# Is harmonic oscillator continuous variable system?

In the literature I have seen that the notions "our system is continuous variable system", "Hilbert space of our system is infinite" were used as if they were equivalent.

For example for harmonic oscillator the Hilbert space is infinite, but I can write my system in discrete basis (with number operator eigenvectors) or continuous basis (position operator eigenvectors).

So I'm a little confused:

• What is the definition of "continuous variable system"?

• Is harmonic oscillator continuous variable system?

• Which literature? Which pages? – Qmechanic Jul 28 '15 at 15:36
• For example I have some paper of Horodecki,Lewenstein (which actually was not published) about bound entanglement for continuous variables. – Agnieszka Jul 29 '15 at 7:25

1. The terminology "continuous variable system" is non-standard, but likely refers to the fact that any canonical quantization of a classical Hamiltonian system (i.e. a system described by a continuous phase space) must have an infinite-dimensional Hilbert space since the canonical commutation relation $$[x,p] = \mathrm{i}\mathbf{1}$$ cannot be realized on finite-dimensional spaces.