We identify the qutrit as the minimal Hilbert-space dimension in which coherence amountand coherence sharing become operationally distinguishable. The starting point is theexact purity decomposition into population predictability, Hilbert–Schmidt coherence, andmixedness. Although this closure is a state-space identity, in dimension three it selects aEuclidean coherence coordinate that can be compared nontrivially with normalized ℓ1coherence. Their mismatch defines a positive deficit Δℓ1 that vanishes if and only if thethree pairwise coherence moduli are equal. The deficit therefore diagnosescoherence-sharing anisotropy rather than total coherence magnitude alone. We show thatΔℓ1 separates two physically distinct mechanisms of coherence reduction: populationanisotropy in symmetric channels and branch asymmetry in balanced qutrits. In a purifiedqutrit–meter model the same Euclidean language acquires a dynamical interpretationbecause reduced mixedness equals normalized linear-entropy entanglement across thesystem–meter cut. We clarify the resource-theoretic status of the diagnostic by showingthat it is not a coherence monotone, but a witness of unequal sharing among pairwisechannels. Finally, we give a reduced-state reconstruction protocol based on populationsand three embedded SU(2) Ramsey blocks, supported by finite-sample simulations. Thesame quantity has a direct three-path interferometric implementation in terms of pairwisefringe visibilities and calibrated path intensities, where it measures anisotropicmutual-coherence sharing rather than a universal violation of classical coherence theory.
We identify the qutrit as the minimal Hilbert-space dimension in which coherence amountand coherence sharing become operationally distinguishable. The starting point is theexact purity decomposition into population predictability, Hilbert–Schmidt coherence, andmixedness. Although this closure is a state-space identity, in dimension three it selects aEuclidean coherence coordinate that can be compared nontrivially with normalized ℓ1coherence. Their mismatch defines a positive deficit Δℓ1 that vanishes if and only if thethree pairwise coherence moduli are equal. The deficit therefore diagnosescoherence-sharing anisotropy rather than total coherence magnitude alone. We show thatΔℓ1 separates two physically distinct mechanisms of coherence reduction: populationanisotropy in symmetric channels and branch asymmetry in balanced qutrits. In a purifiedqutrit–meter model the same Euclidean language acquires a dynamical interpretationbecause reduced mixedness equals normalized linear-entropy entanglement across thesystem–meter cut. We clarify the resource-theoretic status of the diagnostic by showingthat it is not a coherence monotone, but a witness of unequal sharing among pairwisechannels. Finally, we give a reduced-state reconstruction protocol based on populationsand three embedded SU(2) Ramsey blocks, supported by finite-sample simulations. Thesame quantity has a direct three-path interferometric implementation in terms of pairwisefringe visibilities and calibrated path intensities, where it measures anisotropicmutual-coherence sharing rather than a universal violation of classical coherence theory.
A reduced qutrit does not only lose coherence under environmental coupling; it loses coherence with a geometry. This preprint shows that the pairwise coherence structure of a qutrit contains a directly accessible fingerprint of environmental distinguishability. By normalizing the three pair coherences with respect to the corresponding populations, one obtains a reduced-state map that identifies which pairs of qutrit alternatives remain mutually indistinguishable, which pairs are resolved by the environment, and which sectors may be protected or rephased. The method is deliberately economical. It does not attempt full bath reconstruction, process-tensor tomography, or spectral noise reconstruction. Instead, it uses only the reduced qutrit populations and pairwise coherences, or equivalently calibrated pairwise visibilities in an interferometric implementation. The resulting three-component distinguishability map gives the environmental structure as seen by the qutrit. Under additional model assumptions it can be interpreted as an environmental record-overlap map, a scalar coupling-distance diagnostic, a vector-noise metric, or a pair-selective memory fingerprint. The reconstruction is relational rather than microscopic. Different reservoirs may be indistinguishable to the qutrit if they generate the same reduced coherence geometry. The paper therefore identifies the equivalence class of environmental structures visible to the three-level probe. Explicit examples illustrate scalar Gaussian dephasing, protected pairs, vector or correlated-noise geometry, pair-selective recoherence, and safe analysis windows when populations or coherences approach a noise floor.
This preprint introduces a dynamical qutrit coherence-cycle witness based on calibrated pairwise visibility measurements in a three-path interferometer. The central idea is that pairwise visibility data contain a geometry of normalized qutrit coherences that is not captured by scalar coherence measures alone. For nonzero path populations, the population-calibrated two-path visibility equals the normalized off-diagonal coherence of the reduced path qutrit. The squared normalized coherences define the cycle expression Scyc = kappa_12^2 + kappa_23^2 - kappa_13^2. Classical jointly diagonalizable overlap geometries obey the boundary Scyc <= 1. A statistically certified value Scyc > 1 therefore indicates a nonclassical cyclic arrangement of pair coherences, rather than merely a large amount of coherence. The paper proves the physical qutrit range Scyc <= 5/4, gives explicit positive semidefinite qutrit examples with equal scalar coherence but different cycle geometry, constructs a completely positive random-unitary dephasing model with time-dependent geometry windows, and formulates a finite-data certification protocol including calibration and covariance propagation. The static overlap-cycle inequality is imported from prior overlap-cycle work; the new contribution is its calibrated realization as a path-qutrit coherence-geometry witness. The proposed witness is intended as a measurement-efficient diagnostic of normalized coherence geometry in multipath interferometry and open qutrit dynamics. It is not a full state-tomography method and does not by itself constitute a complete contextuality experiment; stronger interpretations require the additional operational assumptions stated in the manuscript.
