Three Fisher-Rao Identities on the Binary Statistical Manifold: Jeffreys Uniqueness, the Gudermannian Bridge, and Isotropic Retention

Abstract

On the one-dimensional Fisher--Rao manifold of binary probability distributions parameterized by the
visibility coordinate $V \in (-1,1)$, the Fisher information equals the squared Lorentz factor:
$I(V) = \gamma^2(V) = 1/(1-V^2)$. 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
$\pi_J(V) = \gamma(V)/\pi$ 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 $s_B = \tfrac{1}{2}\arcsin(V)$
to the Beltrami--Klein arclength coordinate $s_{BK} = \operatorname{arctanh}(V)$ via $2\,s_B = \operatorname{gd}(s_{BK})$; equivalently,
$\arcsin(V) = \operatorname{gd}(\operatorname{arctanh}(V))$. Third, for a mixed qubit state of Bloch radius $r$ undergoing transverse
rotation, the isotropically averaged ratio of classical to quantum Fisher information has the closed form $R(r) = [2r + (r^2-1)\ln((1+r)/(1-r))]/(4r^3)$,
interpolating monotonically between $R(0) = 1/3$ and $R(1) = 1/2$. Each identity is proved from scratch. These three identities are elementary consequences
of $I(V) = \gamma^2(V)$; their joint geometric content positions the binary Fisher--Rao manifold at the intersection of Bayesian statistics, hyperbolic
geometry, and quantum measurement theory