# Information bearing degrees of freedom of a quantum simple harmonic oscillator

I am trying to make sense of arXiv:physics/0210005. I am confused with the concept of information bearing degrees of freedom of a system mentioned at the very beginning. To verify the arguments of the paper, I would like to start with a 1-D quantum simple harmonic oscillator. We know that it has two degrees of freedom, momentum and position. You can also use generalized degrees of freedom if you want. My question is which of these are the information bearing degrees of freedom?

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In your example, the classical harmonic oscillator has two classical degrees of freedom, position and momentum. In the quantum case these two are not independent, because the state in position space is the Fourier transform of the state in momentum space. One finds instead that there is only really one "degree of freedom": the number of excitations. Each number state is orthogonal to every other: $\langle m | n \rangle = \delta_{m,n}$, which means that they are distinguishable in principle. There can be any number of excitations in the system, so in theory one can store an infinite number of different "words" or bit-strings in a single harmonic oscillator. In practice the higher number states require prohibitively large amounts of energy to prepare.