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--- state-vector [May 11, 2026 at 14:58] – created yanevskiv
+++ state-vector [May 14, 2026 at 11:38] (current) – external edit 127.0.0.1
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+# State vector
+**State vector** is the vector representation of a quantum state.
+Quantum state is often written as a linear combination of basis states where the coefficients are  probability amplitudes. For example, a qubit is often written in the following way using Dirac notation.
+$$\lvert\psi\rangle = a\lvert 0\rangle + b\lvert 1\rangle$$
+But it's also a postulate of quantum mechanics that every quantum state lives in a Hilbert space. For qubits, that Hilbert space is $\mathbb{C}^2$. Hilbert space is by definition a complete inner product space. An inner product space is a vector space equipped with an inner product. Since $\mathbb{C}^2$ is a vector space, and $\lvert\psi\rangle$ is postulated to live in it, we can write the qubit as a column vector in $\mathbb{C}^2$:
+$$\lvert\psi\rangle = \begin{pmatrix}a\\b\end{pmatrix}\qquad a,b\in\mathbb{C}$$
+Because Hilbert space satisfies vector space axioms we can also do a linear decomposition:
+$$\lvert\psi\rangle = a\begin{pmatrix}1\\0\end{pmatrix} + b\begin{pmatrix}0\\1\end{pmatrix}$$
+From this, we see a natural definition of $\lvert 0\rangle$ and $\lvert 1\rangle$
+$$\lvert 0\rangle = \begin{pmatrix}1\\ 0\end{pmatrix}$$
+$$\lvert 1\rangle = \begin{pmatrix}0\\ 1\end{pmatrix}$$
+But this is just a convention. We could have chosen any other complex numbers to do define $\lvert 0\rangle$ and $\lvert 1\rangle$ and quantum mechanics would have worked the same way. Thus a state vector is just a linear algebra representation of a quantum state.