Overview This preprint develops a finite-history framework for quantifying how coherently stored environment records reveal logical quantum information. Instead of considering only which error occurred, the analysis treats the full complementary environment channel associated with a labeled family of finite quantum-error histories. For two selected history labels, the leakage is expressed exactly through the numerical-range diameter of the compressed relative operator. This yields global finite-history compression bounds, a zero-leakage characterization equivalent to Knill–Laflamme scalarity, and lower and upper bounds for recovery by a single common channel. Main results For a balanced independent two-step dephasing model on one logical qubit encoded into two physical qubits, the paper solves the worst-case optimization over the full rank-two code Grassmannian: $$\inf_{V\in\mathrm{Gr}(2,4)} \sup_{\phi,\psi} \mathcal L_{\mathrm{hist}}(V) = \frac12.$$ The transverse leakage is determined exactly by $$L_{\mathrm{tr}}(\eta,t) = \max\left\{ \frac12,\, 2\eta,\, \sqrt{\frac14-\eta^2}\sqrt{1+t} \right\}.$$ All global minimizers are classified. Besides the parity-isoclinic orthogonal-axis family, the minimizer set contains a genuine non-isoclinic region characterized by $$0 < \eta \le \frac14, \qquad e_z = o_z = 0, \qquad \vert{}\mathbf e \cdot \mathbf o\vert{} \le \frac{4\eta^2}{1-4\eta^2}.$$ A universal quartic inequality needed for the all-code theorem is reduced analytically to an exact polynomial-positivity problem and closed using rational Bernstein-basis verification. Distributional stability On a fixed labeled history-operator family, the complementary channel, worst-parameter leakage, and all-code minimax value are Lipschitz-stable under changes of the history probability distribution. With $$d_{\mathrm A}(p,q) = \left[ \sum_h \left(\sqrt{p_h}-\sqrt{q_h}\right)^2 \right]^{1/2},$$ the minimax value satisfies $$\vert{}\Lambda(p)-\Lambda(q)\vert{} \le 2d_{\mathrm A}(p,q).$$ Consequently, key quantitative implications of the balanced complete solution—including the minimax value, principal near-minimizer coordinate constraints, and a universal operational recovery lower bound—remain stable under nearby biased or correlated perturbations.
Overview This preprint develops a finite-history framework for quantifying how coherently stored environment records reveal logical quantum information. Instead of considering only which error occurred, the analysis treats the full complementary environment channel associated with a labeled family of finite quantum-error histories. For two selected history labels, the leakage is expressed exactly through the numerical-range diameter of the compressed relative operator. This yields global finite-history compression bounds, a zero-leakage characterization equivalent to Knill–Laflamme scalarity, and lower and upper bounds for recovery by a single common channel. Main results For a balanced independent two-step dephasing model on one logical qubit encoded into two physical qubits, the paper solves the worst-case optimization over the full rank-two code Grassmannian: infV∈Gr(2,4)supϕ,ψLhist(V)=21. The transverse leakage is determined exactly by Ltr(η,t)=max{21,2η,41−η21+t}. All global minimizers are classified. Besides the parity-isoclinic orthogonal-axis family, the minimizer set contains a genuine non-isoclinic region characterized by 0<η≤41,ez=oz=0,∣e⋅o∣≤1−4η24η2. A universal quartic inequality needed for the all-code theorem is reduced analytically to an exact polynomial-positivity problem and closed using rational Bernstein-basis verification. Distributional stability On a fixed labeled history-operator family, the complementary channel, worst-parameter leakage, and all-code minimax value are Lipschitz-stable under changes of the history probability distribution. With dA(p,q)=[∑h(ph−qh)2]1/2, the minimax value satisfies ∣Λ(p)−Λ(q)∣≤2dA(p,q). Consequently, key quantitative implications of the balanced complete solution—including the minimax value, principal near-minimizer coordinate constraints, and a universal operational recovery lower bound—remain stable under nearby biased or correlated perturbations.
