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.
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.
Non-unital noise is a fundamental feature of open quantum systems, governing energy exchange and inducing state translations on the Bloch sphere. While such translations can be useful for tasks such as state preparation and channel capacity, unital dynamics where no translation occurs are often preferred in quantum information processing, particularly for error correction. Phase-covariant dynamics provide a general framework encompassing dissipation, excitation, and dephasing processes in qubit systems; however, commonly used models such as the generalized amplitude damping (GAD) channel offer only limited control over these features. In this work, we present a constructive framework for generating a broader class of phase-covariant dynamics by mixing amplitude-damping and anti-damping channels with unequal decay parameters and time-dependent mixing probabilities. This approach enables independent control over contraction and translation, allows continuous tuning between non-unital and unital regimes, and yields an effective dephasing contribution absent in GAD. To characterize these dynamics, we employ the general theorem for P-divisibility and show that it can be used constructively by evaluating it in suitably chosen Hilbert space bases, leading to simplified conditions for both unital and non-unital cases. The framework captures a wide range of behaviors, including CP-divisible, P-divisible but not CP-divisible, and non-P-divisible dynamics. We further show that appropriate mixing can reduce the deviation from the ideal noiseless evolution and that such improvement persists even after tuning the dynamics to the unital regime. Our results provide a flexible approach to engineering open system dynamics beyond standard thermal models, with potential applications in noise control and quantum information processing.
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.
The contraction coefficient of the trace distance is a central tool in quantum information, quantifying how strongly a quantum channel degrades the distinguishability of states. However, being an extremal ratio, it captures only the most optimistic behaviour of the channel and is often trivial, even for very noisy channels. Moreover, a single scalar is poorly suited to describe how contraction accumulates under channel composition. In this work we introduce the \emph{contraction and expansion values}, two monotone sequences that refine the contraction and expansion coefficients in the same way singular values refine the operator norm. They arise from a min--max variational principle over subspaces of traceless Hermitian operators, admit an operational interpretation in terms of two state-discrimination games, and are shown to coincide with the Gel'fand or Bernstein numbers of the channel restricted to traceless operators. This identification places the sequences within Pietsch's theory of s-numbers and yields, in particular, bounds under channel composition that the contraction coefficient alone cannot provide. We establish their main structural properties and compute or estimate them for single-qubit channels, d-dimensional amplitude damping channels, and direct-sum channels.
We develop a Lie-Jordan geometric formulation of finite-dimensional open quantum dynamics, building on the algebraic framework introduced in our previous work (arXiv:2606.26477). The Hilbert-Schmidt operator space is endowed with an orthonormal Hermitian basis, in which the commutator and anticommutator are encoded by the structure tensors Cμνλ and Bμνλ. Within this formulation, the von Neumann and Gorini-Kossakowski-Lindblad-Sudarshan equations admit a direct component representation. Our central result is the identification of a basis-independent universal trilinear dissipative map, D(X,Y)Z=XZY−21{YX,Z}, whose components define a universal operator-space tensor depending only on the Lie-Jordan structure tensors. The physical dissipator is obtained by contracting this tensor with the expansion coefficients of the Lindblad operators and the Kossakowski matrix, thereby separating the universal algebraic structure from the model-dependent physical information. We further show that the combinations (B+C) and (B−C) generate internal left and right transports, allowing the elementary dissipative map to be expressed as a left-right bimodule action corrected by an ordered Jordan contribution. The universal map satisfies basis-independent trace and Hermitian-conjugation identities, from which trace preservation, Hermiticity preservation of the complete dissipator, and the reality of its component representation in a Hermitian Hilbert-Schmidt basis follow. We also derive its Hilbert-Schmidt adjoint and illustrate the formalism for a qubit with pure dephasing and amplitude-damping channels. This construction provides a tensorial and affine-geometric interpretation of the universal superoperator structure derived in arXiv:2606.26477 while keeping the algebraic and model-dependent sectors explicitly separated.
Abstract The characterization of quantum noise is fundamental to quantum information processing, quantum control and reliable operation of near-term quantum devices. Conventional descriptions of quantum noise are based on completely positive trace-preserving (CPTP) maps acting on density operators. These models describe effective quantum dynamics after the underlying physical interactions have been represented as a quantum channel. However, in practical cases, imperfections such as calibration errors, control-field fluctuations, finite-resolution waveform generation, and analog hardware noise can act upon the probability amplitudes of a quantum state earlier. The effect of such perturbation in quantum state after normalization has largely been unexplored. In this work, we develop a rigorous mathematical analysis of the normalized additive amplitude noise in a finite-dimensional quantum system. Assuming a zero-mean complex perturbation, we derive a second-order closed form expression for the ensemble-averaged density operator. The analysis leads to state-dependent evolution of the quantum state which cannot be represented by a CPTP map. We further prove that in case of isotropic noise, the ensemble-averaged density operator becomes an effective depolarizing channel. For anisotropic perturbations, we derive the orientation-dependent Bloch-vector contraction which is determined by the covariance geometry of the additive noise. Closed-form expressions are further obtained for measurement probabilities, state fidelity, and ensemble purity, showing that fidelity degradation depends only on the component of the perturbation covariance orthogonal to the input state. Monte Carlo simulations closely match the theoretical analysis, especially for small noise regime. The proposed model establishes a connection between classical additive-noise models and effective quantum dynamics generated by normalized state-space perturbations.