Abstract Can pairwise distinguishability determine an ensemble of quantum states? We show that the complete directed max-relative-entropy matrix determines four generic faithful states of a single qubit up to a common unitary or antiunitary transformation. For arbitrary normalized Lorentz-ball ensembles we classify every compatible realization by one scalar; its endpoint uniquely minimizes embedding dimension and, for qubits, maximizes every state's entropy. We establish local stability, a nonuniform-stability obstruction, sparse extension from four anchors, and acquisition rules that cancel specified loss and throughput nuisance parameters. Seven independent 100x100 superconducting-qubit frequency tables at each of 5 and 10 microseconds provide an end-to-end finite-setting test: experimental frequencies are converted directly to a directed divergence and then to reconstructed qubit geometry, with no tomography in the reconstruction path. Median four-state Gram errors are about 2.4e-2 for pooled data and 3.5e-2 under strict split-sample selection/evaluation, with monotonic improvement as the number of measurement directions increases. Independent full-Stokes data separately test acquisition-layer nuisance cancellation. The protocol provides calibration robustness rather than a universal sample advantage
Abstract Can pairwise distinguishability determine an ensemble of quantum states? We show that the complete directed max-relative-entropy matrix determines four generic faithful states of a single qubit up to a common unitary or antiunitary transformation. For arbitrary normalized Lorentz-ball ensembles we classify every compatible realization by one scalar; its endpoint uniquely minimizes embedding dimension and, for qubits, maximizes every state's entropy. We establish local stability, a nonuniform-stability obstruction, sparse extension from four anchors, and acquisition rules that cancel specified loss and throughput nuisance parameters. Seven independent 100x100 superconducting-qubit frequency tables at each of 5 and 10 microseconds provide an end-to-end finite-setting test: experimental frequencies are converted directly to a directed divergence and then to reconstructed qubit geometry, with no tomography in the reconstruction path. Median four-state Gram errors are about 2.4e-2 for pooled data and 3.5e-2 under strict split-sample selection/evaluation, with monotonic improvement as the number of measurement directions increases. Independent full-Stokes data separately test acquisition-layer nuisance cancellation. The protocol provides calibration robustness rather than a universal sample advantage
This computer-assisted preprint proves a sharp reverse relative entropy inequality L >= beta_m chi for uniform ensembles of individually normalized qubit states, for every m >= 2. Here chi is the Holevo quantity and L is the non-optimized quantum lautum information. The proof combines analytic Bloch-vector and Gram-matrix reductions with four interval certificates covering 855,780 boxes. For three matrices in arbitrary finite dimension, the paper also establishes local nonnegativity near faithful pairwise-commuting equality triples, allowing unequal traces and noncommuting perturbations. Further results include normalization bridges, an angular reduction for three qubits, and certified obstructions to proposed pinching and filtering descent arguments. The unrestricted unequal-trace ensemble inequality and the general equality of complete and ordinary modified logarithmic Sobolev constants are not proved. The accompanying archive contains sources, certificate inputs, verifiers, and reproduced results. Developed with AI assistance; not peer reviewed or formally verified. Companion paper: Christoph Hartmann, Reverse relative entropy and finite-time contraction for conditional expectations (2026).
Two pure orthogonal quantum states can be perfectly distinguished by sequential local action of multiple pairs of parties. However, this process typically leads to the complete dissolution of entanglement in the states being discriminated. We propose a protocol that allows an arbitrary number of pairs of parties to distinguish between any two orthogonal, entangled, two-qubit pure states using local quantum operations and classical communication, with a success probability greater than that of random guessing, while ensuring that at each step, the individual ensemble states retain a finite amount of entanglement. Our protocol employs the minimum-error state discrimination approach. For demonstrating the retention of entanglement in the ensemble states at each step, we use logarithmic negativity as well as the concept of entanglement witnessing. For a large family of sets of the two states, the success probability of discrimination can be as close as required to unity, while sustaining a finite amount of entanglement in each step.
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.
