A finite group-valued temporal reference frame defines a correlated random-unitary process rather than a list of independent channels. We identify a regime in which its complete trajectory law is an exact operational coordinate. If the carrier contains every irreducible representation of the finite group $G$, one ancilla-assisted input resolves all group elements, and for arbitrary laws $μ,ν$ on $G^n$ the strategy half-distance equals $d_{\mathrm{TV}}(μ,ν)$. Optimal output-side causal post-processing likewise reduces to classical convolution on $G^n$. This converts temporal hiding into a $q$-ary coding problem. Uniform coset laws of an $[n,k]_q$ code are identical on every set of fewer than $d(C^\perp)$ slots and perfectly distinguishable globally. For $M$ sectors with leakage at most $η$ from $t$ selected slots and global decoding error $ε$, we obtain $(1-ε)\log M \leq (n-t)\log q+η+h_2(ε)$. For nested codes $C\subset D$, the payload $R=\dim D-\dim C$, invisible depth $t=d_{\mathrm{rel}}(C^\perp,D^\perp)-1$, and sector distance $d_{\mathrm{rel}}(D,C)$ obey $R+t+d_{\mathrm{rel}}(D,C)-1\leq n$ and $R+t+2e+f\leq n$, where $e$ and $f$ are adversarial errors and known erasures. Nested generalized Reed-Solomon codes attain the bounds in their existence range. We also give the exact projection-rank leakage profile and show that hiding arbitrary coherent superpositions from $t$ slots is precisely quantum erasure correction, yielding $\log_q K+2t\leq n$. The results separate classical trajectory privacy from coherent temporal privacy and quantify the robust payload hidden from reduced process tomography.
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12 pages, 3 figures. Ancillary verification code included
A finite group-valued temporal reference frame defines a correlated random-unitary process rather than a list of independent channels. We identify a regime in which its complete trajectory law is an exact operational coordinate. If the carrier contains every irreducible representation of the finite group G, one ancilla-assisted input resolves all group elements, and for arbitrary laws μ,ν on Gn the strategy half-distance equals dTV(μ,ν). Optimal output-side causal post-processing likewise reduces to classical convolution on Gn. This converts temporal hiding into a q-ary coding problem. Uniform coset laws of an [n,k]q code are identical on every set of fewer than d(C⊥) slots and perfectly distinguishable globally. For M sectors with leakage at most η from t selected slots and global decoding error ε, we obtain (1−ε)logM≤(n−t)logq+η+h2(ε). For nested codes C⊂D, the payload R=dimD−dimC, invisible depth t=drel(C⊥,D⊥)−1, and sector distance drel(D,C) obey R+t+drel(D,C)−1≤n and R+t+2e+f≤n, where e and f are adversarial errors and known erasures. Nested generalized Reed-Solomon codes attain the bounds in their existence range. We also give the exact projection-rank leakage profile and show that hiding arbitrary coherent superpositions from t slots is precisely quantum erasure correction, yielding logqK+2t≤n. The results separate classical trajectory privacy from coherent temporal privacy and quantify the robust payload hidden from reduced process tomography.
A finite quantum reference token with group-valued misalignment induces a random-unitary channel, but optimal degradation is generally an optimization over all quantum post-processings. For a unitary representation U of a finite group G, we prove that the following conditions are equivalent: U contains every irreducible type; one ancilla-assisted input has an orthonormal G-orbit; signed group measures embed isometrically into channels in diamond norm; and, for every pair of noise laws p,q, infΛ∈CPTP21∥ΦqU−Λ∘ΦpU∥⋄=minr∈P(G)21∥q−r∗p∥1. Thus representation completeness is the exact carrier condition for universal reduction of quantum post-processing to classical convolution. We determine the minimum ancilla dimensions for an orthogonal orbit and for an invariant calibration seed. For an incomplete carrier, with visible Plancherel dimension S(U), we derive the exact conditional-expectation distance 21∥id−ΦuU∥⋄=1−1/S(U) and an explicit quantum--classical deficiency gap. For irreducible carriers the deficiency is obtained in closed form; the faithful two-dimensional representation of S3 yields an exact ten-percent reduction relative to classical convolution. We also characterize law identifiability through the conjugation representation, provide finite linear programs and decision witnesses, and establish both a finite-dimensional obstruction and stable visible-band reconstruction for infinite compact groups. Deterministic ancillary code reproduces the finite-group examples and numerical regression checks.
