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(GQR50)Admissibility Distance as a Resource Monotone for Dynamical Nonlocality of Quantum Channels

JAMES SUTTON

Published Feb 24, 2026• 0 citations
Publisher

Abstract

Operational resource theories of quantum channels require measuresthat quantify intrinsic dynamical structure independently of particularinput states or simulation protocols. We introduce admissibility distance,defined as the diamond-norm distance from a bipartite quantum channelto the free set of locally implementable product channels, and developit as a resource monotone for channel-level dynamical nonlocality. Weprove that the measure is faithful, monotone under local superchannels,and subadditive under both sequential composition and tensor products,providing a compositional operational framework for quantifying deviationfrom local dynamics.To calibrate the framework, we analyse an exactly solvable two-qubitentangling family, recovering the analytical diamond distance in the noiseless limit and deriving rigorous contraction bounds under local unital noise,including depolarising channels, before extending the construction to arbitrary finite-dimensional systems through induced trace-norm contractionon the traceless operator subspace.As an illustrative application, we implement admissibility distance forsimulated Atomic Frequency Comb (AFC) retrieval channels, validatingthe numerical implementation against identity, depolarising and phasedamping benchmarks before applying it to structured retrieval noise. Theresulting numerical experiments indicate that admissibility distance responds substantially more strongly to small structured perturbations thanconventional state-fidelity diagnostics within the benchmark considered,suggesting its potential utility as an operational early-warning metric forquantum-memory benchmarking.Unlike state-based quantities such as entangling power or Choi-stateentanglement, admissibility distance measures the operational separation ofa quantum channel from locally implementable dynamics themselves. Theresulting framework provides a mathematically controlled and operationallyinterpretable geometry for analysing noisy quantum channels and resourcetheoretic aspects of quantum dynamics. Full 70 article programme https://github.com/jamessutton600613-png/GQR/blob/main/Programme_Index.md The programme contains both standalone and self referencing articles. These can all be read as notes in proof of GQR1.

