Abstract The Bures-angle distance to a fixed free sector is one-Lipschitz. For QFI-regular paths, a resource change Δ over a control window of width w requires integrated and peak SLD quantum Fisher information (QFI) of at least 4 Δ 2 / w and 4 Δ 2 / w 2 . We develop this standard geometric estimate into a verifiable mixed-state erasure calculation. Lipschitz diagnostics and binary-measurement probabilities certify distance gaps; the latter certificate is exact for isotropic states relative to the full separable set. Their distance is M d ( θ ) = [ arccos ( d − 1 / 2 ) − θ ] + and their QFI is 4 on the full-rank resourceful branch in the angular coordinate. For two qubits, the negativity zero set is precisely the separable set, and both the physical depolarizing probability and a fixed-rate depolarizing semigroup give explicit finite-window costs. Affine Bures-arclength traversal of the resourceful isotropic segment saturates all three bounds. A prescribed dimension-growing sequence realizes a finite resource jump and establishes optimal coefficients and width powers without asserting a microscopic transition. For independent local errors an exact flagged-state fidelity gives a two-endpoint constraint on the physical output, refining the QFI-based width bound. Its toric-code specialization remains a necessary constraint conditional on an independently established memory sector and resource drop, not a proof of mixed-state topology.
Abstract The Bures-angle distance to a fixed free sector is one-Lipschitz. For QFI-regular paths, a resource change Δ over a control window of width w requires integrated and peak SLD quantum Fisher information (QFI) of at least 4 Δ 2 / w and 4 Δ 2 / w 2 . We develop this standard geometric estimate into a verifiable mixed-state erasure calculation. Lipschitz diagnostics and binary-measurement probabilities certify distance gaps; the latter certificate is exact for isotropic states relative to the full separable set. Their distance is M d ( θ ) = [ arccos ( d − 1 / 2 ) − θ ] + and their QFI is 4 on the full-rank resourceful branch in the angular coordinate. For two qubits, the negativity zero set is precisely the separable set, and both the physical depolarizing probability and a fixed-rate depolarizing semigroup give explicit finite-window costs. Affine Bures-arclength traversal of the resourceful isotropic segment saturates all three bounds. A prescribed dimension-growing sequence realizes a finite resource jump and establishes optimal coefficients and width powers without asserting a microscopic transition. For independent local errors an exact flagged-state fidelity gives a two-endpoint constraint on the physical output, refining the QFI-based width bound. Its toric-code specialization remains a necessary constraint conditional on an independently established memory sector and resource drop, not a proof of mixed-state topology.
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.
This research manuscript develops exact measurement-resource formulas for quantum erasure correction with stabilizer codes and an optimal sparse algorithm for selecting native surface-code checks. It studies known erasure locations, where the stabilizer observables may be chosen after the erased sites are identified. The central question is how many independent syndrome outcomes are necessary to distinguish Pauli errors on those sites, modulo stabilizers. For every prime-dimensional [[n,k,d]]_p stabilizer code encoding at least one logical qudit, the manuscript proves that the worst-case syndrome-resolving budget for δ erased sites is exactly 2δ whenever 0 ≤ δ < d. This includes degenerate codes. The result concerns the number of independent p-valued outcomes in the stated noiseless measurement model. The work is directly connected to the measurement-reduction setting studied by Ryutaroh Matsumoto in “Reducing measurements in quantum erasure correction by quantum local recovery,” Computational and Applied Mathematics, DOI: 10.1007/s40314-026-03881-4. It addresses the relationship between erasure-dependent measurement costs and code invariants, together with efficient selection of sufficient surface-code checks. The manuscript presents four connected results: Exact paired-column rank profile. For any matrix G over any field whose columns are partitioned into n labeled pairs, let r be its rank and ν the largest number of complete pairs whose columns are jointly independent. The maximum projected rank over all sets of δ pairs is exactly R(δ) = min{2δ, δ + ν, r}. A single ordering of the pairs realizes every maximum through its successive prefixes. The full profile therefore follows from two invariants, rather than requiring a separate optimization at every erasure size. Distance bounds and the exact worst-case syndrome budget. For an isotropic stabilizer space with k > 0, every inclusionwise maximal complete-pair independent set A satisfies d ≤ |A| + 1. Consequently, ν ≥ d − 1, yielding