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: $\frac{1}{2}\inf_{Λ\in\mathrm{CPTP}}|Φβ-Λ\circΦα|_\diamond=\frac{α(β-α)}{2-3α}$, 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.

• The paper demonstrates that fixed-boost Wigner noise leads to strict trace-distance contraction between spin channels while violating CPTP degradability.
• The paper analytically parametrizes inversion-symmetric Pauli channels, quantifying deficiencies with an exact diamond-norm measure.
• The paper reveals that observable monotonic decay in trace distance does not imply a full quantum resource ordering, highlighting the need for complete process and Choi analyses.

Fixed-Boost Wigner Noise and the Limits of Trace-Distance Ordering in Quantum Channels

Overview and Context

This paper, "Fixed-Boost Wigner Noise: Strict Trace-Distance Contraction without Quantum Degradability" (2607.12994), investigates the quantum information-theoretic properties of spin decoherence channels generated by Lorentz boosts on massive spin-$\frac{1}{2}$ particles. The focus is on the interplay between operational distinguishability of spin states and the structure of quantum channels arising in relativistic scenarios, emphasizing situations where trace-norm contraction does not imply the existence of a post-processing CPTP map (degradability) between reduced channels.

Characterization of Fixed-Boost, Inversion-Symmetric Spin Channels

The framework begins with Lorentz boosts applied to product-prepared spin-momentum states. Utilizing Wigner rotations, the reduced spin dynamics after tracing over uncertain but calibrated momentum distributions are modeled as convex combinations of unitary conjugations. The resulting qubit channels are unital and have Bloch matrix representations explicitly constructed from symmetrized probability distributions on transverse momentum directions, at a fixed Wigner angle.

A central analytical contribution is the full parametrization of the exact convex cone of inversion-symmetric channels obtainable by averaging over fixed-magnitude transverse Wigner rotations. Within this cone, specific analytic families of diagonal channels of the Pauli form are identified:

$$M_\alpha = \mathrm{diag}(1-\alpha, 1-\alpha, 1-2\alpha), \quad 0 \le \alpha < \tfrac{1}{2}.$$

This structure, unique to fixed observer boosts and physically relevant pure state momentum preparations, provides the foundation for the subsequent comparison theorems.

Monotonicity without Degradability: Construction and Analysis

The core phenomenon established is the existence, for any $0 < \alpha < \beta < \frac{1}{2}$, of pairs of distinct spin channels $\Phi_\alpha$ and $\Phi_\beta$ (both in the above family) such that:

• All binary spin-state trace distances contract strictly more under $\Phi_\beta$ than under $\Phi_\alpha$:

$$\|\Phi_\beta(\rho) - \Phi_\beta(\sigma)\|_1 < \|\Phi_\alpha(\rho) - \Phi_\alpha(\sigma)\|_1 \quad \text{for all}~\rho \ne \sigma.$$

• No CPTP post-processing map $\Lambda$ satisfies $\Phi_\beta = \Lambda \circ \Phi_\alpha$—i.e., quantum degradability fails.
• The unique linear factor $$M_\alpha = \mathrm{diag}(1-\alpha, 1-\alpha, 1-2\alpha), \quad 0 \le \alpha < \tfrac{1}{2}.$$0 always has a negative Choi eigenvalue for such pairs, certifying its non-complete-positivity.

  Figure 1: All three target contrasts lie below the source contrasts for one fixed boost, while the unique factor has a negative Choi eigenvalue.

This result separates “observable monotonicity”—unconditional decay of state distinguishability—from the quantum resource-theoretic Blackwell order, which demands the existence of a physical post-processing channel. Therefore, although every operational binary trace-norm test demonstrates an ordering, the full quantum information structure is not monotonic.