DOI
10.5281/zenodo.18751657

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Operational resource theories of quantum channels require measuresthat quantify intrinsic dynamical structure independently of particularinput states or simulation protocols. We introduce admissibility distance,defined as the diamond-norm distance from a bipartite quantum channelto the free set of locally implementable product channels, and developit as a resource monotone for channel-level dynamical nonlocality. Weprove that the measure is faithful, monotone under local superchannels,and subadditive under both sequential composition and tensor products,providing a compositional operational framework for quantifying deviationfrom local dynamics.To calibrate the framework, we analyse an exactly solvable two-qubitentangling family, recovering the analytical diamond distance in the noiseless limit and deriving rigorous contraction bounds under local unital noise,including depolarising channels, before extending the construction to arbitrary finite-dimensional systems through induced trace-norm contractionon the traceless operator subspace.As an illustrative application, we implement admissibility distance forsimulated Atomic Frequency Comb (AFC) retrieval channels, validatingthe numerical implementation against identity, depolarising and phasedamping benchmarks before applying it to structured retrieval noise. Theresulting numerical experiments indicate that admissibility distance responds substantially more strongly to small structured perturbations thanconventional state-fidelity diagnostics within the benchmark considered,suggesting its potential utility as an operational early-warning metric forquantum-memory benchmarking.Unlike state-based quantities such as entangling power or Choi-stateentanglement, admissibility distance measures the operational separation ofa quantum channel from locally implementable dynamics themselves. Theresulting framework provides a mathematically controlled and operationallyinterpretable geometry for analysing noisy quantum channels and resourcetheoretic aspects of quantum dynamics. Full 70 article programme https://github.com/jamessutton600613-png/GQR/blob/main/Programme_Index.md The programme contains both standalone and self referencing articles. These can all be read as notes in proof of GQR1.
We introduce an operational notion of admissibility for quantum dy-namics defined as the diamond-norm distance of a physical channel froma restricted class of locally implementable channels. For a two-qubit en-tangling family Uθ = e−iθX⊗X , we prove that the noiseless admissibilitydistance to the identity channel equals 2| sin θ|. Extending to locally de-polarized dynamics Eθ,p = (Dp ⊗ Dp) ◦ Uθ , numerical semidefinite-programevaluation reveals a robust factorization δ(θ, p) ≈ 2 sin θ f (p), with f (p)well-approximated by (1 − βp)α over p ∈ [0, 0.6]. The resulting quantitycaptures channel-level nonlocal capability independently of state-specificoutput correlations, as illustrated by probe states that remain uncor-related despite nonzero admissibility distance. These results provide aminimal operational framework for quantifying dynamical deviation fromlocally admissible processes and establish a tractable testbed for furtheranalytic development. Full 70 article programme https://github.com/jamessutton600613-png/GQR/blob/main/Programme_Index.md The programme contains both standalone and self referencing articles. These can all be read as notes in proof of GQR1.
We investigate the interplay between coherent gauge-mediated mixingand decoherence in a minimal bipartite channel model using an opera-tional admissibility distance to product structure. The construction ex-tends previous resource-theoretic formulations of channel-level dynamicalnonlocality by introducing an interleaved open evolution on an auxiliarygauge degree of freedom. Crucially, trace-preserving noise applied only af-ter a full unitary leaves the reduced dynamics invariant; therefore decoher-ence must act during coherent mixing in order to influence channel-leveldeviation.Using a Trotterized interleaving of unitary mixing and gauge dephas-ing, we compute the diamond-norm distance δPROD from product channelsacross the (g, γϕ) parameter plane. The resulting atlas exhibits strict γϕ-invariance at zero mixing, weak sensitivity in the perturbative regime, andnonlinear amplification of dynamical nonlocal deviation for sufficientlylarge mixing and decoherence rates. Rather than a sharp phase bound-ary, the curvature–loss ratio Γ = g2/(κγϕ) emerges as an organizing scalegoverning a smooth crossover between regimes.These results provide a channel-level operational refinement of curvature–coherence balance intuitions in non-Abelian coherence field models andfurnish a tolerance-aware framework for analyzing persistence of entan-gling capability under structured decoherence. Full 70 article programme https://github.com/jamessutton600613-png/GQR/blob/main/Programme_Index.md The programme contains both standalone and self referencing articles. These can all be read as notes in proof of GQR1.
Resource-theoretic measures of quantum channels are useful only ifthey remain mathematically controlled under physically realistic noise. Weinvestigate the behaviour of the operational admissibility distance undernon-unital completely positive trace-preserving maps. Using amplitudedamping as a canonical model of energy relaxation, we show that althoughnon-unital channels possess an affine component in their Bloch representation, this contribution cancels identically on trace-annihilating channeldifferences, leaving contraction governed solely by the linear action onthe traceless operator subspace. This yields an explicit contraction coefficient, a bipartite suppression factor, and an exponential persistence lawunder repeated local noise. These results extend the admissibility-distanceframework from idealized unital channels to dissipative noise models, establishing mathematically controlled resource contraction in physicallyrealistic settings and providing a rigorous foundation for analysing dynamical nonlocality under energy relaxation.
Recent developments in driven quantum matter, including Floquet and time-crystallineregimes, suggest that persistent oscillatory structure may survive far beyond naive equilibriumexpectations under constrained dynamical conditions [1–5]. In parallel, Ref. [6] introduced admissi-bility distance as a resource-theoretic measure of channel-level dynamical nonlocality, establishingcompositional bounds, contraction laws under unital noise, and persistence scales for repeated noisyevolution.Here we discuss a conceptual connection between these frameworks. While admissibility distancedoes not constitute a theory of time crystals, its structural treatment of recurrence, contraction,accumulation, and dynamical persistence provides a natural interpretive language for analysingrecurrent non-equilibrium quantum regimes. In particular, the competition between coherent ac-cumulation and exponential contraction under repeated channel composition mirrors the balancebetween oscillatory persistence and thermalization observed in driven many-body systems.We argue that admissibility-based resource structures may offer a useful channel-centric perspec-tive for discussing metastable oscillatory order, recurrence-protected dynamics, and tolerance-awarepersistence in noisy quantum architectures. The discussion is interpretive rather than theorem-proving, and is intended to outline possible bridges between operational resource theory and emer-gent dynamical order in driven quantum systems.
Abstract Quantifying the number of resources contained in a physical object has been one of the core topics in the resource theory of coherence. In this paper, we introduce the dynamical coherence measures based on a class of norms in the classical channel setting. It is proved that it satisfies faithfulness, decreases monotonically under the maximally incoherent superchannels, and is convex. Moreover, we show that it satisfies subadditivity under both the composition and tensor product of channels. Especially, the diamond measure as a special case is discussed in detail, it can reduce to trace norm of coherence, satisfies amortization inequality, and can be calculated efficiently using a semidefinite program. In addition, we introduce the creation-coherent diamond measure and find that neither the detection coherence nor the creation coherence of a channel exceeds the coherence of the channel, which does not exceed the purity of the channel. Second, we introduce the corresponding dephasing measure, which is a dynamical coherence measure under the dephasing-covariant incoherent superchannels. Meanwhile, we also introduce the dephasing diamond measure as a special case. Third, we use the dephasing diamond measure to accurately calculate the coherence values of some important noisy channels such as amplitude damping channel, phase damping channel, and depolarizing channel, respectively, and give the sufficient and necessary conditions for an unital qubit channel with a parameter probability vector to be a coherent channel. Finally, the operational interpretation of our diamond measure in the binary channel discrimination task is investigated.
The advantage that quantum systems provide for certain quantum information processing tasks over their classical counterparts can be quantified within the general framework of resource theories. Certain distance functions between quantum states have successfully been used to quantify resources like entanglement and coherence. Perhaps surprisingly, such a distance-based approach has not been adopted to study resources of quantum measurements, where other geometric quantifiers are used instead. Here, we define distance functions between sets of quantum measurements and show that they naturally induce resource monotones for convex resource theories of measurements. By focusing on a distance based on the diamond norm, we establish a hierarchy of measurement resources and derive analytical bounds on the incompatibility of any set of measurements. We show that these bounds are tight for certain projective measurements based on mutually unbiased bases and identify scenarios where different measurement resources attain the same value when quantified by our resource monotone. Our results provide a general framework to compare distance-based resources for sets of measurements and allow us to obtain limitations on Bell-type experiments.