Description:A reconstruction theory for complex quantum coherence encoded on graphs using controlled quantum Fisher information. Finite-order data determine edge magnitudes and cycle holonomies up to an explicit integral-lattice ambiguity, revealing continuous and torsion obstructions to identifiability. With physical labelled Schur controls, exact finite-value measurements reconstruct the full coherence matrix up to vertex gauge using at most |E| + 2β scalar QFI values, with explicit injectivity and stability bounds. .
Description:A reconstruction theory for complex quantum coherence encoded on graphs using controlled quantum Fisher information. Finite-order data determine edge magnitudes and cycle holonomies up to an explicit integral-lattice ambiguity, revealing continuous and torsion obstructions to identifiability. With physical labelled Schur controls, exact finite-value measurements reconstruct the full coherence matrix up to vertex gauge using at most |E| + 2β scalar QFI values, with explicit injectivity and stability bounds. .
When does operational distinguishability contain sufficient information to identify physical structure? We address this question through a resource-relative framework of Operational Informational Distinguishability (OID), designed to investigate the mathematical consistency of identifiability across quantum states, information geometry, and physical processes. An operational observer is characterized by (\mathcal{D}=(\mathcal{A},\mathcal{M},\mathcal{R},\delta)), whose labelled outcome distributions define an operational signature and induce equivalence classes of physically indistinguishable realizations. We establish an exact reconstruction criterion: a proposed geometric assignment factors uniquely through the operational signature if and only if it is constant on these equivalence classes. This provides a necessary-and-sufficient condition for operational reconstruction, while separating identifiability from the additional requirements of local non-degeneracy and stability. For faithful finite-dimensional quantum-state manifolds, we establish the measurement–information-geometric hierarchy[g^{(\mathcal{M})}\preceq g^{\mathrm{SLD}}\preceq g^{\mathrm{BKM}}relating measurement-restricted Fisher information, the symmetric logarithmic derivative metric, and the Bogoliubov–Kubo–Mori metric. We clarify the directional attainability of the SLD bound and its distinction from simultaneous multiparameter attainability. As an explicit algebraic result, we characterize the operator span generated by diagonal phase modulation of cyclic-shift eigenstate projectors in the one-excitation sector: it is precisely the constant-diagonal Hermitian subspace, of dimension (N^{2}-N+1), leaving (N-1) independent population directions unresolved. Adjoining site-basis projectors restores informational completeness. This idealized construction is connected to experimentally realized entangled measurements of (W) states. We further extend operational distinguishability to quantum processes through tester-restricted signatures and divergences, explicitly separating signalling from intervention-defined causal influence. Finally, we introduce accumulated information-geometric path length as a candidate operational clock parameter, without identifying it with physical time. By relating these constructions to holographic information-distance bounds, bulk entanglement, conditional mutual information and recovery, and recent approaches to emergent geometry and informational time, we formulate a central testable question: can a common resource-relative distinguishability structure remain identifiable, stable, and compositionally consistent across states, processes, geometry, temporal ordering, and causality? The framework makes both constructive reconstruction and mathematically explicit obstructions possible, providing a falsifiable methodological program rather than presupposing the emergence or unification of physical structure.
Quantum reservoir computing (QRC) uses fixed quantum dynamics to encode a time series and trains only a classical readout, but a benchmark capacity alone does not reveal whether task information is lost in the reservoir, the measurement, feature compression, or finite sampling. We introduce \emph{task-resolved Fisher spectroscopy}, in which prediction targets define orthonormal score coordinates on the stationary distribution of input histories. Reweighting labeled histories along these scores generates an exactly affine family of reservoir states and measurement outcomes. In the same coordinates, we obtain a many-body Fisher-information hierarchy relating the state quantum Fisher information, the Fisher information of the complete measurement record, and moment matrices retained through many-body order r. The quadratic form of each moment matrix is exactly the stationary capacity of the optimal linear readout, while a finite-measurement extension predicts the approach from one-shot records to ideal expectation-value features. The measurement-level quantities require only stationary labeled records and measured outcome strings, not an input model or quantum-state tomography. In a five-spin open reservoir, interactions route fourth-order temporal information into higher-body correlations, so low-order compression can incur orders-of-magnitude sampling overhead even when a conventional parity target remains accessible. For correlated inputs and outputs, record-defined task scores predict held-out capacities, the required feature order, and measurement-budget dependence; optimizing the local measurement axis recovers otherwise hidden task information. The framework connects benchmark performance to experimentally actionable diagnoses of reservoir encoding, measurement choice, classical representation, and shot allocation.
We present a real-time path-integral formulation of the quantum Fisher information for dynamical parameter estimation. For pure states undergoing unitary evolution, we show that the quantum Fisher information can be expressed as a connected symmetrized covariance of a time-integrated action deformation, equivalently as an integrated insertion of ∂λS in the propagator. This reformulation avoids explicit state reconstruction by rewriting the quantum Fisher information in terms of real-time correlators that are natural targets for many-body methods. We further embed the construction into the Schwinger-Keldysh closed-time-path formalism, identifying the quantum Fisher information with the Keldysh component of an appropriate contour-ordered correlator generated by forward and backward propagating sources. Finally, using the Van Vleck-Gutzwiller approximation we re-derive the compact semiclassical quantum Fisher information expression, clarifying how classical trajectory data control leading-order metrological sensitivity.
