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arXiv:2412.17519

Disorder-averaged Qudit Dynamics

Gopal Chandra Santra, Philipp Hauke

Published Dec 23, 2024• 0 citations
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Abstract

Understanding how physical systems are influenced by disorder is a fundamental challenge in quantum science. Addressing its effects often involves numerical averaging over a large number of samples, and it is not always easy to gain an analytical handle on exploring the effect of disorder. In this work, we derive exact solutions for disorder-averaged dynamics generated by any Hamiltonian that is a periodic matrix (potentially with non-trivial base, a property also called ($p,q$)-potency). Notably, this approach is independent of the initial state, exact for arbitrary evolution times, and it holds for Hermitian as well as non-Hermitian systems. The ensemble behavior resembles that of an open quantum system, whose decoherence function or rates are determined by the disorder distribution and the periodicity of the Hamiltonian. Depending on the underlying distribution, the dynamics can display non-Markovian characteristics detectable through non-Markovian witnesses. We illustrate the scheme for qubit and qudit systems described by (products of) spin $1/2$, spin $1$, and clock operators. Our methodology offers a framework to leverage disorder-averaged exact dynamics for a range of applications in quantum-information processing and beyond.

DOI
10.48550/arxiv.2412.17519

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Despite acute interest in the dynamics of non-Hermitian systems, there is a lack of consensus in the mathematical formulation of non-Hermitian quantum mechanics in the community. Different methodologies are used in the literature to study non-Hermitian dynamics. This ranges from consistent frameworks like biorthogonal quantum mechanics and metric approach characterized by modified inner products, to normalization by time-dependent norms inspired by open quantum systems. In this work, we systematically explore the similarities and differences among these various methods. Utilizing illustrative models with exact solutions, we demonstrate that these methods produce not only quantitatively different results but also distinct physical interpretations. For dissipative systems where non-Hermiticity arises as an approximation, we find that simply dividing by the norm in the PT -broken regime closely aligns with the full master equation solutions. In contrast, for quantum systems where non-Hermiticity can be engineered exactly, incorporating metric dynamics is crucial for the probabilistic interpretation of quantum mechanics, necessitating the generalizations of similarity transformations and unitarity to non-Hermitian systems. This study lays the groundwork for further exploration of non-Hermitian Hamiltonians, potentially leveraging generalized transformations for novel physical phenomena.