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arXiv:1310.1981

Fidelity of Mobius matrices related with Lorentz boosts

Sejong Kim

Published Oct 8, 2013• 1 citation
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Abstract

In this article we consider the extended version of a real counterpart of qubit density matrices, called the Möbius matrix, and we see that it is a normalized Lorentz boost. Using the isomorphic gyrogroup structures between the set $\PP$ of all Lorentz boosts and the Einstein gyrogroup on the open unit ball $\B$ of $\R^n$ we give an explicit formula of the fidelity for Möbius matrices in terms of Lorentz gamma factors.

DOI
10.48550/arxiv.1310.1981

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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

•Apr 18, 2026•1 citation
In this article we consider the extended version of a real counterpart of qubit density matrices, called the Möbius matrix, and we see that it is a normalized Lorentz boost. Using the isomorphic gyrogroup structures between the set \PP\PP\PP of all Lorentz boosts and the Einstein gyrogroup on the open unit ball \B\B\B of Rn\R^nRn we give an explicit formula of the fidelity for Möbius matrices in terms of Lorentz gamma factors.
Density matrices for mixed state qubits, parametrized by the Bloch vector in the open unit ball of the Euclidean 3-space, are well known in quantum information and computation theory. By presenting new identities for the qubit density matrix we indicate its intimate relationship with M\"obius addition and scalar multiplication. The latter, in turn, form the algebraic setting for the Poincar\'e ball model of hyperbolic geometry so that, as a result, the qubit density matrix is linked to hyperbolic geometry.
We exploit a well-known isomorphism between complex Hermitian 2 × 2 matrices and 4 , which yields a convenient real vector representation of qubit states. Because these do not need to be normalized we find that they map onto a Minkowskian future cone in 1,3 , whose vertical cross-sections are nothing but Bloch spheres. Pure states are represented by light-like vectors, unitary operations correspond to special orthogonal transforms about the axis of the cone, positive operations correspond to pure Lorentz boosts. We formalize the equivalence between the generalized measurement formalism on qubit states and the Lorentz transformations of special relativity, or more precisely elements of the restricted Lorentz group together with future-directed null boosts. The note ends with a discussion of the equivalence and some of its possible consequences.
Bloch-Messiah Decompositions of Symmetric Ray Transfer Matrices in R4{\mathbb R}^4R4
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