We study identifiability of finite-dimensional Hermitian dynamics from complete site-to-site transition probabilities. On the two-level spectral stratum $H = \alpha I + \beta P$, passive probabilities factor through a simple projector-modulus observer. We give the exact static/visible classification of this observer and then exhibit two sharply different forms of nonidentifiability. For a rank-three harmonic projector on seven sites, one passive law has at least four pairwise diagonal-gauge-inequivalent preimages, while an exact cyclotomic tangent calculation proves local modulus rigidity at each of the four constructed branches. In contrast, a standard one-parameter qutrit SIC family gives a continuous family of gauge-inequivalent zero-diagonal Hamiltonians with identical complete passive probabilities. We formulate the fixed-diagonal tangent map and an equivalent Bargmann-weighted triangle-curl system, making the local/global contrast explicit. Finally, under a calibrated isolated edge-phase intervention model, two cubic-response settings recover a complex rooted triangle invariant and at most $2(n-2)$ settings reconstruct the full diagonal-gauge class on complete support. The results separate passive nonidentifiability, local differential rigidity, and active identifiability without requiring a generic rigidity theorem for projector-modulus fibers.
We study identifiability of finite-dimensional Hermitian dynamics from complete site-to-site transition probabilities. On the two-level spectral stratum H=αI+βP, passive probabilities factor through a simple projector-modulus observer. We give the exact static/visible classification of this observer and then exhibit two sharply different forms of nonidentifiability. For a rank-three harmonic projector on seven sites, one passive law has at least four pairwise diagonal-gauge-inequivalent preimages, while an exact cyclotomic tangent calculation proves local modulus rigidity at each of the four constructed branches. In contrast, a standard one-parameter qutrit SIC family gives a continuous family of gauge-inequivalent zero-diagonal Hamiltonians with identical complete passive probabilities. We formulate the fixed-diagonal tangent map and an equivalent Bargmann-weighted triangle-curl system, making the local/global contrast explicit. Finally, under a calibrated isolated edge-phase intervention model, two cubic-response settings recover a complex rooted triangle invariant and at most 2(n−2) settings reconstruct the full diagonal-gauge class on complete support. The results separate passive nonidentifiability, local differential rigidity, and active identifiability without requiring a generic rigidity theorem for projector-modulus fibers.
Earlier work on preferred decomposition for graph-local Hamiltonians established a layered picture rather than a single theorem: empirical structure selection under locality-sensitive selectors, identification of the correct quotient notion of uniqueness, local rigidity and local quantitative identifiability near exact realizations, stability under sufficiently small forbidden tails, and conditional global uniqueness once the exact zero set is known to be unique. The present paper shows that the unrestricted global step cannot be completed from graph locality alone. It gives an explicit connected counterexample family on star graphs: a commuting qubit Hamiltonian family preserved by an entangling star-CNOT circuit that exchanges leaf fields with edge Ising couplings while producing a genuinely inequivalent tensor-product decomposition. The same abstract Hamiltonian therefore has at least two exact admissible-support realizations, so exact zero-set identifiability fails on the physical quotient. The paper also shows why this does not contradict the earlier positive results. The counterexample lies entirely on a structurally degenerate locus excluded by the nondegeneracy hypotheses used in the local rigidity and exact reconstruction results: visible local directions are at most one-dimensional and the relevant complement algebras are abelian rather than full matrix algebras. The main foundational conclusion is sharp. Graph locality by itself does not determine a unique preferred decomposition even in the exact, static, connected setting. What remains open is a narrower problem: whether exact uniqueness holds on a suitably formulated nondegenerate locus that excludes commuting disentanglable obstruction classes of the type exhibited here.
