# Are Thermal states Harmonic oscillators?

So firstly when we talk about a 2-state system, like a qubit, it has dimension d=2, no?

But what if we consider a 3-state system? A qutrit as far as I know. Is that considered a harmonic oscillator?

Right now I am working on a project which considers thermal states density matrix $$\tau(\beta)$$ with the Hamiltonian $$H = \sum_n \omega n \mid n\rangle\langle n \mid$$ where a the density matrix corresponding to the thermal state generally is defined as $$\tau(\beta) = \frac{e^{-\beta H}}{\mathrm{Tr}(e^{-\beta H})} = \frac{1}{\mathrm{Tr}(e^{-\beta H})} \begin{bmatrix} P_1 & 0 & 0 &.. \\ 0 & P_2 & 0 & ..\\ 0 & 0 & P_3& ..\\ .. &.. & .. & ..\\ \end{bmatrix}$$

Is this matrix also a harmonic oscillator? In fact is any state that $$\rho = \sum_n^d n \mid n \rangle\langle n\mid$$ for d $$\ge 2$$ a harmonic oscillator?

I'm sorry I just have a hard time to conceputally grasp this part.

Edit: Updated the density matrix of $$\tau(\beta)$$
• This seems extremely confused. What definition of the term "harmonic oscillator" are you using? How can a state be a harmonic oscillator? It's like saying, classically, that "$x = 2 \, \mathrm{meters}$" is a harmonic oscillator. – knzhou Jan 11 at 15:27
• Yes sorry, I updated the description. So $\tau(\beta)$ is not a specific state, it is a density matrix. But you are also right, I am extremely confused how/why Harmonic oscillators have anything to do with this. I can do all my calculations with the matrix $\tau(\beta)$ and $H$ and don't need any harmonic oscillators here. But still it seems rather important so I thought I should look into it to understand this – CatoMaths Jan 11 at 15:41