## Overview This paper studies when a reduction of a finite quantum error-history model preserves the local direction of quantum-code optimization, and when it reverses that direction. For finite coactive spectral branches, the paper first proves a second-order local truncation-safety certificate. A strict optimizer of a truncated model persists when the full-versus-truncated effective-jet discrepancy is smaller than the coercivity margin of the truncated objective. Opposite leading jets provide a certificate of local optimizer reversal. The paper then identifies an exact boundary between genuine temporal structure and removable path redundancy. For a finite projective unitary error group, any open-loop path operator has the form Wγ=eiθγUc(γ), where c(γ) is the final group product. Consequently, an arbitrary temporally correlated path law is isometrically equivalent, for complete complementary leakage and for every code, to the single-shot distribution of the final projective product: Lpath(V)=Lπ(V). For Pauli trajectories, this remains true even when ordered errors do not commute, because different orderings can differ only by projective phases after contraction to the final channel. Thus open-loop final-channel leakage cannot identify temporal organization beyond the final projective Pauli-product distribution. Intermediate measurements, controls, syndrome extraction, or adaptive recovery are required for genuinely intervention-resolved multi-time code geometry. ## Distribution-robust optimizer reversal For an arbitrary three-location bit-flip trajectory law, let π denote the induced final-parity distribution on F23, let c=(1,1,1), and define Λπ:=∑t∈F23πtπt⊕c. For the three-qubit repetition code E=span{∣000⟩,∣111⟩}, the complete complementary leakage is exactly Lπ(E)=Λπ. For every fixed full-support final-parity law, the intrinsic Fourier-syndrome graph coordinate w satisfies Lπ(V(w))=Λπ−2Λπ∣w∣2+o(∣w∣2). Hence the repetition code is a strict local maximum throughout this two-real-dimensional graph chart for every full-support final-parity distribution, independently of temporal correlations, stationarity, roundwise independence, identical error rates, or cyclic spatial symmetry. By contrast, a conditioned zero-and-one-fault truncation with positive coarse single-location weights has L≤1(E)=0 and strictly positive leakage at every nonzero sufficiently nearby graph code. The same reference code is therefore a strict local minimum for the truncated model and a strict local maximum for the complete process. The central conclusion is: > Exact aggregation of paths that implement the same projective physical operator preserves the entire code-optimization landscape, whereas deletion of higher-fault history fibers can reverse the local optimizer geometry. ## Scope The results concern random-unitary, open-loop finite-history models and complete complementary-channel leakage. The optimizer reversal is proved in a specified Fourier-syndrome graph chart around the three-qubit repetition code. The paper does not claim: - a full-Grassmannian classification for the three-qubit model;- a decoder or recovery-circuit construction;- a logical-error threshold;- a circuit-level hardware noise model;- collapse of an intervention-resolved process tensor;- equivalence in the presence of intermediate syndrome measurements, controls, or adaptive recovery. These intervention-resolved settings form the natural next boundary of the theory. ## Relation to the preceding research program This paper is a concluding bridge theorem paper connected to a six-paper exact finite-history QEC and spectral-minimax research line. ### 1. Compression-geometric foundation **Sharp Complementarity, Complete Attainable Region, and Metric Rigidity of Rank-Two Quantum Codes under Collective and Differential Dephasing** Version v0.5r2b DOI: 10.5281/zenodo.21523950 This paper determined the complete attainable compression-defect region for rank-two codes under collective and differential two-qubit dephasing, classified the equality geometry, and established sharp metric rigidity. ### 2. Balanced finite-history minimax **Environment-Record Leakage for Finite Quantum Error Histories: Balanced Global Minimax, Complete Minimizer Geometry, and Distributional Stability** Version v0.5r2 DOI: 10.5281/zenodo.21560397 This paper introduced the complete environment-record leakage objective for finite error histories, solved the balanced two-step minimax problem globally over Gr(2,4), classified all balanced minimizers, and established distributional stability. ### 3. Bias-induced geometric selection **Bias-Induced Non-Isoclinic Optimality in Finite Quantum Error Histories: Exact Separation, Thresholdless Bifurcation, and Near-Balanced All-Code Minimax** Version v0.3r2 DOI: 10.5281/zenodo.21598820 This paper proved exact non-isoclinic separation, an open non-isoclinic parameter region, thresholdless loss of isoclinic optimality under arbitrarily small bias, and the leading full-Grassmannian near-balanced minimax selector. ### 4. Fourth-order overlap selection **Flat-Interior Persistence and Singular Overlap Selection in Near-Balanced Quantum Error Histories: Fourth-Order Transition Curves and Full-Grassmannian Minimax Closure** Version v0.1r3 DOI: 10.5281/zenodo.21622029 This sequel resolved the higher-order overlap degeneracy, established flat-interior persistence, derived singular boundary-layer transition curves, and closed the near-balanced full-Grassmannian minimax expansion through fourth order. ### 5. Abstract reflected spectral-jet framework **Reflected Spectral-Jet Realization for Degenerate Minimax Problems: Hard-Normal Selection, Flat Tangential Persistence, and Coactive Boundary Transitions** Version v0.2r1 DOI: 10.5281/zenodo.21636022 This paper extracted a reusable abstract framework for hard-normal isolation, soft tangential persistence, reflected active branches, and coactive boundary transitions in degenerate spectral minimax problems. ### 6. Finite-orbit generalization and first QEC truncation reversal **Beyond Reflected Pairs: Finite-Orbit Spectral Jets, Quadratic Isolation, and Multi-Branch Coactivity in Degenerate Minimax Problems** Version v0.4r1 DOI: 