θ / √(1 - r² cos²θ), equal to 1 exactly when r = 1. We prove a Heisenberg-picture data-processing inequality, an integrated Bures-distance bound, an operator-norm version ‖[K_M, O]‖op ≤ 2η_M ‖O‖op saturating for off-axis Pauli operators, and a minimum-time relation τ⊥ = π/(2η_M) for orthogonal evolution iff n̂·m̂ = 0. The Bloch vector of ρ_E precesses around n̂ at angular frequency 2η_M; the half-cycle work satisfies |W| ≤ 2r sin θ |h⊥| for any observable Hamiltonian H = h₀𝟙 + h·σ. The bound vanishes at V_M → 1 when ρ_E = ρ_M despite divergent η_M, a finite-dimensional third-law analog. The decomposition ρ_M^{it} = e^{iφ(t)} exp(it η_M n̂·σ) identifies ρ_M^{it} as an SL(2,ℂ) boost element with imaginary spinor rapidity ζ = 2itη_M, a finite-dimensional Bisognano-Wichmann analog in which the universal QFT factor 2π is replaced by the state-dependent 2η_M. All numerical coefficients verified below 4×10⁻¹⁵.

DOI
10.5281/zenodo.20245544

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Modular automorphisms generated by Tomita–Takesaki theory are canonical butstate-dependent structures. In special QFT regimes (Bisognano–Wichmann wedgesectors, holographic entanglement wedges), modular flow coincides with geometriccausal evolution (Lorentz boosts, bulk modular flow). We define a state-independentsignaling defect ηt(A → B) that measures how much a candidate evolution αt (inparticular modular flow) fails to preserve the commutation structure between twocommuting algebras A and B. The defect is the operator-norm distance from theevolved algebra αt(B) to the commutant A′. We prove a sharp equivalence: ηt(A →B) = 0 if and only if no operational signaling channel exists from lab A to lab B underevolution αt, for all input states and all A-localized interventions. Quantitatively, ηt ≤ε implies the maximal signaling advantage is at most 2ε. In geometric regimes wheremodular flow is causal (Bisognano–Wichmann), the defect must vanish; nontrivialdefect is a hard falsification of the localization–evolution assignment. This yieldsa universal operational causality test applicable to any candidate dynamics, withimmediate consequences for emergent spacetime and holography.
We analyze the angular dependence of the purity-loss rate in a restricted class of open quantum systems: a qubit coupled to a thermal bosonic bath through a commuting interaction operator. Within the Born–Markov approximation, we derive the dissipative generator in closed form andestablish four results. First, the restricted induced trace norm of the generator (on the Hermitian traceless subspace) isindependent of the coupling angle, ‖ℒ(𝐷)𝜃 ‖1→1,ℬ0 = 4𝜋𝜂𝑘𝐵 𝑇,and we give a closed-form proof of this fact based on the rotational structure of the coupling family, together with an independent numerical cross-check.Second, the initial purity-loss rate of a pure reference state depends on the coupling angle through the geometric quantity Γ(𝜃) = |⟨𝜓|𝐴(𝜃)|𝜓⟩|2, which for the qubit family 𝐴(𝜃) = cos 𝜃 𝜎𝑥 + sin 𝜃 𝜎𝑧 takes the exact form Γ(𝜃) = (𝑒𝑥 cos 𝜃 + 𝑒𝑧 sin 𝜃)2. Third, the local curvature of Γ(𝜃) at its maximum defines a curvature-derived angular scale 𝜎geo = 1/√2, universal within this coupling family and independent of every physical parameter of the model; the two forms agree to within ∼ 5% at |𝜃 − 𝜇| = 𝜎geo (Appendix E). Fourth, we quantify explicitly the algebraic (non-exponential) decay of the Markov tail integral for the Ohmic–Lorentzian bath considered here, and we show that a slowly time-dependent coupling angle, 𝜃(𝑡), admits a well-defined adiabatic-Markovian generator; we then prove that the resulting purity-loss functional is not proportional, in general, to the trajectory-integrated Frobenius-norm witness of Ref. [1], identifying the specific obstructions (coherent-precession contamination and a linear-versus-quadratic mismatch in Bloch-vector language) responsible, and we pose as an open problem the narrower question of whether any restricted or reformulated version of the witness admits a well-defined relation to the present results. These results provide an exact microscopic benchmark for trajectory-dependent effective decoherence models: the angular dependence originates from the geometry of the system–environment coupling and the evolving state, not from an intrinsic angular preference of the bath, while a full reduction of the phenomenological model of Ref. [1] to this microscopic setting remains open.
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