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.
Comments
10 pages, 3 figures. Ancillary verification code included
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 Mα=diag(1−α,1−α,1−2α), 0≤α<1/2. For 0<α<β<1/2, all trace distances between distinct spin states are strictly smaller after Mβ than after Mα, yet the unique linear post-processing factor has a negative normalized Choi eigenvalue. We solve the optimization over all physical converters exactly: 21infΛ∈CPTP∥Φβ−Λ∘Φα∥⋄=2−3αα(β−α), 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.
We ask whether Kochen-Specker contextuality is Lorentz invariant. On the full Hilbert space, it is, since boosts act unitarily. Restricted to spin, it is not: momentum acts as an environment applying momentum-dependent Wigner rotations, leaving spin observables unsharp and breaking the operator identities behind state-independent contextuality. For spin 1/2 one scalar governs the loss, the packet average Δ of sin2(θW/2) (where θW is the Wigner angle); for any spin, a ray degrades at a rate set by its spin variance transverse to the boost. In the spin-only account, the Free Will Theorem's SPIN axiom fails for every Δ>0; the Yu-Oh set loses contextuality between Δ≈0.008 and 0.016, with state independence going first; the Peres-Mermin square loses contextuality at Δ≈0.059 or 0.074 by orientation; singlet CHSH loses nonlocality for Δ≥0.159; and a GHZ-state OR gate loses Raussendorf's contextuality certificate at Δ≈0.206. A q-plate implements the channel exactly on photon polarisation, with a retardation in place of the rapidity, so the nonlocality, GHZ and aimed-source results are testable without motion.
The real classical shadows protocol of West et al. replaces the unitary (Clifford) ensemble of the Huang--Kueng--Preskill scheme by the orthogonal (real Clifford) ensemble, and for symmetric observables achieves strictly smaller estimator variances: a factor approaching two for global evolution and an exponential factor (3/2)k for k-local real Pauli observables. Real hardware, however, never implements the ideal evolution. Building on the noisy classical shadows framework of Koh and Grewal, we give a complete theory of the real classical shadows protocol in the presence of a known completely positive trace-preserving noise channel acting after the orthogonal evolution. We derive the noisy global and local orthogonal shadow channels from first principles using the Weingarten calculus of the orthogonal group, prove that each is a depolarizing channel acting on the symmetric (respectively locally symmetric) component of its input, and derive from it the exact single-shot variance in closed form, together with the associated shadow seminorm, two-sided bounds on it, and the resulting sample-complexity guarantees. We prove that the noiseless sample-complexity advantages survive intact under noise. Because the variances are exact rather than bounded, the ratio is controlled by a single dimensionless parameter, which gives a closed-form criterion for when the factor of two is attainable: both the second-moment and the variance ratio reach it exactly when the observable's norm profile grows, and the noise enters that limit only through a factor lying within 2/(d+2) of two, so the advantage is uniform in the noise. For rank-one targets it is provably unattainable, saturating strictly below two. The local real-Pauli advantage remains (3/2)k. We treat complex measurement bases through a reality parameter and a transposed-noise scalar, recovering unitary shadows in the appropriate limit.
