# How should I interpret $|2\rangle|3\rangle$?

I am a beginner at QC. I was going through the basics of multi-qubits I encountered a state $$|2\rangle|3\rangle$$.

I want clarification on the following points:

1. Can I write $$|2\rangle$$ as $$|10\rangle = |1\rangle|0\rangle$$ always?
2. If $$|2\rangle = |10\rangle = |1\rangle|0\rangle$$, Can I write $$|2\rangle|3\rangle = |10\rangle|11\rangle = |1011\rangle$$?

Assume $$|0\rangle$$, $$|1\rangle$$ in computational basis. $$|0\rangle = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$$, $$|1\rangle = \begin{bmatrix} 0 \\ 1 \end{bmatrix}$$

• I've never seen decimal notation like $|2\rangle$; it is not clear whether it means 2-qubit state $|10\rangle$ or 3-qubit state $|010\rangle$ or other multiqubit state. – kludg Dec 1 '19 at 10:41
• @kludg: It is generaly basis vector representing decimal number 2. You are right that it can be $|011\rangle$ as well, however, it should be clear from dimensionality of a problem. – Martin Vesely Dec 1 '19 at 11:44

1) $$|2\rangle$$ is $$|10\rangle$$, or vector $$\begin{bmatrix}0\\0\\1\\0\end{bmatrix}$$ in Hilbert space representing two q-bits system.
2)Yes, it is possible as well. The first two q-bits in in state $$|10\rangle$$ and last two q-bits in state $$|11\rangle$$, hence the state of all four q-bits is $$|1011\rangle$$, or $$|11\rangle$$ (eleven!) in decimal but I would recommend to avoid writing decimal numbers in this case as they can be confused with binary numbers easily.