## Overview This theorem-focused paper studies one logical qubit encoded into two physical qubits and exposed to two independent temporally ordered dephasing events, one collective and one differential, whose binary history labels are coherently stored by an environment. The objective is to minimize the worst-case logical information leaked into the environment’s history register over all rank-two quantum codes. A probability-adapted Bernoulli basis reduces the complementary output difference for arbitrary independent event probabilities to a traceless three-dimensional Hermitian arrow matrix. Phase-reflection and event-exchange symmetries then show that the fixed-code worst-phase leakage and its all-code minimax depend on the event probabilities only through their Bernoulli variances. ## Main results At equal event probabilities μ+=μ−=1/5, an explicit transverse non-isoclinic code satisfies L1/5,1/5wc(V)≤25099<52, where 2/5 is the exact optimum over the complete parity-isoclinic manifold. The strict upper bound is certified using exact rational tensor-Bernstein arithmetic. Distributional stability extends this separation to an explicit two-parameter rectangle whose open interior is a region of forced non-isoclinic optimality. On the symmetric independent line, the paper proves a thresholdless bifurcation: every nondeterministic nonbalanced distribution destroys parity-isoclinic global optimality. Near the balanced point, probability bias selects the universal finite anisotropy η0=832−4, and the minimax over the full rank-two code Grassmannian satisfies Λdiag(δ)=21−(21−62)δ2+o(δ2),δ=1−2μ. Every asymptotically minimizing code accumulates on the transverse balanced-minimizer stratum with r=21,η=η0,ez=oz=0, and with relative transverse-axis overlap bounded by ∣e⋅o∣≤32−1. Thus probability bias breaks the balanced continuum of globally minimizing codes and selects a genuinely non-isoclinic geometric phase with a universal leading anisotropy over the full code Grassmannian. ## Scope The paper does not claim an exact finite-bias formula for the complete minimax, a determination of the order at which the remaining transverse-axis overlap is selected, or a complete off-diagonal phase boundary for unequal event variances. These are identified as follow-up problems. ## Reproducibility package This record contains: - the manuscript PDF;- the complete LaTeX source;- a standalone reproducibility bundle;- seven deterministic exact-arithmetic theorem verifiers;- machine-readable JSON verification reports;- a SHA-256 manifest and verification utility. No floating-point comparison enters a verifier’s PASS decision. The scripts certify the exact rational or symbolic identities used in the proofs. Compactness, localization, perturbative-envelope, and Grassmannian liminf arguments remain analytic parts of the manuscript and are not replaced by computation. ## Version Version v0.3r2, July 2026. ## Related public records - Directional code-geometry foundation: DOI `10.5281/zenodo.21523950`- Balanced finite-history benchmark and complete minimizer geometry: DOI `10.5281/zenodo.21560397`
## Overview This work develops a fourth-order perturbative theory for the minimax protection of finite two-event quantum error histories by rank-two quantum codes in the near-balanced regime. The central phenomenon is an unusual form of degenerate selection. Within every compact subset of the balanced overlap interior, the fully optimized worst-case leakage remains exactly independent of the overlap parameter. Consequently, the interior degeneracy is not lifted by the second-order term, the fourth-order term, or any finite Taylor coefficient within the same analytic phase. Overlap selection instead occurs through a singular boundary layer near the balanced overlap cap. The paper establishes: - exact fixed-interior overlap persistence;- the singular overlap-cap boundary-layer law;- a locally unique analytic fourth-order transition curve;- the bias-induced displacement of the optimal code anisotropy;- the optimized transverse minimax expansion;- and the fourth-order minimax closure over the full Grassmannian of rank-two codes. ## Main result Let $$c_*=\frac12-\frac{\sqrt2}{6},\qquadc_4=\frac{479\sqrt2-492}{3024}.$$ For the symmetric near-balanced bias parameter δ, the full-Grassmannian minimax leakage satisfies Λdiag(δ)=21−c∗δ2−c4δ4+o(δ4). Equivalently, in the variance coordinate A↑1/4, Λind(A,A)=21−c∗(1−4A)−c4(1−4A)2+o((1−4A)2). The optimized transverse family supplies the matching recovery sequence, while a reflected one-axis lower jet, balanced-normal coercivity, compact localization, and a full-space liminf argument show that nontransverse rank-two codes cannot improve the minimax value through fourth order. ## Singular overlap selection The balanced overlap cap is t∗=32−1. In the second-order boundary layer t=t∗−ρδ2+o(δ2), the leakage coefficient is governed by CBL(ρ)=min{c∗,41+κ02ρ}, where κ02=83(2−2). At fourth order, the locally unique transition curve at fixed selected anisotropy is tc(η0)(δ)=32−1−92−2δ2+1134181−1492δ4+O(δ6). The optimal anisotropy moves according to ηδ=η0+η023523762−379δ2+O(δ4), with η0=832−4. ## Full-space geometry For O(δ4)-near minimizers, the full-space lower bound gives the mixed-scale localization η−η0=O(∣δ∣), while the three balanced hard-normal coordinates obey $$r-\frac12=O(\delta^4),\qquade_z=O(\delta^4),\qquado_z=O(\delta^4).$$ The corresponding fourth-order effective lower functional is Qfull(α,β)=−c4+β+278(22+3)α2,β≥0. Its minimum is −c4, attained by the optimized transverse recovery family. ## Scope The results are perturbative and apply in the symmetric near-balanced regime. The paper does not claim: - an exact finite-bias formula for the full minimax value;- a complete finite-bias classification of all minimizing codes;- sixth-order full-Grassmannian geometry;- or the full unequal-variance phase diagram. The leakage minimax is operationally related to recovery obstruction through complementary-channel duality, but the present theorem does not identify it with a particular recovery-fidelity optimizer. ## Reproducibility The accompanying research bundle contains: - the manuscript PDF and LaTeX source;- theorem-facing proof notes;- deterministic SymPy verification scripts;- machine-readable JSON verification reports;- a tested Python/SymPy environment specification;- a portable SHA-256 manifest verifier;- and a complete SHA-256 manifest. The symbolic verifiers check exact algebraic identities, characteristic-polynomial expansions, radical constants, Hessian signs, transition coefficients, and the algebraic inputs to the full-space effective functional. Compactness, analytic continuation, branch compatibility, and liminf/recovery arguments are supplied in the manuscript and proof notes rather than claimed as fully machine-verified. This paper is an independent fourth-order sequel to: **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`.
