Bell state

The Bell states form an entangled orthonormal basis for a two particle system

\begin{align*} \ket{\phi_0} &= \ket{\phi^+} = \frac{1}{\sqrt 2} \left( \ket{\upup} + \ket\downdown \right) \\ \ket{\phi_3} &= \ket{\phi^-} = \frac{1}{\sqrt 2} ( \ket\upup - \ket\downdown ) \\ \ket{\phi_1} &= \ket{\psi^+} = \frac{1}{\sqrt 2} (\ket\updown + \ket\downup) \\ \ket{\phi_2} &= \ket{\psi^-} = i \frac{1}{\sqrt 2} ( \ket\updown - \ket \downup ). \end{align*}

\begin{align*} f(x) = \int dx\, g(x) \end{align*}