Quantum adiabatic theory has been developing through a productive interplay between applications in physics and formalisation in mathematics. It originated in the work of Born and Fock [1] on the physics of quantum theory, and then mathematically systematised in Kato’s adiabatic perturbation theory [2]. Kato’s mathematical framework was highly influential and since its inception was used in dozens of distinct applications in physics. Each new application extended the theory, which led to new developments of its mathematical toolkit. The article of Burgarth, Facchi, Nakazato, Pascazio, and Yuasa fits this trend. In providing the first systematic application of adiabatic theory to Zeno dynamics, the authors also produce a hitherto unexplored version of Kato’s theorem. I discuss their results and the new directions these open up below.

The authors prove that
\begin{equation}
\label{1}
\exp(t (\gamma B + C)) = \exp(t \gamma B) \exp(t \bar{C}) + O(\gamma^{-1}),
\end{equation}
where $\bar{C}$ is the diagonal part of $C$ with respect to the operator $B$. In the Zeno language this equation says that the evolution is frozen on the spectral subspaces of $B$, and that $\bar{C}$ is the generator of the dynamics inside the Zeno subspace. This is a well-known result in two cases: (1) when $B$ and $C$ are Hamiltonian and (2) when $B$ corresponds to a measurement. The novelty of the article is in providing a unified theory for these past results and generalising the statement to include new cases that combine (1) and (2).

Within the existing mathematical theory, the above equation was established by Davies [3] in the case where $B, C$ are Hamiltonians. The standard proof uses the interaction picture with respect to $B$, a method that is not available when $B$ is not a Hamiltonian, e.g., when it is a Lindbladian, and the corresponding extension is not known. The authors circumvent this problem by deriving the equation using a new version of Kato’s theory [2].

I see two productive avenues for leveraging this result, the first in further applications to Zeno dynamics and the second in extending the mathematical theory. Adiabatic theory has well-established methods for handling unbounded operators. The authors’ formulation of Zeno dynamics in terms of adiabatic theory might now allow transferring these methods and study Zeno dynamics on infinite dimensional Hilbert spaces with unbounded generators. As for theory development, the derivation of the above equation uses a version of adiabatic theory that weakens the standard assumption that $B$ is a generator of contractions to that $\exp(t B)$ is bounded. Adiabatic theory with such a weakened assumption is by and large unexplored, and this work provides impetus to look into it. Also, taking a closer look at the connections between this work and the methods used by Davies [3] might reveal some interesting mathematical relationships and lead to knew insight.

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