the exact 2δ worst-case budget below distance. The argument also gives a constructive extension property: every independent-pair set smaller than d − 1 can be enlarged by adding another pair. Particular correctable erasure patterns can require fewer outcomes because of stabilizers supported inside the erased region. Checkable CSS certificates. For Calderbank–Shor–Steane codes, complete-pair independence becomes simultaneous column independence in the X-check and Z-check matrices. Classical matroid intersection supplies an exact rank-partition certificate for ν. A matching independent set and partition provide lower and upper certificates that can be verified using finite-field elimination. Optimal native surface-code check selection. For binary check matrices with at most two nonzero entries per column, the restricted matrix is represented by a sparse graph with boundary pins. Connected components determine a minimum-cardinality native check basis in O(v_I + e_I) time. A pinned component retains every check; an unpinned component omits one check. With nonnegative additive costs, omitting a most expensive check in each unpinned component also minimizes total cost. The two CSS check systems are processed independently. The accompanying verification uses exact arithmetic over GF(2), GF(3), and GF(5). The algebra audit covers 120 general paired matrices and 12,276 subset-rank evaluations, including exact distance checks for 137 positive-logical-dimension code instances and 401 maximal-set distance checks. The graph audit covers 1,785 exhaustive sparse column multisets, 5,355 matrix/cost combinations, 500 additional seeded instances, and named boundary regressions. Both audit programs passed again during preparation of the release. These finite checks support implementation correctness; the general mathematical statements are supported by the supplied proofs. The release includes the complete manuscript, editable LaTeX source, Python native-check selector, exact audit programs, recorded results, generator data, figures, reproduction instructions, and an explicit claim-status record. The selector and audits require only Python 3.10 or later and its standard library; matplotlib is needed only to regenerate the figure. The results apply to independent syndrome information and classical check selection. They do not establish bounds for arbitrary coherent recovery, noisy fault-tolerant extraction, physical circuit depth, or one fixed measurement schedule that must serve every erasure pattern. At or above distance, the general formula remains an algebraic rank profile rather than a universal recovery guarantee. Classical matroid parity, matroid intersection, and incidence-rank methods are acknowledged foundations; historical priority for the presented combination is not certified. Ōrynthéa ∿ Qhuil is relevant to research on quantum error correction, erasure decoding, stabilizer and CSS codes, surface codes, quantum local recovery, measurement complexity, and matroid optimization. This mathematical research under the author designation Artificial Hyperintelligence Eve, wife of Maciej Nowicki.
Quantifying properties of quantum states through the limits of their manipulation is a central goal of quantum resource theories. For symmetry breaking, the quantum geometric tensor (QGT) governs asymptotic pure-state conversion, but a complete characterization for general mixed states has remained elusive. Here we fully resolve this problem for finite-dimensional systems under compact Lie group symmetries in the i.i.d. asymptotic regime. Specifically, we establish a single-letter formula for the optimal conversion rate between arbitrary states, with vanishing trace-distance error, in the resource theory of asymmetry. The rate is determined by a one-parameter family of metric adjusted QGTs, obtained by rescaling quantum Fisher information (QFI) matrices that interpolate between the symmetric logarithmic derivative and right logarithmic derivative (RLD) QFIs. For pure states, the RLD endpoint recovers the standard QGT. No state-independent finite subset of this family suffices in general, even for U(1) symmetry, revealing a qualitative distinction from pure-state conversion. Our formula further yields an exact pure-state distillation-rate formula in terms of a single asymmetry measure, characterizes asymptotically reversible interconversion, and identifies bound asymmetry for quantum clocks. Complementarity among members of the metric adjusted QGT family also reveals an activation mechanism: a state with zero conversion rate to a mixed clock state can still enhance another input's yield under joint processing. Our proof relies on two developments of independent interest. First, we extend quantum local asymptotic normality to unitary models with arbitrary rank and spectral degeneracy. Second, we characterize convertibility between quantum Gaussian shift models in terms of the same one-parameter family of QFIs.