The coherent ergotropy of a quantum system—the work extractable by unitary control that is stored in energy-basis coherences—is usually treated as a purely thermodynamic quantity. This preprint establishes its exact geometric content for a two-level system and precisely delineates the limits of that connection. First, it proves the exact identity C = ε(√(z² + F_Q) − |z|), where z is the population imbalance and F_Q is the quantum Fisher information conjugate to the level phase, equivalently the squared Ramsey visibility. Coherent work is therefore a joint function of the population imbalance and the phase QFI, rather than a function of the QFI alone. Second, an exact two-level bound C ≤ ½√(F_Q^H) is established for the Hamiltonian-generated quantum Fisher information, with saturation on the Bloch equator. An explicit qutrit counterexample, extended by embedding, demonstrates that this bound fails for every dimension d ≥ 3. The work further corrects the target state entering the quantum speed-limit formulation of work extraction, showing that unitary extraction must drive the system toward its passive state rather than its dephased state. Finally, the exact identity predicts an anisotropic coherence horizon under pure dephasing: the 1/e lifetime of coherent work changes continuously from T2 on the Bloch equator to T2/2 near the poles. Coherent ergotropy can consequently be reconstructed experimentally from three calibrated quantities—the population imbalance, Ramsey visibility, and level splitting—without a work-extraction protocol, full state tomography, or an ancilla.
We study an inverse problem for quantum cellular automata (QCA) and strictly locality-preserving discrete quantum dynamics: what can an observer infer about a finite local update from physical data, and which inferences remain impossible even with exact observations? We separate general interacting locality-preserving automorphisms from homogeneous finite-dimensional linear excitation sectors. In the latter, finite range implies a matrix Laurent-polynomial update and algebraic quasienergy branches. We derive a reciprocal-lattice monodromy obstruction: if repeated Brillouin windings generate infinitely many distinct analytic germs of a target quasienergy phase, the target cannot be an exact band of any finite-Laurent update; moreover, M distinct continuations require internal block dimension q ≥ M. Massive relativistic dispersion violates this criterion, and generic transverse slicing extends the obstruction to isotropic massive and massless dispersion in d ≥ 2. Independently, positive minimum site separation, exact finite-speed causality, and arbitrarily divisible microscopic time imply ultralocal dynamics. We then formulate the inverse direction. In an independently specified operational locality basis, finite full-matrix Fourier support plus pointwise unitarity is constructively equivalent to a homogeneous finite-range update, whereas spectra alone are insufficient. Exact reconstruction generally fixes only a blocking/refinement equivalence class. Conditional on such reconstruction, reachability is a compact-group control problem. Counterexamples accompany each nontrivial converse, and a complexity-controlled evidence hierarchy separates weak spectral consistency from process-level reconstruction.
Holonomy-first quantum frameworks treat closed-loop transformations — holonomies — as the primitive operations from which native gates are mined, rather than as derived quantities. We develop this viewpoint in two parts. First, we construct a gauge-invariant commutator-strength matrix S(ε) from trace data of short group-commutator words and use it to drive an explicit native-operation promotion protocol. The bracket-norm interpretation of S(ε) is asymptotic with a general O(ε) remainder; for inversion-symmetric primitive families all odd-order corrections vanish, giving O(ε2) accuracy, and a counterexample shows the symmetry cannot be dropped. The promotion criteria are operationalized as declared statistical tests — sampling-cost laws for interferometric and readout-floor regimes, a bias-corrected scale-equivalence rule, and multiplicity control — and a deterministic simulation campaign reports decision replication, false-promotion control, inverse-defect detection, and execution cost under finite-shot readout, calibration error, imperfect inverses, and visibility loss. Second, for two-particle preparations of Schmidt rank two, the common-basepoint return-amplitude residue factorizes exactly as ΔPθ = C(θ)HE(χ, γ), with C(θ) the concurrence and HE = sin2(2γ) sin4(χ/2) a calibrated loop factor, determining negativity, entanglement entropy, product fidelity, and geometric entanglement within the family. A two-boson construction realizes the encoded geometric gate −σx on span{|20⟩, |02⟩}; under the same two-mode, same-dimension projected-scalar condition, no corresponding one-particle holonomic cycle exists unless the one-body Hamiltonian is scalar. The mathematical ingredients are established; the contribution is the invariant-readout protocol, its quantified cost and uncertainty analysis, and precise statements of what can and cannot be inferred without full operator reconstruction. This record contains the abstract only. It is not the full text of the article. The complete peer-reviewed article is published open access in Discover Quantum Science and is freely available from the publisher at https://doi.org/10.1007/s44464-026-00037-5 — including all figures, tables, appendices and the supplementary code and data. Cite as: Kazadi J. Holonomy-first native operations geometric quantum framework and entanglement-sensitive multi-particle holonomic gates. Discover Quantum Science. 2026;2:35. https://doi.org/10.1007/s44464-026-00037-5