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

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 $M_\alpha=\operatorname{diag}(1-\alpha,1-\alpha,1-2\alpha)$, $0\leq\alpha<1/2$.

For $0<\alpha<\beta<1/2$, all trace distances between distinct spin states are strictly smaller after $M_\beta$ than after $M_\alpha$, yet the unique linear post-processing factor has a negative normalized Choi eigenvalue.

We solve the optimization over all physical converters exactly: $\frac{1}{2}\inf_{\Lambda\in\mathrm{CPTP}}\|\Phi_\beta-\Lambda\circ\Phi_\alpha\|_\diamond=\frac{\alpha(\beta-\alpha)}{2-3\alpha}$, whereas the reverse deficiency is $\beta-\alpha$.

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.