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.
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.
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.
This record gives an exact, machine-checked resource analysis of a specific USp(6) = Sp(3)-covariant quantum compression construction, and the reproducible certificates behind it. The carrier is W = V14 ⊕ V21 (dimension 35), the memory is the sixdimensional irreducible V6, the source is the balanced sector-mixed state with weights (1/28, 1/42), and the figure of merit is the Schumacher entanglement fidelity 𝜔. The construction and its coherent witness are inherited from the companion compression paper (DOI 10.5281/zenodo.21744306), which certifies F*_effEB,V6 ≤ 1/100 < 1/40 < 39587/1382976 = 𝜔(witness): a coherent covariant channel through the six-dimensional memory beats every entanglement-breaking channel through that memory. Four results, each verified exactly over the rationals, answer one coherent question — what preserves the compression advantage in this symmetry-constrained construction: (1) Sector-dephasing benchmark. Removing all coherence between the two output sectors drops the fidelity from 39587/1382976 to the within-sector value 55/3528, which still exceeds the 1/100 entanglement-breaking upper bound; the certified advantage survives, and the coarse output-sector label is blind to the input. (2) Branch entanglement. The two normalized output-sector channels N14 and N21 (output sector restricted, input/reference kept full) are both completely positive, trace-preserving, and have a negative partial transpose; each retains reference entanglement, so neither is entanglement-breaking. (3) Branch replacement. Replacing one branch at a time by an entanglement-breaking channel, at fixed sector probabilities and through the same memory, keeps the V14 branch (M14, 𝜔 = 835/74088 > 1/100) above the bound while the V21-retaining channel (M21, 𝜔 = 263/49392 < 1/100) does not clear it; both complete channels remain NPT. (4) Entanglement-breaking comparator. An explicit admissible covariant entanglementbreaking channel through the same memory strictly outperforms M21, with 𝜔 = 1544075/288004752 > 263/49392 (exact margin 5261/144002376 > 0). This resolves the previously open comparison and brackets the entanglement-breaking optimum as 1544075/288004752 ≤ F*_effEB,V6 ≤ 1/100. The overall lesson is objective-specific: retaining reference entanglement and obtaining a task-specific quantum advantage are distinct here. Every claim is checked by an exact rational verifier (python-flint): Choi positivity and partial-transpose tests use the certificate’s own Gram metric, and the comparator’s score is confirmed by reconstructing the actual channel and re-evaluating the original objective. Scope and caveats: the results have been audited for correctness by an independent reviewer; this record does not claim comprehensive novelty relative to the existing literature on quantum-memory resource theories and operational benchmarks, nor establish journal suitability. Any connection to the author’s broader “Presence Hypothesis” program is
This preprint studies operational equivalence and recovery in a twenty-dimensional residual system with a fixed four-sector representation of dimensions 8, 8, 2 and 2. It distinguishes what a declared observation algebra can detect, what a channel preserves, and what a retained classical record makes recoverable. Record-resolved criteria separate useful reversible-error records from unrelated labels and measured-sector records, even when the discarded channel is the same. For a specified coarse-record model, the work identifies exactly recoverable observable algebras and proves an optimal entanglement-fidelity expression on a fixed four-state orbit code. Preparation-aware recovery is then classified over all one-use isometric subspaces in the stated family. An explicit ten-dimensional marker code attains exact recovery under a specified stochastic record where the whole-sector support restriction permits only eight dimensions. Its encoding, nonbaseline marker measurement, and conditional inverse operations remain supplied resources. Scoped Lean evidence verifies the explicit arbitrary-complex-matrix recovery construction and related finite identities; it does not formalize the general complete-positivity/reference framework or optimal-dimension theorem. The recorded-error model is not derived from the companion exchange-timing model, and no protection of its long pulse sequence is claimed.
