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In a paper I am reading, it states:

For open-loop coherent controllability a quantum system with Hamiltonian $H$ is open-loop controllable by a coherent controller if and only if the algebra $\mathcal{A}$ generated from $\{ H, H_i \}$ by commutation is the full algebra of Hermitian operators for the system.

How would you produce an algebra from the set $\{ H, H_i \}$ using commutation? What is the basic idea in this regard?

Frederik vom Ende
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John Doe
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1 Answers1

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In general, an algebra $\mathcal{A}$ generated from a set $\{H_1, H_2,..., H_n\}$ by commutation refers to the algebra whose generators are $H_1,H_2,...,H_n$, all their first-order commutators $C_{ij} = [H_i,H_j]$, and all their second-order commutators $C_{ijk} = [[H_i, H_j],H_k]$ and so on.

Sanchayan Dutta
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