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--- basis-state [May 11, 2026 at 13:07] – yanevskiv
+++ basis-state [May 14, 2026 at 11:38] (current) – external edit 127.0.0.1
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+# Basis state
+**Basis states** are set of two or more quantum states out of which all other quantum states can be built through linear combination.
+A qubit has two basis states. The standard basis states are $\lvert 0\rangle$ and $\lvert 1\rangle$. This notation is called the ket-notation. It's a way to give a name to the basis states. The names inside kets have no meaning. We could have just as easily said $\lvert\text{cat}\rangle$ and $\lvert\text{dog}\rangle$. What's important is that there are exactly two distinct quantum states. These are the states we associate complex probability amplitudes to. If we had more than two states, we'd have a [qudit](https://en.wikipedia.org/wiki/Qudit). A qubit is just a special case when the number of states is $d = 2$.
+We've seen that the qubit state $\lvert\psi\rangle$ is described by two complex numbers $a,b\in\mathbb C$. We might as well write it as a column vector in $\mathbb C^2$, since this will lets us take advantage of linear algebra. The space of all possible states is called a Hilbert space. In this case, Hilbert space is the space of all possible column vectors in $\mathbb C^2$.
+$$ \lvert\psi\rangle = \begin{pmatrix}a\\b\end{pmatrix},\quad\lvert\psi\rangle\in\mathbb C^2$$
+We can then write the column vector $\lvert\psi\rangle$ in the following form. In which case, we recognize the familiar expression $\lvert\psi\rangle = a\lvert 0\rangle + b\lvert 1\rangle$:
+$$\lvert\psi\rangle = a\begin{pmatrix}1\\0\end{pmatrix} + b\begin{pmatrix}0\\1\end{pmatrix}$$
+So, when it comes to a qubit, everything just boils down to 2-dimensional linear algebra over complex numbers. Basis states are then just special type of vectors in this space:
+$$\lvert 0\rangle = \begin{pmatrix}1\\0\end{pmatrix},\quad\lvert 1\rangle = \begin{pmatrix}0\\1 \end{pmatrix}$$
+In fact, in Hilbert space $\mathbb C^2$, all kets $\lvert \psi\rangle$ are column vectors, all bras $\langle \phi\lvert$ are just row vectors, and all operators $H$ are just 2x2 matrices (the elements are, of course, complex numbers). This comes from the fact a qubit only has two states and the Hilbert space $\mathbb C$ was enough to capture all possible states.
+In quantum mechanics more broadly, there may be infinite number of states. In that case, states are not 2-dimensional vectors, but rather infinite dimensional. At that point, $\mathbb C^2$ is not enough to capture all possible states, so other kinds of Hilbert spaces are used.