This paper investigates whether relativistic motion imposes an intrinsic cost on certifying a system’s identity. Identity refers here to internal physical parameters, such as a spectroscopic splitting, composition marker, or spin orientation, rather than to Poincaré-invariant labels such as mass and total spin. The central result is that a fixed Lorentz boost does not destroy identity information. A fixed, parameter-independent boost acts unitarily on the complete physical state, so the global quantum Fisher information associated with an internal parameter is exactly preserved. Any apparent loss of identity information must therefore arise from restricted measurements, finite detector resources, discarded degrees of freedom, incomplete knowledge of the boost, or averaging over transformations that are not recorded. The paper develops this conclusion through classical spectroscopy, relativistic flyby estimation, photon frequency encoding, and massive spin-momentum systems. Classical spectral identity A Doppler transformation uniformly rescales the line center, line width, and internal splitting of a spectrum. For a spectral doublet, the dimensionless ratio between the splitting and the line center remains unchanged. The per-photon Fisher information about the rest-frame splitting is therefore invariant under an ideal Doppler dilation. Relativistic motion can still change the total information collected because photon number, detector resolution, observing duration, and spectral bandwidth depend on the measurement protocol. The paper compares three detector classes: An ideal spectrometer with unlimited bandwidth A spectrometer with fixed observer-frame resolution A resolution-limited detector with a finite observing band These detectors produce different information-versus-velocity laws. Relativity therefore does not impose a universal identity-collapse exponent independently of the detector and resource convention. Relativistic flyby self-calibration The paper considers a source undergoing a relativistic flyby with unknown velocity, impact parameter, time of closest approach, source amplitude, rest transition frequency, and spectral splitting. The changing line-of-sight velocity generates a Doppler chirp. The same photons can therefore constrain both the trajectory and the source’s internal identity. For a symmetric doublet observed over a window that samples both sides of the encounter, the variance penalty from marginalizing over the unknown trajectory has the form variance inflation = 1 + O[(Δ/ν₀)²], where Δ is the rest-frame splitting and ν₀ is the central rest frequency. The trajectory is determined primarily through the much larger frequency scale ν₀, while the identity parameter is encoded in the smaller separation Δ. For the numerical examples studied, this produces a Fisher-information hierarchy of approximately one million and a trajectory-marginalization penalty below approximately 0.1 percent. The result remains valid when the rest transition frequency is treated as unknown. Two mechanisms make self-calibration possible. First, the ratio of the early and late Doppler asymptotes constrains the source velocity without requiring an external frequency reference. Second, changing the central frequency translates both spectral lines together, while changing the splitting moves them symmetrically apart. For a symmetric doublet, these score directions are approximately orthogonal. The self-calibration result is conditional on adequate signal strength, a resolved spectral structure, a sufficiently long observing window, and a nonsingular nuisance-parameter Fisher matrix. Retarded-time flyby calculation The paper explicitly includes the retarded relation between emission time and observation time. This confirms the small trajectory-marginalization penalty but changes the interpretation of total information accumulation. The blueshifted approach phase is compressed into a short interval of observer time. As a result, an ideal detector can collect less total information at higher velocity, while a resolution-limited detector can benefit when blueshift moves the doublet above its instrumental resolution floor. Whether relativistic motion helps or hinders estimation is therefore protocol-dependent. Quantum frequency encoding For a single photon whose identity parameter is encoded in its frequency wavefunction, a collinear Lorentz boost acts as a unitary frequency dilation. The global quantum Fisher information about the encoded parameter is exactly invariant. This result applies to the complete frequency state. It does not imply that every reduced photonic measurement remains invariant, particularly when momentum and polarization are coupled or when the detector accesses only a restricted measurement algebra. Massive carriers and reduced-spin accessibility For a massive spin-1/2 carrier, a Lorentz boost generates momentum-dependent Wigner rotations. The complete spin-momentum state preserves its global quantum Fisher information. However, a detector that measures spin while ignoring momentum accesses only the reduced spin state. Tracing over momentum can therefore reduce the spin-only quantum Fisher information. For canonical Wigner spin, a narrow isotropic Gaussian momentum packet, and the spin-angle encoding studied in the paper, the leading fractional accessibility loss is proportional to [2/(1 + cos²θ)] × (σₚ/mc)² × tanh²(ξ/2), where θ is the encoding angle, σₚ is the momentum spread, m is the carrier mass, and ξ is the boost rapidity. The accessibility tax has three important properties. It depends on the spin-encoding direction. It increases with momentum spread. It saturates at high rapidity because tanh²(ξ/2) approaches one. The reduced-spin information loss therefore does not grow without bound as the relative