We determine the exact worst-direction Fisher-information cost of using a parameter-independent readout to estimate an unknown qubit direction at known Bloch-vector length η. Every fixed-local architecture, including recorded classical randomization, heterogeneous single-copy measurements, and arbitrary outcome spaces, reduces exactly to a zero-barycenter probability measure on the Bloch ball. For every full-rank qubit and every trace-balanced spectral Fisher loss, the resulting minimax problem is rigid: the unique optimal aggregate design is the spin-coherent Haar positive-operator-valued measure (POVM). For N copies, inverse-Fisher A loss has the exact value 2/[Nf(η)], where $
f(η)=
\frac{2η-(1-η^2)\log[(1+η)/(1-η)]}{4η}$. This uniqueness has an immediate finite-readout consequence. No finite-support measurement attains the unrestricted mixed-state optimum, while at the smallest globally regular support the tetrahedral symmetric informationally complete (SIC) measurement is uniquely A- and D-minimax, with exact worst-direction values. Relaxing the fixed-readout constraint separates the asymptotic resources: one-way local operations and classical communication (LOCC), unrestricted LOCC, and separable measurements have A-loss coefficient 4/η2, whereas collective measurements attain 2(1+η)/η2. We further classify the rigidity conditions for unequal contrasts and show that Haar uniqueness survives at the nonregular pure-state endpoint.
We develop a theory of approximate quantum error correction (QEC) based on the error-set model, complemented by general methods for code construction. Exact QEC has a powerful error-set structure: by the Knill-Laflamme conditions, a code correcting a given error set automatically protects against every channel whose Kraus operators lie in their linear span. This linearity gives rise to code distance, the equivalence between erasures and general errors, and a theory of asymptotically good codes. A longstanding view has been that these features do not extend to AQEC, leaving the theory essentially channel-by-channel. We show instead that, although full Knill--Laflamme linearity fails, a restricted form survives and suffices to extend all three structural features to the approximate setting. Specifically, a common error-set criterion governs families of channels whose Kraus operators are linear combinations of a given error set and whose coefficient matrices satisfy a spectral constraint. Using the Bény-Oreshkov worst-case and Petz average-case frameworks, we derive uniform fidelity guarantees for these families in terms of two new code parameters--the \emph{environment-leakage distance}, controlling worst-case performance, and the \emph{Knill-Laflamme Hellinger distance}, characterizing the average-case performance of Petz recovery.
To demonstrate the scope of this model, we develop partition-based constructions across diverse quantum systems and geometries, placing exact and approximate correction on equal footing. These constructions lead to a metric--error alignment hierarchy for Hilbert spaces, metrics, and error families, which in turn characterizes the resulting recovery guarantees. They yield the first known asymptotically good code families for fermionic systems, one-dimensional Rydberg-blockaded systems, and deletion errors, and extend to other physical platforms.
## 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.
Universal recovery -- the existence of a single recovery map that corrects an entire family of error channels -- is a central feature of quantum error correction (QEC). In exact QEC, linearity guarantees that a code correcting a given error set also corrects every channel whose Kraus operators lie in its linear span, and that a single recovery map suffices for all such channels. Approximate quantum error correction (AQEC), which relaxes perfect recovery to recovery with controlled error, has traditionally lacked this structure. In a recent paper (arXiv:2607.22995), we developed a theory of approximate quantum error correction showing that a restricted form of linearity persists in the approximate setting, yielding uniform AQEC guarantees for the family of channels controlled by a given error set.
In this work, we complete the picture by establishing the second half of universal recovery in the approximate setting: a single recovery map can simultaneously correct every channel controlled by a given error set. The error-set theory we proposed quantifies approximate correctability through two parameters: the environment-leakage distance, governing worst-case performance, and the Knill--Laflamme Hellinger distance, governing average-case performance. We show here that both quantities also control universal decoding. We further study the Petz map naturally associated with an error set as an explicit universal recovery, and obtain uniform average- and worst-case guarantees across the entire family of channels.