We establish a global, worst-case non-identifiability theorem for periodically driven finite-dimensional quantum systems. On the unrestricted smooth periodic Hamiltonian class, we show that the period-averaged Fubini-Study metric component of a fixed initial state cannot, in general, be reconstructed from exact one-period propagators indexed by every starting time and external parameter. Thus, even complete starting-time-resolved endpoint data are insufficient to determine this intra-period geometric quantity. The obstruction is characterized exactly. The starting-time-indexed endpoint data determine the conjugation path of the monodromy, but not its particular unitary lift. On each fixed-monodromy slice, the observational fibres are precisely the right orbits generated by smooth parameter-dependent based loops taking values in the pointwise centralizer of the monodromy. The Fubini-Study functional is not invariant under this fibre action and therefore does not factor through the endpoint observation map. An explicit real-analytic two-level witness demonstrates the obstruction within a commuting, one-generator Hamiltonian family, so neither non-Abelian time ordering nor Floquet-logarithm ambiguity is required. A continuous family of Hamiltonians produces identical starting-time-indexed one-period endpoint data while yielding different, and on the unrestricted class arbitrarily separated, period-averaged Fubini-Study geometry. Non-identifiability persists under any prescribed uniform bound on the parameter derivative of the Hamiltonian. The result is a deterministic global statement, not a claim of generic non-identifiability or experimental impossibility. It identifies a precise information gap between complete one-period endpoint data and full intra-period dynamics: the missing information is the unitary lift of the observed monodromy path rather than ordinary scalar phase freedom.
We develop a finite operational response theory for controlled quantum systems in which observer closure, hidden microscopic structure, and higher-order response are treated within a single algebraic framework. The central distinction is between generic higher-order observability and Closure–Residue (CR)-constrained generative response. Standard finite-dimensional quantum theory does not, by itself, impose the specific CR accessibility condition introduced here; physically realizable controls are ordinarily determined by the microscopic Hamiltonian, hardware connectivity, conservation laws, and environmental couplings of the system under study. We therefore formulate First-Residue Accessibility as an additional finite selection principle for the coherent-control sector studied here: the infinitesimal deformation of the observer closure algebra generated by any admissible tangent in the declared target control manifold must lie in the first-residue derivation space inherited from an underlying microscopic CR sector. The algebraic consequences are rigid. Within this coherent-control sector, the maximal abelianness of the population algebra implies that the infinitesimal action of a control on the closure algebra determines its anti-Hermitian generator uniquely modulo a diagonal population gauge. First-Residue Accessibility therefore forces the declared coherent control class, up to this gauge, to lie in a canonical graph-internal subtype extracted from the microscopic CR sector. For diagonal prepared states, the visible first-order population response is silent, while the second variation factorizes through a graph-divergence map of a microscopic bilinear edge flow. A three-site single-excitation benchmark realizes the factorization exactly and yields an even response law of the form Sigma(epsilon X) = epsilon^2 Q_CR(X,X) + O(epsilon^4). We also separate exact discrete support transport from continuous-time quasilocality. A hard one-step support cone has a singular continuous-time limit whenever the limiting generator genuinely spreads local observables; finite propagation in the continuous limit must therefore be formulated by soft tails rather than exact support. We introduce a normalized-trace visible-leakage profile and derive a soft causal readout bound. In the coherent single-excitation sector this profile is uniformly equivalent, up to fixed observer-volume constants, to intrinsic operator-norm quasilocality. For finite-range exchange with simultaneous site dephasing we prove an N-uniform normalized-trace quasilocality estimate without using rank preservation. This genuinely nonunitary extension establishes a common finite-time exponential locality class with standard open-system quasilocality, while the stronger two-sided intrinsic equivalence is claimed only for the coherent unitary subclass. We further show that divergence-image membership is non-discriminating for connected full-population readout, construct an explicit standard CPTP control family that is first-order silent but violates the CR component constraint on a disconnected graph, and isolate experimentally measurable null tests. We distinguish a declared target-control manifold from the larger independently calibrated manifold of physically available coherent controls, thereby separating a conditional declared-manifold accessibility hypothesis from the stronger Physical First-Residue Accessibility statement. Under star-compatible oriented provenance we also derive an observable cross-prediction bound that links independently fixed static CR data to a measurable quadratic response coefficient. A framework-discriminating test requires both that the CR structure be fixed without using the target Hessian data and that the resulting constraint be stronger than what follows from the same calibrated microscopic information in a standard quantum model. We further exhibit a connected three-site collective-mode candidate in which CR Accessibility predicts an exact left–right quadratic null, while a same-support standard comparator constrained only by independently fixed aggregate tangent norms permits a nonzero response envelope. This yields a strict prediction-set separation under an explicit Dataset-A information condition, while also making clear that the separation disappears if standard microscopic calibration independently fixes the same collective tangent relation. We map this strict comparator onto a single-electron linear triple quantum dot, construct a four-point mixed-Hessian estimator, and propagate swap-odd instrumental asymmetry, finite-difference bias, and shot noise into a preregistered experimental envelope. The deeper derivation of the accessibility principle from the full Generative Map is left to the subsequent axiomatic stage of the program.