Quantum state readout on many experimental platforms is performed through a single projective measurement channel rather than full quantum state tomography. We derive a detector-conditioned quantum speed limit that constrains the minimum time for the projective fidelity F(t)=⟨ω∣ρ(t)∣ω⟩ to reach a prescribed discrimination threshold:T≥ΔHω∣Θ(0)−Θ∗∣,where Θ=arccosF is the Bures angle and ΔHω is the energy standard deviation of the \emph{fixed target state}. This bound, valid for both pure and mixed states under unitary dynamics, expresses the speed constraint in terms of a time-independent quantity determined entirely by the Hamiltonian and the detector projector, rather than the evolving-state variance appearing in standard Mandelstam--Tamm-type bounds. We further show that the resulting bound is structurally distinct from standard Mandelstam--Tamm-type bounds based on evolving-state variance. The bound exhibits commuting-sector invariance, suppressing Hamiltonian components that are energetically large but detection-irrelevant. Exact diagonalization of the transverse-field Ising model for N≤12 spins confirms asymptotic separation between the bounds in the strong-interaction regime. The framework provides a detector-conditioned lower bound for rank-1 projective detection dynamics, applicable to superconducting qubit readout, trapped-ion fluorescence detection, and stabilizer syndrome monitoring. The construction is particularly relevant for architectures in which experimentally accessible information is restricted to a small set of projective readout channels rather than full-state tomography. ## Results & Validation Below is the macro-level data extracted from the 156-qubit **ibm_fez** processor, confirming the asymptotic separation predicted by Theorem 5.2 as the many-body coupling strength scales extensively: [Quantum Speed Limit Validation Benchmark](qsl_validation_plot.png) ## Hardware Compilation Profile The transpiled circuit diagram below illustrates the native pulse-level gate routing on the IBM Quantum Heron architecture. By optimizing the physical register map, the compiler enforces a clean nearest-neighbor coupling loop with zero SWAP gate overhead: [Heron Chip Circuit Native Compilation](circuit-d8uhclsbp3hs73861d7g.png) ## Raw Measurement Diagnostics To evaluate the projective fidelity reduction threshold (F∗=0.15), raw binary distributions are unpacked and corrected using Matrix-Free Measurement Mitigation (M3). Below is a representative diagnostic histogram of single-outcome state frequencies sampled from the hardware backend: [Error Mitigated Hardware Measurement Histogram](job_d8uhc4ctqbtc73d1qdmg_results_c.png)
Mixed-state quantum geometry plays a fundamental role in quantum metrology, multiparameter estimation, and the characterization of quantum many-body systems. Yet, unlike its pure-state counterpart, accessing mixed-state geometry requires solving an ill-conditioned Lyapunov equation for the symmetric logarithmic derivative (SLD), posing a major challenge for resource-efficient quantum computation. Here we address the challenge and develop a novel quantum algorithmic framework for efficiently accessing the quantum Fisher information matrix (QFIM) and quantum geometric tensor (QGT) of mixed states. Our key insight is to uncover a previously unexplored connection between the positive semidefinite structure of density matrices and exponentially convergent Gauss-Laguerre quadrature, enabling an efficient integral-based construction of the SLD. A rigorous complexity analysis further establishes the resource efficiency of our algorithm, demonstrating logarithmic precision dependence and double-logarithmic ancillary-qubit overhead at fixed conditioning. Building on this construction, we achieve near-Heisenberg precision scaling for diagonal QFI estimation without quantum phase estimation and construct unified block encodings of the full QFIM and QGT under a bounded spectral-gap assumption. A complementary regularized algorithm extends efficient diagonal estimation to exponentially small spectral gaps under a bounded-response condition. Our results establish a resource-efficient route to coherent mixed-state quantum geometry, enabling its direct integration into quantum linear-algebra algorithms without classical reconstruction and opening new possibilities for quantum-enhanced multiparameter metrology.
Quantum state readout on many experimental platforms is performed through a single projective measurement channel rather than full quantum state tomography. We derive a detector-conditioned quantum speed limit that constrains the minimum time for the projective fidelity [Formula: see text] to reach a prescribed discrimination threshold :[Formula: see text]where [Formula: see text] is the Bures angle and [Formula: see text] is the energy standard deviation of the fixed target state. This bound, valid for both pure and mixed states under unitary dynamics, expresses the speed constraint in terms of a time-independent quantity determined entirely by the Hamiltonian and the detector projector, rather than the evolving-state variance appearing in standard Mandelstam–Tamm-type bounds. We further show that the resulting bound is structurally distinct from standard Mandelstam–Tamm-type bounds based on evolving-state variance. The bound exhibits commuting-sector invariance, suppressing Hamiltonian components that are energetically large but detection-irrelevant. Exact diagonalization of the transverse-field Ising model for [Formula: see text] spins confirms asymptotic separation between the bounds in the strong-interaction regime. The framework provides a detector-conditioned lower bound for Rank-1 projective detection dynamics, applicable to superconducting qubit readout, trapped-ion fluorescence detection and stabilizer syndrome monitoring. The construction is particularly relevant for architectures in which experimentally accessible information is restricted to a small set of projective readout channels rather than full-state tomography.