# What is the $\left| 22\right>$ state?

I came across with a problem that involves $2$ quantum trits in state $\left| 22 \right>.$ What is it's tensor product interpretation and a matrix interpretation?

• I doubt you mean quantum bits since that implies that the quantum system only has levels 0 and 1. Do you know if they're definitely qutrits, i.e. 3-level systems (0,1 and 2)? – DaftWullie Jun 11 '18 at 10:33
• Where did you come across that ? Depending on the context there may be different interpretations of it. – Sanchayan Dutta Jun 11 '18 at 10:34
• Sorry, I should have written qutrits, not quantum bits. I thought I had to interpret it with 3 vectors of size 2 each, so I was stuck there. – Archil Zhvania Jun 11 '18 at 11:03

It is worth emphasising that the stuff that you write inside a ket is completely arbitrary. It's just a label you're attaching to something, so it should have a proper definition somewhere. Now, usually, we're talking about quantum spins with a certain size of Hilbert space, say $d$. Probably here you're talking $d\geq 3$, and perhaps specifically $d=3$. Then, one set of basis states is often written as $|i\rangle$ for $i=0,1,\ldots d-1$. You can choose to represent these as vectors $(0,0,\ldots,0,1,0,0,\ldots 0)$ where the 1 is the $i+1$th entry, and there are $d$ entries. So, I guess you're talking about $$|2\rangle\equiv\left(\begin{array}{c} 0 \\ 0 \\ 1 \end{array}\right)$$ The tensor product $|2\rangle\otimes|2\rangle$ then has the standard meaning.