Non-Markovian optical channels can redistribute interferometric visibility,which-path information, and entanglement in ways that are not captured byconcurrence dynamics alone. Motivated by cavity-QED, waveguide-feedback,and related photonic reservoir architectures, we study an exactly solvabletwo-qubit mini-reservoir model and interpret its reduced dynamics as an effectivelocal photonic feedback channel. For the reduced two-qubit state, predictabilityPk, visibility Vk, and concurrence C obey the mixed-state relationP2k (t)+V 2k (t)+C2(t) = Bk(t) ≤ 1, where Bk(t) is the locally accessible complementaritybudget. Within the Hilbert–Schmidt formulation of completecomplementarity, we show that the corresponding deficit is closed exactly byMk(t) ≡ 1 − Bk(t) = 2Cnlhs(t) − C2(t). The remainder therefore identifiesthe part of the nonlocal optical coherence sector that is not already encodedin concurrence. Exact trajectories for Bell, nonmaximally entangled parity,superposition-of-Bell, and Werner initial states reveal memory-inducedcollapse-and-revival structure in the accessible budget, in the visibility sectorwhen local coherence is present, and in the remainder itself. Comparisonswith one-sided discord and with the purity deficit show that Mk(t)is a distinct reduced-state diagnostic rather than a relabeling of entanglement,discord, or mixedness. Because all quantities are reconstructed fromthe same reduced two-qubit state, the framework is naturally compatible with interferometric and tomographic readout in memory-bearing quantumopticalplatforms.
Quantum resource theories usually begin after a localization decision has already beenmade: the resource is assigned to a state, channel, measurement, process, circuit, code,contextuality scenario, or other formal object. Once that arena is fixed, standard resourcetheory supplies powerful tools for quantification, conversion, witnessing, and operationalanalysis. This paper addresses the prior localization problem: which stabilized distinction iserased by a tested descent, where is that distinction retained, and how is it later reactivatedinto a task-visible separation?The analysis is developed within Finite-Resolution Physics (FRP). Starting from finiteresolutionacts, stabilized histories, plateaux, continuation carriers, codecs, regimes, andSIB/SILM boundary coherence, the paper reconstructs quantum resources as task-visiblereactivations of retained distinctions erased by specified descents. The relevant structure isnot treated as primitive nonclassicality or as a universal obstruction object. It is diagnosedby asking whether a descent to a simpler target erases stabilized carrier–codec content thatlater admissible acts of the protocol can still make operationally visible.This yields a carrier–residual localization principle: a resource is localized at the minimalcarrier–codec layer where the relevant distinction must be retained, cannot be removed,descended, or relocated without loss, and can be reactivated into the task-visible separation.Entanglement, magic, contextuality, coherence, measurement adaptivity, quantumerror correction, and topological protection are then reformulated as distinct descent failures:product descent, Clifford–stabilizer descent, noncontextual global-assignment descent,reference-insensitive descent, nonadaptive-history descent, noise-degraded descent, and localor contractible descent.The paper does not replace ordinary quantum resource theory. Rather, it suppliesan upstream diagnostic layer that precedes ordinary quantification. Standard monotones,witnesses, conversion rates, simulation bounds, and thresholds remain applicable afterthe retained distinction, descent target, localization layer, and reactivation route havebeen identified. The resulting framework clarifies why representation does not determinelocalization: a resource represented in a Hilbert-space codec may be retained in a state,channel, measurement context, history, code, boundary/SIB structure, compact or discretecarrier, or stabilizer dynamics. FRP resource localization therefore asks not merely whethera resource is present, but where the task-relevant distinction is retained and how the protocolreactivates it.
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
Complementarity residuals quantify local-information budgets that can coincide with entanglement remainders in calibrated pure-state settings. In noisy and mixed multipartite systems, however, the same residual can remain positive even after the resource interpretation inherited from the pure-state limit is no longer justified. This preprint analyzes that change of meaning and introduces a support-layer diagnostic for complementarity residuals under mixedness. The paper separates three logical levels: the residual itself, the independent support layer, and the resource-specific interpretation that is allowed only when both are present. In the pure three-qubit calibration limit, the residual agrees with a known residual-tangle interpretation. In mixed four-qubit white-noise families, however, the residual can remain finite after the chosen all-cut NPT support layer has disappeared. A separable product-state construction further shows that local mixedness alone can generate a positive residual, so the residual is not an entanglement witness and not a genuine multipartite entanglement witness. The result is an operational hierarchy for noisy multipartite quantum states. A positive complementarity residual flags unresolved local-information budget, while independent support tests determine whether an entanglement-related interpretation is justified. The approach is intended as a safe diagnostic protocol: it preserves the usefulness of complementarity residuals while preventing overinterpretation in regimes where mixedness, local decoherence, environmental correlations, or classical mixing contribute to the observed residual.