# Are Thermal states Harmonic oscillators?

Excuse me if I use somewhat wrong terminology. But I've always been confused about this.

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.

Thanks in advance !

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
• Can you just explain what you think the term "harmonic oscillator" means? Otherwise the only possible answer to your question is just -- "no, that question doesn't even make sense". – knzhou Jan 11 at 15:42
• Well from what I've gathered is that Harmonic oscillators can be used to somehow model physical systems. And I'm working on a project where I am given a thermal state (which has minmal energy) and am supposed to increase the energy of said system via 2-level rotations. Quote 'However, for larger systems such as registers of many qubits, arbitrary global operations may be difficult to realize and call for more specialized practical solutions. In particular, this applies to infinite-dimensional quantum systems such as (ensembles of) harmonic oscillators.' – CatoMaths Jan 11 at 15:49