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--- basis-state [May 14, 2026 at 11:38] – external edit 127.0.0.1
+++ basis-state [June 13, 2026 at 03:13] (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.
+**Basis state** is a state vector that is part of a basis. A basis is 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 $d = 2$.
+A qubit is a two-level quantum system so has two basis states. The standard basis $\lvert 0\rangle, \lvert 1\rangle}$. 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 $\mathbb C^2$.