Among the surprising features of quantum measurements, the problem of distinguishing and antidistinguishing general quantum measurements is fundamentally appealing. Unlike classical systems, quantum theory offers entangled states and peculiar state update rule of the postmeasurement state, which gives rise to four distinct scenarios for distinguishing (and antidistinguishing) quantum measurements: (i) probing single systems and without access to the postmeasurement states, (ii) probing entangled systems and without access to the postmeasurement states, (iii) probing single systems with access to the postmeasurement states, and (iv) probing entangled systems with access to the postmeasurement states. In these scenarios, we consider the probability of distinguishing (and antidistinguishing) quantum measurements sampled from a given set in the single-shot regime. For some scenarios, we provide the analytical expressions of distinguishability (and antidistinguishability) for qubit projective measurements. We show that the distinguishability of any pair of qubit projective measurements in scenario (iii) is always greater than its values in scenario (ii). Interestingly, we find certain pairs of qubit nonprojective measurements for which the optimal distinguishability in scenario (ii) is achieved using a nonmaximally entangled state. It turns out that, for any set of measurements, distinguishability (and antidistinguishability) in scenario (i) is always less than or equal to in any other scenario, while it reaches its highest possible value in scenario (iv). We establish that the relations form a strict hierarchy, and there is no hierarchical relation between scenarios (ii) and (iii). In particular, we introduce different variants of the well-known ``trine'' qubit measurement to construct pairs (and triples) of qubit quantum measurements such that they are perfectly distinguishable (and antidistinguishable) in scenario (ii) but not in scenario (iii) and vice versa. Additionally, we present qubit measurements that are perfectly distinguishable (and antidistinguishable) in scenario (iv) but not in any other scenario.
ARCHANGEL develops a constructive mathematical framework for the common realization of positive matrix moments, compatibility testing across directional data, and rigorous certification of operator-valued responses. Its principal contributions connect matrix inequalities, constrained realization theory, spectral geometry, semidefinite certification, and finite-dimensional quantum tomography. The central result is a necessary-and-sufficient common-realization criterion within a specified finite positive Gaussian model class. Under explicit kernel and positive-curvature assumptions, the construction determines whether complete joint moment data admit one realization satisfying a prescribed spectral cutoff constraint. It provides an explicit extremizer attaining the resolvent lower bound Lₐ = Q(Q + aR)⁺Q simultaneously for every positive parameter a. The auxiliary dimension is minimal and equals rank(Q); in the three-direction, ten-dimensional formulation, at most 30 auxiliary modes are required. A particularly distinctive result is an exact rational counterexample to the inference from separate directional feasibility to joint realizability. Every real direction individually passes its optimized feasibility test, yet no single common realization satisfies all directional constraints together. The obstruction survives every compatible complex completion of the unmeasured imaginary entries. This establishes a precise limitation of reconstructing a shared system from separately successful directional tests. Additional results include: Sharp directional coverage bounds. A general coverage parameter quantifies how much finite directional testing can underestimate the largest response. The optimal missed-peak factor is 4/3 for planar trigonal sampling and 3/2 even when every direction in all three coordinate planes is tested. Sharp reconstruction stability. Three planar trigonal measurements reconstruct the complete quadratic directional operator, with an optimal 5/3 amplification factor for uniformly bounded measurement errors. Ten-dimensional holonomy certification. Two Weyl transformations detect every nonscalar component of a Hermitian response operator. A sharp Hilbert–Schmidt robustness inequality has the golden ratio as its optimal constant. Exact continuum positivity certification. A planar operator-valued trigonometric inequality is reduced to a finite semidefinite feasibility problem using matrix Fejér–Riesz factorization. For the ten-dimensional code, the degree-one certificate uses a 20 × 20 block matrix. Explicit tomography and combined bounds. A construction using 100 preparations reconstructs any ten-dimensional Hermitian response. Combining spatial reconstruction with holonomy contrasts gives a sharp joint certification inequality. The mathematical value lies in distinguishing independently compatible observations from data that admit one shared realization, constructing extremal realizations with minimal auxiliary dimension, and replacing incomplete sampling arguments with quantitative or exact certificates. These questions are relevant to matrix moment problems, inverse spectral modelling, constrained system identification, and quantum response certification. The release includes a 95-page compendium, complete proofs, five preserved research studies, correction and theorem indexes, reproducible Python code, three synthetic figures, and 32 passing computational verification groups. Exact arguments establish the continuum statements; numerical calculations provide supplementary consistency checks. Status and scope: The stated finite-matrix problems are resolved under their explicit assumptions. The work is an AI-assisted theory and methods contribution using established tools, including Schur complements, Weyl operator methods, and matrix polynomial factorization. External peer review and a comprehensive literature-priority assessment remain outstanding. No experimental validation is claimed.
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.