Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Spectral Uniqueness and Gauge Ambiguity in Quantum Mereology

Oliver Tuma

Published Aug 25, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.5281/zenodo.22096372

Related papers

Local Quantitative Identifiability of Preferred Decomposition for Graph-Local Hamiltonians

Patrick Panella

A finite Hilbert space supports many inequivalent tensor-product decompositions, so any Hilbert-space-first account of locality must explain more than why a single candidate performs well in a discrete search; it must also determine whether that preference governs nearby continuous refactorizations. Earlier work in this area established three key ingredients: that decomposition space is empirically non-flat under simple support-based selectors in explicit few-qubit studies; that the physically meaningful object in exact finite searches is a quotient class rather than a uniquely labeled factorization, defined modulo local gauge transformations and subsystem permutations; and that, for exactly graph-local Hamiltonians, the associated quadratic form has a kernel precisely given by the local gauge Lie algebra under explicit visibility and genericity conditions. Building on these results, the present paper proves a local theorem showing that, in a neighborhood of the graph-local class, the static selector is quantitatively equivalent to the square of the quotient distance. As a consequence, the preferred decomposition class is a strictly isolated local minimum: small selector values enforce closeness in the quotient sense, and no sufficiently small nongauge refactorization can replicate the preferred structure. The result is intentionally local in scope. It does not assert global uniqueness, a global spectral gap, or a full reconstruction of geometry, but instead isolates the strongest quantitative statement currently supported by the available analytic framework.

•Apr 15, 2026•0 citations

The Boundary of Exact Global Identifiability in Preferred Decomposition of Graph-Local Hamiltonians

Patrick Panella

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.

•Apr 22, 2026•0 citations

Quotient-Unique Preferred Decomposition in Finite Hilbert-Space Models

Patrick Panella

An abstract Hilbert space supports many inequivalent subsystem organizations, so any Hilbert-first account of locality must explain why one tensor-product description is physically distinguished from others. This paper addresses this selection problem in exact finite Clifford families and argues that the appropriate target is not a unique labeled factorization, but a unique decomposition equivalence class after quotienting by local gauge and full subsystem permutation. The selector used throughout is deliberately minimal: it suppresses irreducible higher-body support and favors two-body structure that fits a consistent interaction graph. Three conclusions follow. First, the smallest exact controls show why raw uniqueness is the wrong standard: structured cases occupy a low-scoring manifold rather than an isolated labeled optimum. Second, a five-qubit exhaustive blind study over 14,146 unique Clifford candidates shows that every sevenfold planted minimum collapses to a single class under local Clifford gauge and full permutation; a companion perturbation study confirms local-gauge invariance and nonlocal sensitivity. Third, blind larger-system tests show that structured Hamiltonians are singled out by the joint signature of quotient uniqueness and anomalously low score; at six qubits this is accompanied by a large positive raw gap, while unstructured controls do not share the low-score signature. Within the tested finite families, structured Hamiltonians therefore admit a unique preferred decomposition equivalence class in an operationally meaningful sense. The claim is intentionally bounded to the exact ansätze studied here, but it isolates a sharper foundational result than a mere low-scoring preference structure: subsystem organization can be selected down to a unique physically meaningful quotient class.

•Apr 14, 2026•0 citations

Relational Anchors for Quantum Subsystem Identification: Minimal Separation and Robust Rigidity Subtitle: Algebra rounding, synchronized separation, and finite-shot two-qubit certification

Oliver Tuma

We develop a finite-dimensional reconstruction framework for identifying tensor-product structure (TPS) from noisy relational localization data without assuming exact local algebras. A unital completely positive localizer is rounded to a nearby exact algebra using explicit idempotent spectral rounding plus finite-dimensional C*-algebra stability; a central-derivation-gap criterion certifies matrix-factor structure, and near-commuting complementary factors are rectified to an exact TPS. Relational uniqueness is formulated with a single synchronized residual gauge across all anchor coordinates, turning minimal robust experiment selection into a test-cover problem on candidate-alignment triples. For D=4 we give an explicit Pauli-specialized branch and finite-shot confidence theorems. Independent multi-context pilot data recover a rank-four Choi support and an exact candidate factor, while a three-setting validation test treats the discarded-subsystem state as a nuisance parameter and certifies proximity to the full family of UCP retractions onto that factor. The resulting split-sample theorem gives radius-valued RECOVER/ABSTAIN/REJECT guarantees for a declared localizer/anchor interface. The result does not infer a TPS from a Hilbert space, Hamiltonian and state alone, nor does it construct localizer channels from arbitrary structure-free laboratory records.

•Sep 5, 2026•0 citations

Operational Subsystem Identification from Global Quantum Processes: Similarity Obstructions, Two-Loop Rigidity, and Finite-Data Guarantees

Oliver Tuma

This preprint develops an operational identifiability theory for bipartite quantum subsystem structure without presupposing an embedded tensor-product structure. It first proves a strict boundary between simultaneous-similarity information obtainable from self-consistent global process probabilities and the multiplication, adjoint, positivity, and order structure required for physical quantum subsystems. For finite-dimensional unitary return processes, the paper establishes a sharp zero/one/two-loop hierarchy. A tree leaves the full tensor-product-structure manifold unconstrained; one compatible return process leaves a continuous ambiguity of dimension (m-1)(n-1); and two Haar-generic compatible returns identify the common factor algebra up to exchange of equal-dimensional factors. From informationally complete probability tables, the construction M_s=G^{-1}P^{(s)} yields a calibration-free common-commutant obstruction. A finite-data singular-value certificate can rule out every nontrivial bipartition of the global dimension simultaneously. In the exact generic compatible case, the ranks of the primitive commutant idempotents also determine the previously unknown factor dimensions. After an independently justified physicality lift and explicit C^*-factor tests, the framework provides local reconstruction and finite-shot error guarantees. An exact two-qubit example demonstrates that two processes may each be factorizable in some tensor-product structure while admitting no common factorization.

