Convexity and non-Markovianity of Weyl Maps

Abstract: We investigate the emergence of non-Markovian dynamics in finite-dimensional open quantum systems governed by Weyl dynamical maps and their convex combinations. Using the Hermite normal form, we provide a complete classification of the subgroups of the discrete phase space $\mathbb{Z}_d \times \mathbb{Z}_d$, establishing the algebraic framework underlying the Weyl maps. We characterize isotropic Weyl dynamical maps that generate Markovian semigroups and show that anisotropic Weyl maps with nonuniform weight distributions cannot possess the semigroup property. Furthermore, we analyze the role of convexity in the generation and suppression of memory effects. Remarkably, we prove that convex combinations of eternally non-Markovian Weyl dephasing maps can generate Markovian semigroups, demonstrating that non-Markovianity is not additive under mixing. Conversely, we establish a general condition under which convex mixtures of $N$ distinct Weyl semigroups exhibit eternal non-Markovianity. In contrast to the qubit Pauli setting, we further identify the existence of irreducible eternally non-Markovian Weyl dephasing maps, namely, individual dynamical maps that display eternal memory effects without requiring any mixing mechanism. Finally, explicit qutrit examples illustrate the transition among Markovian, non-Markovian and eternally non-Markovian regimes. Our results uncover a fundamental connection among finite phase-space algebra, convex structures, and quantum memory effects, thereby extending the theory of non-Markovian dynamics beyond the Pauli framework.

• The paper presents an algebraic framework using Hermite normal form to classify Weyl maps, extending qubit Pauli channel analysis to qudit systems.
• It establishes that only isotropic Weyl maps yield Markovian semigroups while anisotropic mixtures result in eternal non-Markovian dynamics, as shown in qutrit examples.
• The research highlights how convex mixtures and subgroup algebra influence quantum memory effects, offering insights for quantum control and error correction in higher-dimensional systems.

Convexity and Non-Markovianity of Weyl Maps: Structural Characterization and Dynamical Implications

Introduction

This paper provides a systematic analysis of quantum dynamical maps constructed from discrete Weyl operators on finite-dimensional Hilbert spaces, emphasizing their convexity and non-Markovianity properties. The authors extend the foundational framework established for qubit Pauli channels to the more general setting of qudit Weyl channels, enabling a deeper understanding of quantum memory effects in higher-dimensional systems. The algebraic classification of subgroups of the discrete phase space $\mathbb{Z}_d \times \mathbb{Z}_d$ via Hermite normal form underlies the structural foundation of the presented results.

Algebraic Foundation: Classification of Subgroups

The dynamical maps considered are random unitary channels defined by Weyl operator bases. Utilizing the Hermite normal form, every subgroup $G \subseteq \mathbb{Z}_d \times \mathbb{Z}_d$ is uniquely generated by lattice-theoretic constructs, classified into cyclic, split rank-2, and non-split rank-2 types. The paper derives an explicit polynomial formula for the enumeration of subgroups of a given order, showing unimodality and maximality at $|G|=d$, leveraging the divisor sum function $\sigma_1(d)$ and prime factorization arguments.

This classification enables precise determination of the algebraic support for Weyl maps and their commutation structures, crucial for the spectral analysis of dynamical properties and for the mapping between the discrete phase space and the set of dynamical maps.

Markovianity and Semigroup Properties

The authors distinguish Markovianity by CP-divisibility (RHP criterion), based on non-negativity of decay rates in the time-local master equation's Lindblad form. They detail the conditions for an isotropic Weyl map (with uniform weights over a subgroup $G$) to define a quantum dynamical semigroup, concluding that only isotropic Weyl maps can generate Markovian semigroups with exponential eigenvalue decay and positive, time-independent decay rates.

In contrast, anisotropic Weyl maps (non-uniform weights) cannot possess the semigroup property if more than one nontrivial eigenvalue is present, reflecting the rigidity of the algebraic structure required for semigroup evolution.

Eternal Non-Markovianity and Convex Mixtures

A significant claim in the paper is the existence of irreducible eternally non-Markovian Weyl dephasing maps in higher dimensions (qudits), a phenomenon not present in the Pauli–qubit setting. Specifically, individual Weyl dephasing maps can display eternal non-Markovianity (ENM) for odd subgroup order $\ell=d/s\geq3$, or for even $\ell$ with $0<r\leq1/2$ in the mixing parameter, where $r$ is tied to the probability distribution amplitude. This is established via technical lemmas involving generating function identities and symplectic structures.

Moreover, the paper demonstrates that convex mixtures of ENM Weyl maps can yield Markovian semigroups if the subgroup orders align appropriately, highlighting the non-additivity of non-Markovianity under mixing. Conversely, mixing $N$ distinct Weyl semigroups below a subgroup coverage threshold (dictated by $G \subseteq \mathbb{Z}_d \times \mathbb{Z}_d$0, $G \subseteq \mathbb{Z}_d \times \mathbb{Z}_d$1, and combinatorial counts) generically produces ENM, and this number $G \subseteq \mathbb{Z}_d \times \mathbb{Z}_d$2 can be substantially larger than in the Pauli case, reflecting a richer structure in qudit systems.

Explicit Qutrit Examples and Comparison with Generalized Pauli Maps

For $G \subseteq \mathbb{Z}_d \times \mathbb{Z}_d$3, the authors analyze explicit two-, three-, and four-way mixtures of subgroups, illustrating the transition between Markovian, non-Markovian, and ENM regimes, dictated by the coverage of the phase space and the choice of mixing coefficients. They also demonstrate that generalized Pauli maps, defined via mutually unbiased bases (MUBs), are special instances of Weyl maps—a result that embeds the convex analysis of generalized Pauli maps within the broader Weyl framework.

Strong numerical results include:

• For $G \subseteq \mathbb{Z}_d \times \mathbb{Z}_d$4, three-way mixtures yield multiple negative decay rates, i.e., ENM, corroborating and extending previous findings [wudarski2015], while four-way uniform mixtures recover Markovianity.

Theoretical and Practical Implications

The findings have substantial implications for the theory of quantum open systems and quantum information:

• Intrinsic ENM: High-dimensional systems harbor irreducible ENM dynamics, implying fundamentally higher memory capacity and more complex noise structures in qudit systems.
• Convex Structure: Convexity properties directly impact divisibility and operational criteria for channel mixing, with practical consequences for quantum control protocols and error correction, especially in the presence of memory effects.
• Resource Theory: The interplay between subgroup algebra and convexity is likely relevant to resource theories of non-Markovianity and operational tasks involving reversible dynamics and information backflow.

Future directions suggested include analyzing time-dependent subgroup structures, noninvertible dynamics, and operational consequences of irreducible ENM in communication, metrology, and error correction. The connection with symplectic geometry and generalized resource theories is highlighted as an avenue for further foundational exploration.

Conclusion

This work rigorously extends the structural theory of quantum dynamical maps to the Weyl–qudit setting, providing explicit algebraic classification, semigroup conditions, and a nuanced analysis of convex mixtures and non-Markovianity. The results reveal qualitative distinctions between qubit and qudit systems regarding memory effects, and highlight scenarios where mixing ENM channels can suppress non-Markovianity, or conversely, where extensive mixing preserves it. The algebraic approach via the Hermite normal form and discrete symplectic structures sets the stage for future research in high-dimensional quantum systems, offering new perspectives on noise modeling and quantum memory resources (2605.23852).