# Why are the basis unit vectors |0⟩ and |1⟩ written as column vectors [1, 0] and [0, 1], respectively?

I am starting to learning quantum computing and am currently reading Quantum Computing: A Gentle Introduction. As I started reading into the first section after the Introduction, it talks about basis vectors |0⟩ = |↑⟩ and |1⟩ = |→⟩ along with this figure:

However, what confuses me is why |0⟩ = |↑⟩ and |1⟩ = |→⟩ are, respectively, represented as $$\begin{pmatrix}1 \\ 0\end{pmatrix}$$ $$\begin{pmatrix}0 \\ 1\end{pmatrix}$$

Shouldn't |↑⟩ be represented as $$\begin{pmatrix}0 \\ 1\end{pmatrix}$$

and shouldn't |→⟩ be represented as $$\begin{pmatrix}1 \\ 0\end{pmatrix}$$, since those appear to be their heads' coordinates on the circle?

• Just rotate coordinates system 90 degrees and everything will be right. There is no reason why you cannot do so. Commented May 8, 2023 at 20:52

Which physical state or one on the sphere we call $$|0\rangle$$ and which we call $$|1\rangle$$ has no impact whatsoever on any sort of analysis, as the labels 0 and 1 are completely arbitrary.