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?

  • 2
    $\begingroup$ Which literature? Which pages? $\endgroup$ – Qmechanic Jul 28 '15 at 15:36
  • $\begingroup$ For example I have some paper of Horodecki,Lewenstein (which actually was not published) about bound entanglement for continuous variables. $\endgroup$ – 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.

  2. Every Hilbert space encountered in physics is separable, that is, it admits a countable (or "discrete") basis in the sense that every vector can be written as an infinite linear combination of the basis vectors. The uncountable (or "continuous") basis of position "eigenvectors" does not, in fact, lie inside the Hilbert space, but every state in the Hilbert space may be written as an integral ("continuous sum") over that basis. The position/momentum eigenvector only lie in a larger space (which is not a Hilbert space), requiring the introduction of a rigged Hilbert space/Gel'fand triple. A discussion of this an other subtleties in the QM formalism can be found in, for example, "Mathematical surprises and Dirac's formalism in quantum mechanics" by Gieres.

  • $\begingroup$ The terminology "continuous variable system" is certainly standard in quantum information. $\endgroup$ – Norbert Schuch Jul 29 '15 at 11:08

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