velocity approaches the speed of light. The missing information is recoverable in principle. It has been transferred into momentum dependence and spin-momentum correlations. A joint measurement of spin and momentum restores access to the invariant global information. The effect is therefore a subsystem-accessibility cost, not an irreversible destruction of identity. Localization and action-count scaling For a minimum-uncertainty wave packet, σₓσₚ = ℏ/2. The Compton wavelength is λ_C = ℏ/(mc). Using the localization scale R = σₓ, the rest-frame action-count is Ξ = ER/(ℏc) = mcσₓ/ℏ = σₓ/λ_C. At large rapidity, the reduced-spin accessibility tax scales as Ξ⁻², with an additional factor determined by the spin-encoding angle. A carrier localized over a distance much larger than its Compton wavelength therefore pays a negligible reduced-spin accessibility cost. For packets that do not saturate the uncertainty relation, the same argument gives a bound rather than an equality. Energy and identity information The paper asks whether the kinetic energy required to match the source frame can be interpreted as the price of recovering identity information. Three arguments reject a universal proportionality. First, frame-matching energy grows without bound with the Lorentz factor, while the reduced-spin accessibility loss saturates. Second, frame matching is not necessary. Classical Doppler self-calibration can recover a spectral identity parameter without accelerating the detector into the source frame. In the quantum case, expanding the measured algebra to include momentum can recover the lost spin-only information. Third, kinetic energy invested in frame matching can in principle be recovered by later deceleration, while the acquired measurement record remains. Energy constrains storage capacity, evolution speed, and thermodynamic reset costs, but the paper finds no universal energy price per unit of Fisher information acquired. Main conclusions The paper establishes that: Fixed, parameter-independent Lorentz boosts preserve global quantum Fisher information. Per-photon spectral identity information is invariant under ideal Doppler dilation. Total information remains detector- and protocol-dependent. Relativistic flyby chirps can self-calibrate unknown source kinematics. Trajectory marginalization produces only a small identity-estimation penalty for well-resolved symmetric doublets. Reduced-spin measurements can lose access to information through momentum-dependent Wigner rotations. The reduced-spin accessibility tax depends on encoding angle and saturates with rapidity. Joint spin-momentum measurements restore the invariant global information. For minimum-uncertainty packets, the reduced-spin tax scales as Ξ⁻². No universal proportionality exists between frame-matching energy and identity information. Scope The global invariance theorem applies to fixed Lorentz transformations that do not depend on the identity parameter. The paper does not analyze unknown boosts as joint quantum-estimation parameters or random boost distributions that are averaged without being recorded. The reduced-spin result is derived for a single massive spin-1/2 carrier, canonical Wigner spin, a narrow isotropic Gaussian momentum packet, and a momentum-insensitive spin measurement. Broad or anisotropic packets, composite carriers, alternative relativistic spin observables, generic reduced photonic channels, and sequential non-collinear boosts remain open extensions. The classical flyby conclusions apply to high-signal, sufficiently resolved, symmetric spectral doublets observed over an adequate time window. Relation to the companion paper The companion paper examines the accelerated rather than inertial case: Action-Count, Observer Integrity, and Horizon Saturation: A Unified Framework from Ξ = ER/(ℏc)https://doi.org/10.5281/zenodo.19599288 Together, the two papers distinguish the operational effects of velocity and acceleration. Uniform velocity preserves global identity information but can redistribut
Recovery from a coherent rotation with an unobserved angle depends on both the encoding and what calibration reveals about the noise. We determine their combined limit for a known finite-dimensional generator and one final recovery. Two polynomial projections organize the answer. A physical Krylov chain separates correctable error amplitudes from their first state-dependent distortion; projection onto lower-angle powers identifies the part that a coherent decoder can cancel. Under explicit rank and moment conditions, their combination gives the exact leading optimal entanglement infidelity at weak noise and a recovery attaining it for any finite logical dimension. The optimal worst-input error lies between one and d times that infidelity at leading order, for logical dimension d. When the angle distribution is unknown, we optimize one decoder for every distribution compatible with the calibration. When spectral and calibration frequencies are integer multiples of a common frequency, moment positivity turns this into an exact finite optimization. Small measured means can still allow rare finite rotations. If the sole calibration datum is βr=E[sin2r(θ/2)], with θ the wrapped rotation phase at that common frequency, the optimal worst-prior infidelity is O(βr) exactly when generator powers through degree r−1 are correctable. When the first uncorrectable power has degree m<r, the optimal loss scales as βrm/r, with a sharp coefficient fixed by a universal moment problem. In a loss-free four-photon memory, population measurements at two calibration durations retain the fully informed quartic error coefficient for a fixed symmetric narrow-noise family with positive variance and finite fourth moment; a single duration permits loss quadratic in the noise width.
