I'm sorry in advance if my question is unclear (as it is also unclear for me).
I'm dealing with 2 different sets of generators, namely
- $\{p, Q_n\}$ ($n \in [0,N]$ ) with $Q_n = \sum_{n'=0}^n |n'><n'|$, $p$ being the momentum operator and N the dimension of the truncated algebra associated to the harmonic oscillator.
- $\{p\sigma_z, q\sigma_z, \sigma_x, \sigma_y\}$ where $q$ is the position operator and $\sigma_i$ are Pauli's matrices.
I'm interested in understanding how "fast" these two representations generate the algebra associated with the harmonic oscillator (i.e. I'm not interested in what's the status of the 2-level system), so the only part I'm interested in for the second representation is the one where powers of $p$ and $q$ appear.
I computed the generators of both representations, and in the first case I have $N$ independent commutator, where $N$ is the dimension of the truncated Fock space. In the second case I only have 2 independent commutators if I only consider the fock space (4 if I include the fact that $p$ and $q$ are multiplied by different Pauli's matrices).
So I'd conclude that the first set generates my algebra faster. Moreover, a basis of a truncated fock space is either the power set of $\{p,q\}$ or $\{|l><m|\}$ with $l,m \in [0,N]$. Since the first set generates this second basis in a finite number of steps, while the second set needs infinite amount of steps to generate the first basis, that would also make me think that the first basis is "superior". Although I know this is not rigorous, that's why I'd be grateful if someone could give me some directions about where to go and how to prove this.
I'm mainly interested in the set that generates my algebra faster, not on how simple my set is! Moreover, I'm also interested in understanding how this difference scales with $N$. And that part is also unclear to me.
Thanks in advance, and sorry again if the question is not clear.