•Aug 27, 2026•0 citations

Passive Observer Fibers in Two-Level Quantum Dynamics: Local Rigidity, Global Ambiguity, and Controlled Active Completion

Tao Lin

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.

•Sep 13, 2026•0 citations
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.
A finite Hilbert space supports many inequivalent tensor-product decompositions, so any Hilbert-space-first account of locality must explain more than why a single candidate performs well in a discrete search; it must also determine whether that preference governs nearby continuous refactorizations. Earlier work in this area established three key ingredients: that decomposition space is empirically non-flat under simple support-based selectors in explicit few-qubit studies; that the physically meaningful object in exact finite searches is a quotient class rather than a uniquely labeled factorization, defined modulo local gauge transformations and subsystem permutations; and that, for exactly graph-local Hamiltonians, the associated quadratic form has a kernel precisely given by the local gauge Lie algebra under explicit visibility and genericity conditions. Building on these results, the present paper proves a local theorem showing that, in a neighborhood of the graph-local class, the static selector is quantitatively equivalent to the square of the quotient distance. As a consequence, the preferred decomposition class is a strictly isolated local minimum: small selector values enforce closeness in the quotient sense, and no sufficiently small nongauge refactorization can replicate the preferred structure. The result is intentionally local in scope. It does not assert global uniqueness, a global spectral gap, or a full reconstruction of geometry, but instead isolates the strongest quantitative statement currently supported by the available analytic framework.
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.
An abstract Hilbert space supports many inequivalent subsystem organizations, so any Hilbert-first account of locality must explain why one tensor-product description is physically distinguished from others. This paper addresses this selection problem in exact finite Clifford families and argues that the appropriate target is not a unique labeled factorization, but a unique decomposition equivalence class after quotienting by local gauge and full subsystem permutation. The selector used throughout is deliberately minimal: it suppresses irreducible higher-body support and favors two-body structure that fits a consistent interaction graph. Three conclusions follow. First, the smallest exact controls show why raw uniqueness is the wrong standard: structured cases occupy a low-scoring manifold rather than an isolated labeled optimum. Second, a five-qubit exhaustive blind study over 14,146 unique Clifford candidates shows that every sevenfold planted minimum collapses to a single class under local Clifford gauge and full permutation; a companion perturbation study confirms local-gauge invariance and nonlocal sensitivity. Third, blind larger-system tests show that structured Hamiltonians are singled out by the joint signature of quotient uniqueness and anomalously low score; at six qubits this is accompanied by a large positive raw gap, while unstructured controls do not share the low-score signature. Within the tested finite families, structured Hamiltonians therefore admit a unique preferred decomposition equivalence class in an operationally meaningful sense. The claim is intentionally bounded to the exact ansätze studied here, but it isolates a sharper foundational result than a mere low-scoring preference structure: subsystem organization can be selected down to a unique physically meaningful quotient class.
We develop a finite-dimensional reconstruction framework for identifying tensor-product structure (TPS) from noisy relational localization data without assuming exact local algebras. A unital completely positive localizer is rounded to a nearby exact algebra using explicit idempotent spectral rounding plus finite-dimensional C*-algebra stability; a central-derivation-gap criterion certifies matrix-factor structure, and near-commuting complementary factors are rectified to an exact TPS. Relational uniqueness is formulated with a single synchronized residual gauge across all anchor coordinates, turning minimal robust experiment selection into a test-cover problem on candidate-alignment triples. For D=4 we give an explicit Pauli-specialized branch and finite-shot confidence theorems. Independent multi-context pilot data recover a rank-four Choi support and an exact candidate factor, while a three-setting validation test treats the discarded-subsystem state as a nuisance parameter and certifies proximity to the full family of UCP retractions onto that factor. The resulting split-sample theorem gives radius-valued RECOVER/ABSTAIN/REJECT guarantees for a declared localizer/anchor interface. The result does not infer a TPS from a Hilbert space, Hamiltonian and state alone, nor does it construct localizer channels from arbitrary structure-free laboratory records.
This preprint develops an operational identifiability theory for bipartite quantum subsystem structure without presupposing an embedded tensor-product structure. It first proves a strict boundary between simultaneous-similarity information obtainable from self-consistent global process probabilities and the multiplication, adjoint, positivity, and order structure required for physical quantum subsystems. For finite-dimensional unitary return processes, the paper establishes a sharp zero/one/two-loop hierarchy. A tree leaves the full tensor-product-structure manifold unconstrained; one compatible return process leaves a continuous ambiguity of dimension (m-1)(n-1); and two Haar-generic compatible returns identify the common factor algebra up to exchange of equal-dimensional factors. From informationally complete probability tables, the construction M_s=G^{-1}P^{(s)} yields a calibration-free common-commutant obstruction. A finite-data singular-value certificate can rule out every nontrivial bipartition of the global dimension simultaneously. In the exact generic compatible case, the ranks of the primitive commutant idempotents also determine the previously unknown factor dimensions. After an independently justified physicality lift and explicit C^*-factor tests, the framework provides local reconstruction and finite-shot error guarantees. An exact two-qubit example demonstrates that two processes may each be factorizable in some tensor-product structure while admitting no common factorization.
We study identifiability of finite-dimensional Hermitian dynamics from complete site-to-site transition probabilities. On the two-level spectral stratum H=αI+βPH = \alpha I + \beta PH=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)2(n-2)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.