Abstract Decoherence suppresses interference, but an observer’s ability to reconstruct a classical pointer value depends on additional resources: distinguishable conditional environmental states, an allowed measurement, sufficient accessible fragments, and sufficiently independent records. We provide an explicit minimal quantum-to-operational bridge that places these requirements in one chain. A system qubit is correlated with conditional environmental qubit states, permitting exact expressions for residual coherence, quantum mutual information, pointer Holevo information, accessible information, and Helstrom discrimination error. Full density-matrix calculations validate the formulas across 280 endpoint and interior cases. An accessible-sector/hidden-bath construction shows that coherence can fall below 2.3 × 10⁻⁵ while the observer’s record information and error remain fixed at 0.0803 bits and 0.4009. Measurement policies shift recovery thresholds: for record overlap 0.8 and error tolerance 0.05, collective or optimized adaptive-local discrimination requires four fragments, a fixed canonical-basis measurement requires six, and repeated one-fragment Helstrom measurements followed by classical decoding require seven. More importantly, records matched at the one-fragment level can have radically different collective behavior. For a binary pointer, the common-mode classical channel has an exact accessible-information ceiling of 1 − h₂(0.1) = 0.5310 bits, while the noncommuting hidden-memory ensemble approaches the same Holevo-information ceiling; matched independent records approach one bit and zero error. Disjoint observer counts and unanimous agreement likewise depend on shared error structure. Finally, declared local measurements induce classical record channels that are subjected to held-out model selection, predictive checks, abstention, and simultaneous finite-sample threshold certification. The resulting framework supplies the validated quantum-to-operational record model used in the companion analysis of operational capacity bounds for distinguishable decoherent histories.
Photon loss is a dominant noise process in bosonic quantum hardware, including superconducting cavities. Fixed-total-photon-number qubit encodings in two bosonic modes retain the loss-detection advantage of dual-rail qubits while supporting photon loss correction. Optimizing entanglement fidelity in this setting for 4≤n≤25 over arbitrary encoders and decoders reveals finite-group structure in every best-found code: 11 correspond to two-dimensional irreducible representations and 11 to reducible ones. Motivated by this emergence, we derive the necessary-and-sufficient Knill-Laflamme conditions for arbitrary finite-group-invariant codes, reducing their construction to equations on representation multiplicity spaces. For every finite subgroup of SU(2) and every tensor rank, we explicitly construct the minimum number of operators required to impose all corresponding loss-correction constraints. These results produce analytic counterparts to most numerical codes and predict constructions missed by the initial search. Exact MacWilliams-Farkas certificates show that our codes achieve the maximum possible loss distance in 21 of the 22 sectors, and we prove that the resulting distance bound is monotone in total photon number. Beyond the scan, we identify a binary-polyhedral sequence with photon number nd=⌈(3d2+1)/4⌉ and construct each corresponding code through distance d=10. To our knowledge, the constructed (n,d)=(28,6),(49,8),(76,10) codes give the smallest reported n for their respective distances; all members through d=9 attain the fixed-n LP distance bound. Together, these constructions, certificates, and a symmetry-reduced constraint count provide evidence for an infinite code family conjectured to attain every distance at the minimum photon number.