Two labelled rank-one quantum measurement contexts can have identical static transition probabilities while differing in their relative complex realization. This preprint studies reconstruction in a specified one-system access model with state preparation, context readout, and independently calibrated projector-phase controls. It gives a finite-pulse extraction of fourth-order Bargmann invariants, constructive reconstruction of the relative unitary matrix modulo row and column phases, a common-unitary completion condition, stability bounds, and a fit-free four-setting composition identity. Combined with simultaneous finite-sample confidence regions, the latter yields a conservative level-alpha rejection rule for the stated stable-control model class. Numerical evidence is synthetic and limitations, blind spots, and unequal hardware access in comparisons are reported explicitly.
An isolated group of k bands in an N-level quantum system defines a rank-k spectral projector and hence a map into the complex Grassmannian Gr(k,N). For a smooth gapped Hamiltonian, this projector is globally smooth and periodic even when band topology obstructs a globally smooth periodic, or symmetry-compatible, Bloch frame. We take the globally defined differential $\dd P$ as the central object: it is the tangent field of the Grassmannian map, removes unphysical rotations within the selected subspace, and retains the physical interband transition. The interband block of this tangent data simultaneously determines the quantum metric and Berry curvature; the associated horizontal generator produces finite Grassmannian motion, while wedge products of $\dd P$ enter topological forms. We construct a shortest path between two projectors in the ambient Grassmannian and show that the singular values of its horizontal generator block are the principal angles between the endpoint subspaces. This construction provides a piecewise-geodesic interpretation of discrete geometric phases. In a separate development, we derive a basis-independent expression for the determinant of a multiband Wilson loop from traces of powers of an ordered projector product, without decomposing a degenerate band multiplet into individual bands. Finally, the same tangent-vector calculus organizes Chern characters, chiral winding numbers, and the time-reversal Z2 index. The resulting framework unifies local quantum geometry, finite subspace distance, holonomy, and topology while avoiding global gauge fixing.
This work separates two notions of uniqueness that are often conflated in spectral approaches to quantum subsystem reconstruction. For a fixed bipartite dimension (D=mn), Hamiltonian eigenvalues are placed on an (m\times n) grid and scored by their least-squares distance from an additive table (E_{ij}=a_i+b_j). It is proved that, for Lebesgue-almost every spectrum, the optimal spectral labeling is unique modulo row and column permutations and, for square grids, factor exchange. The exact non-uniqueness locus is a semialgebraic subset of a finite union of proper quadratic hypersurfaces. A positive selection gap additionally provides an explicit perturbation bound under which the optimal labeling remains unchanged. This generic spectral uniqueness does not determine a unique embedded tensor-product structure. For a nondegenerate Hamiltonian and a fixed labeling, the residual centralizer orbit is classified exactly as[\mathbb T^{(m-1)(n-1)}.]Its complete invariants are gauge-invariant plaquette phases. The additive spectral residual and the phase ambiguity are shown to arise from the same cycle structure of the complete bipartite graph (K_{m,n}): the former is its real cycle space, while the latter is its (U(1)) character torus. Finally, continuous operational anchoring is shown to require at least ((m-1)(n-1)) real response coordinates for local identification, and strictly more for global identification.