A qubit carries one of two non-orthogonal label states, exchanged by a Pauli-𝑍 symmetry. A channel must produce a record of the label while returning the qubit, with the returned qubit’s deviation from the label held within a fixed disturbance budget; the record’s deviation from the label is the error to be minimised. We compare three implementation classes at one operating point (label overlap 3/5, budget 1/10): environment-free isometries intertwining the symmetry; channels that may use and discard an environment on which the symmetry acts trivially (equivalently, classical mixtures of the first class, with the budget on the mixture); and all channels. The optimal errors are bracketed by exact rational certificates: a feasible channel and an exactly dual-feasible point for each of the two convex classes, and an exact one-variable reduction with a rigorous interval enclosure for the nonconvex environment-free class. The unrestricted class lowers the minimal error by between 17.2789% and 17.3105% relative to the environment-free class; the environment-free optimum and the mixture-class optimum differ by less than 1.09 ⋅ 10−6, so more than 99.968640% of the reduction lies in the step from the mixture class to the unrestricted class. Equality of the first two optima is not established. All certificates are verified by exact arithmetic from standalone files; the verification requires no optimisation solver.
A qubit carries one of two non-orthogonal label states, exchanged by a Pauli-𝑍 symmetry. A channel must produce a record of the label while returning the qubit, with the returned qubit’s deviation from the label held within a fixed disturbance budget; the record’s deviation from the label is the error to be minimised. We compare three implementation classes at one operating point (label overlap 3/5, budget 1/10): environment-free isometries intertwining the symmetry; channels that may use and discard an environment on which the symmetry acts trivially (equivalently, classical mixtures of the first class, with the budget on the mixture); and all channels. The optimal errors are bracketed by exact rational certificates: a feasible channel and an exactly dual-feasible point for each of the two convex classes, and an exact one-variable reduction with a rigorous interval enclosure for the nonconvex environment-free class. The unrestricted class lowers the minimal error by between 17.2789% and 17.3105% relative to the environment-free class; the environment-free optimum and the mixture-class optimum differ by less than 1.09 ⋅ 10−6, so more than 99.968640% of the reduction lies in the step from the mixture class to the unrestricted class. Equality of the first two optima is not established. All certificates are verified by exact arithmetic from standalone files; the verification requires no optimisation solver.
We study a minimal coexistence task: a single qubit of memory must be used, through one unassisted encode–decode factorization, both to report a classical binary label (which of two orthogonal "sectors" a carrier occupied) and to preserve quantum information within one sector, in the presence of a Z₂ symmetry. We quantify the first by a label accuracy L — which we show is exactly a two-state discrimination success probability for a decoder-induced measurement — and the second by the unconditional same-sector entanglement fidelity F_same. Our main result is the exact optimal trade-off: for every required accuracy τ in [1/2, 1], the maximum attainable F_same is a single closed-form curve Φ(τ), and this maximum is the same whether or not the factorization is required to be covariant under the symmetry (at the fixed memory representation V = Z). An explicit two-parameter covariant CPTP family attains Φ at every threshold, so covariance imposes no optimal-fidelity penalty anywhere on the trade-off. The frontier has a structural transition at τ = 5/8, above which every optimal code-restricted encoder is necessarily nonunitary, and the attainable score region is nonconvex. A directly applicable prior hybrid-memory inequality (Kuperberg) specializes here to F_same ≤ (5 − 2τ)/4, which is valid but strictly loose for τ > 1/2 and tight at τ = 1/2. Whether the exact frontier, the transition, or covariant optimality follow from earlier results remains open. The results are proved analytically, with selected exact symbolic identities, explicit constructions, and regression tests. This record includes the manuscript and a verification supplement of seven standalone SymPy scripts, a README, and a runner (run_all.sh) that reproduce the exact checks (each exits 0 on success; tested with Python 3.11.15, SymPy 1.14.0, NumPy 2.4.4). The results are mathematical; any interpretive applications are developed separately and are not part of this work.
## 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`
When does Gaussian encoding miss the communication limit? This paper identifies an encoding class that attains the asymptotic capacity of arbitrary quantum block codes, and an exact resource threshold below which individual Gaussian signal states cease to be optimal. The result gives communication theorists a source-selection rule: classical mixtures of Gaussian packets suffice, while the shape of their displacement distribution can be essential. Channel and operating regime. One use is a two-mode additive Gaussian-noise channel with covariance Yn,c=(nI2cZcZnI2), where Z=diag(1,−1) and vacuum covariance is I2/2. For fixed y>0 and resource budget r>0, cn=n2−y2,En≥cn+2n+21,n→∞. The mean photon budget is 2En per pair. The resource is relative entropy to normalized PPT states across the two input streams, averaged over transmitted messages at at most r nats per pair. Arbitrary mixed, non-Gaussian and cross-use-entangled messages and joint decoding are allowed. The coding limit is taken at each fixed channel before the strong-correlation limit. 1. Exact asymptotic capacity reduction. With g(x)=(x+1)ln(x+1)−xlnx and An=2g(En+n), C=Chull=An−lnn−1−H(r)+o(1). Here Chull restricts every transmitted message to the trace-closed Gaussian convex hull. The regularized classical variational function H is the lower boundary, at cost at most r, of the closed convex upper hull of (mD(f∥w),mh(f)−lnπ),w=Wσ≥0,m≥1. The displacement law f is a classical probability density; its reference w is the nonnegative Wigner function of a physical state. The exact dual is H(r)=supα>1α−1−P(α)−αr,P(α)=supm≥1,Wσ≥0m1ln[πm(α−1)∫Wσα]. This converts optimization over quantum block codes into an entropy variational problem with an explicit quantum admissibility condition. 2. A sharp decision boundary for Gaussian signals. Define k−1−lnk=r, 0<k≤1, T(r)=−lnk and V(r)=1−H(r). Then CG=An−lnn−2+T(r)+o(1),rc=ln2−21,V(r)=T(r)(r≥rc),T(r)<V(r)≤1−ln2+2r(0<r<rc). Thus a Gaussian density is asymptotically optimal above the threshold. Below it, non-Gaussian displacement shapes strictly improve the rate, while Gaussian-packet mixtures still attain the all-state optimum. The low-resource offset is characterized by the variational formula. 3. Sub-vacuum Wigner entropy yields a communication advantage. For messages that are exactly PPT, CPPT=An−lnn−1−hW++o(1),limn→∞(CPPT−CG(0))=1−hW+>0, where hW+=infm,σ[h(Wσ)/m−lnπ], over finite-energy physical states with Wσ≥0, satisfies ln2≤hW+<1. The comparator CG(0) allows Gaussian messages with vanishing resource per use. The achieving messages are simultaneously PPT, Wigner-positive and mixtures of Gaussian states. All rates and logarithms use nats.