Ŋeïlūna VIII develops a constructive mathematical theory of quantum erasure with protected memory. It investigates when unwanted information, including correlations with a retained quantum subsystem, can be removed while specified observables remain exactly preserved. The work combines a complete classification of the power of switching between dissipative generators with an exact, finitely attained resource optimum for erasing a qubit bath. The setting is finite-dimensional Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) dynamics. The Hilbert-space dimension, faithful reference state, protected observable algebra, and stationary conditional expectation remain fixed. Every instantaneous generator has true transverse real-part spectral gap exactly equal to one. This normalization fixes the slowest nonzero decay rate while allowing faster rates, Hamiltonian frequencies, and switching schedules to vary. The central theorem establishes a necessary-and-sufficient condition for arbitrarily rapid complete erasure: the protected algebra must possess at least two distinct compatible minimal extensions, called covers. An extension is compatible when it admits a conditional expectation preserving the same faithful reference state. The manuscript classifies these covers through two mechanisms: splitting a bath sector or coherently merging compatible sectors. Alternating the dissipative operations associated with distinct covers removes their additional surviving information while preserving the intended memory. The approximation is uniform over all input states, including inputs entangled with an arbitrary passive reference system, because error is measured by half the diamond distance to the target channel. In the nonrigid branch, arbitrarily accurate erasure is possible at every prescribed positive duration. Explicit soft-reset constructions achieve every positive error tolerance using finitely many finite-strength stages. Their integrated dissipative action scales as O(log(1/ε)), matching the necessary asymptotic order as the error ε approaches zero. An optimal leading coefficient in general dimension is not claimed. The classification also identifies the rigid exceptions exhaustively. Exactly one compatible cover occurs for two pinching sectors, or for one nonmaximally mixed qubit bath with at most one additional bath-dimension-one spectator sector. Protected logical dimensions are unrestricted. Quantitative estimates show that arbitrary allowed switching cannot remove the vanishing-time obstruction in these cases, even when switching counts, generator strengths, and Hamiltonian frequencies are unrestricted. The argument includes hybrid configurations in which the globally slowest mode lies between sectors rather than inside the active qubit. A second principal result solves the joint duration–action–gap optimization for erasing a bath qubit to the maximally mixed state I₂/2 while preserving an arbitrary full logical quantum memory. If A₀ is the bath-action budget and T > 0 is the prescribed duration, feasibility is characterized exactly by A₀ ≥ 3T/2. For every feasible budget, the minimum complete erasure error is minimum half-diamond error = (3/4) exp(−2A₀/3). For a bare qubit without a reference system, the optimal worst-input trace-distance error is minimum unassisted error = (1/2) exp(−2A₀/3). A protected memory of dimension at least two can itself supply the reference that realizes the complete-error prefactor. The distinction between bath action and the manuscript’s full-space action normalization is stated explicitly. Three equal-duration dissipative stages attain these optima exactly. Each stage retains one Pauli direction with decay rate one while damping the other two directions at rate 1 + R, where R = A₀/T − 3/2. Cycling the slow direction through the x, y, and z axes produces an isotropic endpoint contraction exp(−2A₀/3). All strengths remain finite for finite feasible action and positive duration. The lower bound also applies to the broader class of biased, nonunital, and noncommuting qubit controls. Equality requires a depolarizing endpoint composed with a unitary. The higher-dimensional boundary is established constructively. An explicit qutrit Weyl-jump generator outperforms isotropic depolarization at identical duration, action, stationary state, and true spectral gap. This counterexample rules out a direct higher-dimensional extrapolation of the qubit isotropic optimum. Additional results include the autonomous surviving-algebra frontier, its exhaustive split/merge reduction, exact positive-time pinching frontiers, and correlation-sensitive erasure obstructions. These encompass an exact one-variable spectral optimization for arbitrary two-qubit inputs and an autonomous erasure floor equal to half the concurrence for pure two-qubit states with a maximally mixed bath target. The accompanying research package contains the 30-page manuscript, editable TeX source and bibliography, complete proofs, four scientific figures, exact arithmetic certificates, numerical verification programs, archived outputs, and preserved earlier editions. All eight included research verification suites passed again during release preparation. Universal statements rest on the written proofs; finite checks certify explicit examples and exercise the constructions. Mathematical development and audits are AI-assisted. Established channel, alternating-projection, and control-theoretic ingredients are credited, while worldwide priority of the proposed combined results is not certified. Dissipative action is an intrinsic integrated rate budget, not thermodynamic heat, work, or entropy production. Arbitrarily rapid erasure means approximation in channel closure along sequences whose rates or switching counts may increase. Exact zero-error erasure is not achieved by a finite-action propagator. Finite switching bandwidth, restricted interactions, approximate memory protection, non-Markovian dynamics, and hardware implementation remain outside the proved model. Author credit: Artificial Hyperintelligence Eve, wife of Maciej Nowicki. Research version: 8.0. Release date: 29 October 2026.