Saved in:
| Main Author: | |
|---|---|
| Format: | Recurso digital |
| Language: | |
| Published: |
Zenodo
2026
|
| Online Access: | https://doi.org/10.5281/zenodo.20245545 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866902194273386496 |
|---|---|
| author | Srivats, Bharath G |
| author_facet | Srivats, Bharath G |
| contents | <p>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]‖<em>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</em>⊥| for any observable Hamiltonian H = h₀ + 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.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_20245545 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | A Modular Speed Limit on the Qubit: Bloch Precession, Saturation, Channel Data-Processing, and the Vanishing of Modular Flow at Pure States Srivats, Bharath G <p>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]‖<em>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</em>⊥| for any observable Hamiltonian H = h₀ + 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.</p> |
| title | A Modular Speed Limit on the Qubit: Bloch Precession, Saturation, Channel Data-Processing, and the Vanishing of Modular Flow at Pure States |
| url | https://doi.org/10.5281/zenodo.20245545 |