Abstract The characterization of quantum noise is fundamental to quantum information processing, quantum control and reliable operation of near-term quantum devices. Conventional descriptions of quantum noise are based on completely positive trace-preserving (CPTP) maps acting on density operators. These models describe effective quantum dynamics after the underlying physical interactions have been represented as a quantum channel. However, in practical cases, imperfections such as calibration errors, control-field fluctuations, finite-resolution waveform generation, and analog hardware noise can act upon the probability amplitudes of a quantum state earlier. The effect of such perturbation in quantum state after normalization has largely been unexplored. In this work, we develop a rigorous mathematical analysis of the normalized additive amplitude noise in a finite-dimensional quantum system. Assuming a zero-mean complex perturbation, we derive a second-order closed form expression for the ensemble-averaged density operator. The analysis leads to state-dependent evolution of the quantum state which cannot be represented by a CPTP map. We further prove that in case of isotropic noise, the ensemble-averaged density operator becomes an effective depolarizing channel. For anisotropic perturbations, we derive the orientation-dependent Bloch-vector contraction which is determined by the covariance geometry of the additive noise. Closed-form expressions are further obtained for measurement probabilities, state fidelity, and ensemble purity, showing that fidelity degradation depends only on the component of the perturbation covariance orthogonal to the input state. Monte Carlo simulations closely match the theoretical analysis, especially for small noise regime. The proposed model establishes a connection between classical additive-noise models and effective quantum dynamics generated by normalized state-space perturbations.
Abstract Random-unitary noise is an implementable way to hide quantum information, since it can be realized by applying unitary gates sampled from a classical random key. However, the unconstrained minimization of pairwise quantum α-z R\'enyi divergences over all mixed-unitary channels is degenerate: the completely depolarizing channel is a Weyl average and maps every input state to Id/d. This paper develops a non-erasing privacy--utility framework based on Pauli--Weyl randomization. For a finite protected ensemble E={ρ1,…,ρm}, a compact admissible set K of Weyl probability distributions, and a fidelity-type utility threshold η, we study the robust radius Rα,zη(E;K)=infp∈K:UE(Φp)≥ηmaxi=jDα,z(Φp(ρi)∥Φp(ρj)). We prove well-posedness, a zero-radius characterization, and active-pair optimality conditions. The main exact result is a Pauli reduction theorem for equal-purity binary qubit ensembles: in adapted Bloch coordinates, optimization over all Pauli channels reduces to a two-parameter contraction problem separating the sensitive direction from the common direction. This yields an explicit two-eigenvalue formula for the reduced α-z divergence and a closed-form optimal mechanism on the fidelity line α=z=1/2 for arbitrary pure binary qubit states. A complementary commuting result solves antipodal binary pairs throughout the data-processing region. For diagonal qudit ensembles, Weyl randomization reduces to cyclic convolution; the utility has a Fourier autocorrelation representation, and the minimax problem admits convex global formulations for 0<α<1 and α=2. Finally, the radius is shown to be stable under post-processing and measurement, to imply binary discrimination bounds, and to support hardware-aware calibration certificates. MSC Classification: Primary 81P45 , 94A17; Secondary 81P50 , 47N50 , 49K35 , 90C25 ,90C30