What does $\mathbb{C}^2$ represent?

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I'm getting interested in quantum computing but I lack the mathematical background so I try to catch up.

I've seen that a qubit is represented by a two-dimensional vector space over the complex numbers $\mathbb{C}^2$, but I don't know what $\mathbb{C}^2$ is. The complex number type squared? What does it mean?

Also when are you supposed to be confronted to this notion in school, high school or above ?

When I try to google this google simplify my search by $\mathbb{C}^2$ and all I get are results about speed of light because of the $E=mc^2$ equation.

Thank you

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1 Answer

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$\Bbb{C}^2$ is the set of all ordered pairs of complex numbers, i.e. $(z,w)$ where $z$ and $w$ are complex numbers. I don't think this notation is typically used in high school classes.

More generally, if $A$ and $B$ are sets, then $A \times B$ is called the Certesian product of $A$ and $B$, defined as the set of all ordered pairs $(a,b)$ where $a\in A$ and $b \in B$. For example, if $A = \{1,2\}$ and $B = \{4,5,6\}$, then $A\times B$ has $6$ elements: $(1,4),(1,5),(1,6),(2,4),(2,5),$ and $(2,6)$.

$\Bbb{C}^2$ is just short for $\Bbb{C}\times\Bbb{C}$.

(in case you don't know: a set is a collection of objects, usually numbers; these objects in the set are called elements of that set; if $A$ is a set, then $a\in A$ means "$a$ is an element of $A$").

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