I am trying to understand the jump operators in the Lindblad equation. Specifically, if there is any condition of boundedness we need to impose on them. I ask this because, as was pointed out in this post, there should be no assumption on the norm of the jump operators. But for example in this review, the jump operators live in the space of bounded operators. So is boundedness of the jump operators a necessary condition? In Preskill's notes, there is a derivation of the Lindbladian in terms of the operator-sum representation which gives some normalization condition that the jump-operator should fulfill in terms of the Kraus operators but I am not sure if this is relevant to my question.
1 Answer
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Note that this question only arises in infinite dimensions as for finite-dimensional systems all linear operators are bounded automatically. Indeed, in the post you linked the example of an unbounded jump operator was the annihilation operator which acts on the (necessarily infinite-dimensional) Hilbert space $\ell^2(\mathbb N)$.
So if you're only interested in finite-dimensional systems, then boundedness is not something you have to care about as it holds trivially. However, if you do care about systems of infinite dimension, then boundedness of the jump operators leads to an important distinction for the dynamics: there is uniform continuouity ($\lim_{t\to 0^+}\|\Phi_t-{\rm id}\|_{1\to 1}=0$) as well as strong continuouity ($\lim_{t\to 0^+}\|\Phi_t(A)-A\|_1=0$ for all $A$) of a semigroup $\{\Phi_t\}_{t\geq 0}$, cf. also this answer. In particular, uniform continuity is a strictly stronger requirement than strong continuity in infinite dimensions. This is exemplified by the result that $L$ is the generator of a uniformly continuous quantum-dynamical semigroup if and only if $$ L=-i[H,\cdot]-\sum_{j\in J}\Big( \frac12(V_j^*V_j(\cdot)+(\cdot)V_j^*V_j)-V_j(\cdot)V_j^* \Big)\tag{1} $$ for some bounded, self-adjoint operator $H$ and some family $\{V_j\}_{j\in J}$ of bounded operators such that $\sum_{j\in J}V_j^*V_j$ converges in the weak operator topology whereas, in contrast, there exist generators $L$ of a strongly continuous quantum-dynamical semigroup which are not of the "standard form" (1), even when allowing for unbounded $H,V_j$, cf. this article by Siemon et al. (arXiv)
