Fixed-Boost Wigner Noise: Strict Trace-Distance Contraction without Quantum Degradability

Abstract

A Lorentz boost acts on the canonical spin of a massive particle through a momentum-dependent Wigner rotation. We show that, for one fixed observer boost, reducing over an uncertain momentum can strictly contract every pairwise spin-state trace distance without producing a channel that is degradable from the less contracted one. For spin $1/2$, we first characterize the exact inversion-symmetric channel cone generated by a fixed Wigner angle and transverse momentum directions. Inside this cone lies the Pauli family $M_α=\operatorname{diag}(1-α,1-α,1-2α)$, $0\leqα<1/2$. For $0<α<β<1/2$, all trace distances between distinct spin states are strictly smaller after $M_β$ than after $M_α$, yet the unique linear post-processing factor has a negative normalized Choi eigenvalue. We solve the optimization over all physical converters exactly: $\frac{1}{2}\inf_{Λ\in\mathrm{CPTP}}\|Φ_β-Λ\circΦ_α\|_\diamond=\frac{α(β-α)}{2-3α}$, whereas the reverse deficiency is $β-α$. Thus the identity dominates the family, while all positive-noise members are pairwise incomparable under CPTP post-processing. The ideal construction is realized as the narrow-packet limit of pure, normalizable five-component momentum states, and explicit perturbation and finite-shot tomography bounds certify an open set of examples. Separately, every nonidentity member fails embedding in a time-homogeneous Pauli-diagonal Lindblad semigroup. Hence ordering all unassisted spin distinguishabilities does not determine the quantum statistical post-processing order.