Deciding finiteness of bosonic dynamics with tunable interactions

TL;DR

This work tackles the problem of deciding when the bosonic dynamics generated by tunable interactions admit a finite-factorization, by embedding the problem in the skew-hermitian Weyl algebra $\hat{A}_n$ and developing a constructive, algebraic framework. It introduces a decomposition of $\hat{A}_n$ into meaningful subspaces and derives degree-based controls on commutators, complemented by two types of commutator chains (Type I and II) that reveal when generated chains remain finite or become infinite. The main contribution is a practical four-condition criterion (i)–(iv) that exactly characterizes finiteness for driftless Hamiltonians with independently tunable free terms, enabling a fast algorithm to assess dimensionality from a finite generator set. The results connect quantum-control problems with deep Weyl-algebra theory, offering a principled route to identify which bosonic systems admit exact finite-factorizations and guiding the design of quantum-control strategies and simulations with bosonic modes.

Abstract

In this work we are motivated by factorization of bosonic quantum dynamics and we study the corresponding Lie algebras, which can potentially be infinite dimensional. To characterize such factorization, we identify conditions for these Lie algebras to be finite dimensional. We consider cases where each free Hamiltonian term is itself an element of the generated Lie algebra. In our approach, we develop new tools to systematically divide skew-hermitian bosonic operators into appropriate subspaces, and construct specific sequences of skew-hermitian operators that are used to gauge the dimensionality of the Lie algebras themselves. The significance of our result relies on conditions that constrain only the independently controlled generators in a particular Hamiltonian, thereby providing an effective algorithm for verifying the finiteness of the generated Lie algebra. In addition, our results are tightly connected to mathematical work where the polynomials of creation and annihilation operators are known as the Weyl algebra. Our work paves the way for better understanding factorization of bosonic dynamics relevant to quantum control and quantum technology.

Figures (5)

• Figure 1: Factorization algorithm: A pictorial representation of the general idea behind the approach to verifying whether a given Hamiltonian $H(t)$ will lead to a finite factorization of the quantum dynamics. First, the individual components of the Hamiltonian must be identified. These components will form the set $\mathcal{G}$ of generators of the Lie algebra $\mathfrak{g}$. Then, a set of conditions is applied to the generators $\mathcal{G}$, and to a subset of their commutators $[\mathcal{G},\mathcal{G}]$. If all conditions are verified, the Lie algebra $\mathfrak{g}$ has a finite dimension. Violation of any condition guarantees an infinite-dimensional Lie algebra $\mathfrak{g}$.
• Figure 2: Commutator Chains of Type I: Visualization of how a Commutator Chain of Type I, which is denoted by $C_\sigma^I(\gamma,\tilde{\gamma})$ and has elements $z^{(\ell)}_\sigma$ (highlighted in red), is constructed starting from $z^{(0)}_\sigma:=g_\sigma^{\gamma}$ and $g_+^{\tilde{\gamma}}$ with $\gamma=(\alpha,\beta)$, $\tilde{\gamma}=(\tilde{\alpha},\tilde{\alpha})$, and $\sigma \in\{+,-\}$. The arrows indicate the location of the respective element in a commutator.
• Figure 3: Commutator Chains of Type II: Visualization of a Commutator Chain of Type II which is denoted by $C_\sigma^{II}(\gamma)$ and has elements $y^{(\ell)}_\sigma$ (highlighted in red). It is constructed starting from $y^{(0)}_\sigma:=g_\sigma^{\gamma} \in \hat{A}_n$ with $\gamma=(\alpha,\beta)$ and $\sigma \in\{+,-\}$.
• Figure 4: Flow chart for one direction in the proof of our main result as given by Theorem \ref{['main:theorem_final']}. We proceed by showing that negating any condition as outlined in the claim results in a infinite-dimensional Lie algebra $\mathfrak{g}$. The reverse direction is not depicted as a more direct proof is possible.
• Figure 5: General properties of Commutator Chains of Type I and II. Different classes occurring as commutator chains are highlighted to provide a qualitative understanding of these mathematical objects. For both types, there are three major behaviors: (i) red chains are those for which $z^{(\ell)}_\sigma=0$ or $y^{(\ell)}_\sigma=0$ for $\ell\geq l_*$. For type I, it always occurs that $\ell_*=1$, while $\ell_*$ can take other values for type II. (ii) Blue chains are those for which the degrees $\deg(z^{(\ell)}_\sigma)$ and $\deg(y^{(\ell)}_\sigma)$ increase monotonically and without bound as a function of $\ell$. (iii) Finally, the last type of chain differentiates between the green chains of Type I for which $|\tilde{\alpha}|=1$ (or $|\tilde{\gamma}|=2$) and the purple chains of Type II for which $|\gamma|=2$. Note that here $\mathfrak{sp}(2n,\mathbb{R})$ is the Lie algebra of the symplectic group.

Theorems & Definitions (82)

• Definition
• Example 1: Hamiltonian with drift
• Example 2: Hamiltonian without drift
• Example 3
• Definition 4
• Lemma 5
• proof
• Definition 6
• Example 7: Single elements
• Example 8: Commutator