Quantitative Deficiency and Its Operational Meaning

The paper proceeds to derive the exact directed CPTP post-processing deficiency (the diamond-norm optimal error) between $$M_\alpha = \mathrm{diag}(1-\alpha, 1-\alpha, 1-2\alpha), \quad 0 \le \alpha < \tfrac{1}{2}.$$1 and $$M_\alpha = \mathrm{diag}(1-\alpha, 1-\alpha, 1-2\alpha), \quad 0 \le \alpha < \tfrac{1}{2}.$$2, showing

$$M_\alpha = \mathrm{diag}(1-\alpha, 1-\alpha, 1-2\alpha), \quad 0 \le \alpha < \tfrac{1}{2}.$$3

which quantifies by how much the simulated channel falls short when attempting to physically post-process $$M_\alpha = \mathrm{diag}(1-\alpha, 1-\alpha, 1-2\alpha), \quad 0 \le \alpha < \tfrac{1}{2}.$$4 into $$M_\alpha = \mathrm{diag}(1-\alpha, 1-\alpha, 1-2\alpha), \quad 0 \le \alpha < \tfrac{1}{2}.$$5.

Figure 2: Exact directed CPTP deficiency $$M_\alpha = \mathrm{diag}(1-\alpha, 1-\alpha, 1-2\alpha), \quad 0 \le \alpha < \tfrac{1}{2}.$$6 on the Pauli line. The diagonal is the only zero set in the positive-noise interior; the edge $$M_\alpha = \mathrm{diag}(1-\alpha, 1-\alpha, 1-2\alpha), \quad 0 \le \alpha < \tfrac{1}{2}.$$7 is also zero because every target is a post-processing of the identity.

This deficiency provides a direct operational meaning via channel discrimination: the excess success probability over random guessing for distinguishing $$M_\alpha = \mathrm{diag}(1-\alpha, 1-\alpha, 1-2\alpha), \quad 0 \le \alpha < \tfrac{1}{2}.$$8 from any post-processed $$M_\alpha = \mathrm{diag}(1-\alpha, 1-\alpha, 1-2\alpha), \quad 0 \le \alpha < \tfrac{1}{2}.$$9 is exactly $0 < \alpha < \beta < \frac{1}{2}$0.

Robustness: Wave Packets, Tomography, and Model Error

Analytic results are shown to persist beyond idealized sharp-momentum preparations. The strict contraction/no-go effect extends to open neighborhoods of the fixed-boost Pauli line under perturbations from finite-width, normalizable wave packets—even under model or tomographic error. Explicit quantitative stability radii are provided. For instance, small confidence intervals on Bloch matrix entries derived from finite-shot process tomography can certify non-degradability if they exclude the Pauli-Markov semigroup region.

Failure of Markovian Semigroup Embedding

A complementary impossibility result is established: none of the nontrivial fixed-boost Pauli channels can be realized as positive-time points in a time-homogeneous, Pauli-diagonal Lindblad semigroup. The necessary Markov-embedding inequality, $0 < \alpha < \beta < \frac{1}{2}$1, is violated everywhere except at the identity channel.

Analytic Description of Nondegradability Margin

The negative Choi eigenvalue, which quantifies the margin by which the candidate post-processing map fails to be CPTP, is computed exactly and illustrated throughout the strict interior of the (trace-distance) contraction-ordered region.

Figure 3: Analytic nondegradability margin from the explicit negative Choi eigenvalue formula; the effect persists for every nonzero Wigner angle and every $0 < \alpha < \beta < \frac{1}{2}$2.

Implications and Outlook

This work rigorously demonstrates that, within fixed-relativistic kinematics, total ordering of all pairwise spin-state trace distances does not induce a corresponding CPTP information ordering of the channels. This uncovers a critical and subtle limitation of using observable monotonicity as a surrogate for operational quantum noise ordering, even in natural physical constructions.

Practically, the results imply that experimental or process-tomographic identification of spin-damping in relativistic quantum systems must not infer degradability (and the feasibility of reversed or simulated noise) solely from decay curves or trace-norm comparisons; full process or Choi analyses are essential.

Theoretically, these findings expose the inherent incompleteness of local distinguishability monotones as quantum resource orderings in relativistic noise. Extensions to higher spins, more complex momentum correlations, or different symmetry classes represent meaningful directions for further work.

Conclusion

The paper provides a comprehensive analytic treatment of fixed-boost Wigner spin noise, showing a strict separation between operational monotonicity and quantum degradability. The construction, stability, and certificate results not only generalize known information-theoretic orderings but highlight the necessity for refined quantum resource monotones in relativistic contexts. As such, monotonic decay of all binary spin contrasts does not characterize the information ordering of relativistic quantum channels even under strict physical constraints.