10.5281/zenodo.21644364 This paper generalized the reflected-pair theory to finite transitive orbits, developed multi-branch coactivity and representation-theoretic jet obstructions, and exhibited an exact three-location QEC realization in which restoring omitted multi-fault histories reverses the local optimizer geometry. ### 7. Present bridge theorem The present paper determines which open-loop temporal path information is exactly removable and which history reductions remain unsafe. It extends the earlier single-process reversal from independent or cyclic models to arbitrary correlated projective-unitary paths and arbitrary full-support final-parity laws. The resulting research progression is compression geometry⟶balanced global minimax⟶bias-induced selection⟶fourth-order closure⟶reflected spectral jets⟶finite-orbit coactivity⟶projective path collapse and distribution-robust reversal. ## Reproducibility package The accompanying bundle contains: - the manuscript PDF;- the LaTeX source;- an exact symbolic verification script;- a machine-readable verification report;- a README describing the proof-artifact contract;- a SHA-256 manifest;- version and package-audit records. The verifier performs 72 exact symbolic checks, including projective Pauli phase identities, phase-covariant path compression, arbitrary-parity polar factors, translation orthogonality, Fourier-syndrome graph compressions, and coefficient bookkeeping for the distribution-robust local expansion. The verifier deliberately rejects optimized Python mode. PASS decisions do not rely on floating-point tolerances. Compactness, maximizing-direction localization, trace-norm differentiability, and uniform asymptotic assembly remain analytic parts of the manuscript proof rather than delegated computational claims.
This survey introduces the role of frame phase retrieval in pure-state identification and information preservation by quantum channels. For unit vectors, the lift x↦xx∗ identifies vectors that differ only by a global phase with the same rank-one quantum state, converts frame intensities into linear functionals of the lifted state, and makes the adjoint pullback of output observables an operator-valued measurement on the input system. From this viewpoint, a channel is phase retrievable exactly when it is injective on pure states. We develop this correspondence through Choi-rank-two linear combinations of Kraus operators, higher-rank relative spectra, structural obstructions, and constructions with prescribed Choi rank. We distinguish injective identification from full tomography, perfect one-shot discrimination, zero-error classical communication, and exact quantum correction. Twirling channels then provide a structured setting in which commutants, irreducible dimensions, multiplicities, orbit-frame orthogonality, coding indices, and phase-retrievable subspaces can be read from group representations.
Quantum reference frame (QRF) transformations are usually presented as unitary or gauge-fixing changes of coordinates. That picture is exact for ideal frames but fails for finite clocks, gyroscopes, POVM-defined tokens, observers correlated with the targets they describe, and relativistic frames with quantum boost and clock degrees of freedom; the obstruction is loss of distinguishability. We model a non-ideal frame by a normal covariant perspective channel. A deterministic transformation A → B then exists exactly when the kernel of A's channel is contained in the kernel of B's channel, and is unique on the preparable image. Ancilla-stable transformations are complete-order quotient morphisms, while a laboratory implementation on a chosen output space is a separate Choi-Arveson extension problem. The common invariant content of a family of frames is a Heisenberg operator system, carrying an algebraic product only on its product-compatible core. For finite frames in the regular representation, concrete covariant degradability A → B is decided by a group-convolution linear program with explicit dual witnesses ; for Abelian groups this becomes a Fourier mode-support resource lattice. Our main result is that for every finite group this convolution linear program already equals the full diamond-norm deficiency: coherent, multiplicity-exploiting post-processings never outperform random translations, so the resource theory of frame degradation is entirely classical. The same quotient principle governs process tensors, protected relational sectors, entangled observers, relativistic covariance — the boosted-spin perspective being an explicit U(1) dephasing channel — and observer networks. Reproducible numerical certificates accompany every analytic claim.
For classical--quantum wiretap channels with a transitive covariant input action, both output ensembles admit a description as unitary group orbits of fixed seed states. We use this structure to construct finite-block preprocessing schemes whose private information is an entropy difference evaluated at two distributions related by group convolution. Applying the construction to product groups allows correlations across any fixed number of channel uses. We give a sufficient condition for the associated uniform-input bound to equal the single-use private information. Constant-composition channel codes and universal2 hashing achieve the resulting rates, with trace-distance leakage controlled by sandwiched Rényi information. For uniform outer randomization, the representation structure reduces the comparison-state optimization to Eve's invariant states. For the hybrid binary channel considered by Tikku, Berta and Renes, numerical optimization at three and four uses gives rates above our numerical estimate of the optimized noisy-repetition benchmark wherever that estimate is positive on the tested grid. The improvement comes from noise distributions beyond the family of independent physical flips and a symmetric logical flip. We also prove nonadditivity at an explicit channel parameter using a two-use encoder, an analytic single-use converse, and certified entropy bounds. Finally, we relate the hybrid model to binary phase-shift keying with a fixed receiver measurement.
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