How much quantum storage must a hidden implementation retain when experimental observations leave its structure undetermined? This paper establishes exact minimum quantum-storage requirements across all implementations compatible with specified channel observations. The resource is the worst-input expected logarithm of retained quantum dimension across one controlled temporal cut, with unlimited classical records available at no cost. For the channel dimension measure introduced by Cope and Uola, the paper proves their suggested minimax equality relating minimum storage to output Schmidt measure and extends it to every nonempty compact convex set of compatible channels. One common input enforces the resource minimum across all compatible implementations. A finite optimal implementation exists, although an optimal entangled probe need not. For qubit channels, four measured coefficients determine an exact, attained minimum without assuming channel covariance or unitality. An explicit five-Kraus construction attains 99/169, while a separate analytical certificate proves that every compatible implementation must use nontrivial quantum storage with at least that probability on the same specified input. A second exact formula uses only two Pauli-transfer coefficients and yields a minimum storage requirement of 0.58 when both coefficients are jointly constrained to the interval [0.79, 0.81]. The paper also presents semidefinite optimization over compatible experimental uncertainty regions. The results require calibrated quantum inputs and outputs, concealed preparation labels, and accounting for every quantum carrier across the storage cut. They constrain storage resources without reconstructing the hidden implementation or certifying environmental non-Markovianity. Motivated by Shadow Theory’s source/readout distinction, the proofs use standard quantum mechanics and quantum-information mathematics. Version 1.1 includes complete proofs, explicit attaining constructions, worked examples, and discussion of experimental assumptions and limitations. No experimental data are reported.
A central task in quantum information science is state certification: testing whether an unknown state is є1-close to a fixed target state, or є2-far. Recent work has shown that surprisingly simple measurement protocols – comprising only single-qubit measurements – suffice to certify arbitrary n-qubit states. However, these certification protocols are not robust: rather than allowing constant є1, they can only positively certify states within є1=O(1/n) trace distance of the target. In many experimental settings, the appropriate error tolerance is constant as the system size grows, so this lack of robustness renders existing tests inapplicable at scale, no matter how many times the test is repeated.
We study the canonical USp(4)-covariant quantum compression problem in which the fifteen-dimensional carrier W = V5 ⊕ V10 is encoded through a four-dimensional covariant memory M = V4 and decoded, and we ask for the optimal entanglement fidelity F*(p) of the composite channel on a trace-one, sector-weighted reference state ρ_p. This problem is the operational face of a structural fact: the same pair of sectors 5 ⊕ 10 that makes USp(4) on C^4 state-homogeneous but not observable-uniform is exactly the modal content a minimal covariant memory is asked to carry. Our contribution is not a resolution of the underlying capacity question but a demonstration that this problem admits an exact reduction and a machine-checkable global bound: the reduction and the final lower- and upper-bound certificates are exact, while only the intermediate relaxation-solving step is numerical. We give an exact representation-theoretic reduction — Wedderburn splitting of the covariant commutant over Q(i), an induced-involution positivity bridge, and an exact trace-preservation reduction to fifteen free rational coordinates — turning F*(p) into a bilinear polynomial optimization over a product of two explicit semialgebraic bodies, each the affine preimage of the product cone (R+ × SOC4 × SOC4). Exploiting correlative sparsity we build an order-two sum-of-squares relaxation, solve it numerically for discovery, reconstruct the resulting SOS upper bounds in exact rational arithmetic, and separately certify exact rational feasible lower bounds from explicit witnesses. At p = 1/2 and p = 7/8: 0.0749 ≤ F*(1/2) ≤ 0.21, and 0.1546 ≤ F*(7/8) ≤ 0.643 (exact rationals in the paper). The upper bounds are Positivstellensatz certificates whose polynomial identity and positive semidefiniteness are verified with no optimizer in the loop by a standalone verifier we provide. Throughout we keep three visibly separate registers — exact theorems, numerical diagnostics, and open or conditional interpretations — and we are explicit that the exact optimum, the exactness of the relaxation, and the width of the certified interval remain open. Package: this record includes the manuscript, a single 30-page combined archival PDF (cover + manuscript + auditable reduction-supplement listing) inside the package, and a verification package with two independent machine-checkable obligations (the exact reduction chain and the certified two-sided enclosure). Manuscript licensed CC BY 4.0; code licensed MIT.