Bell State
- [[quantum-state]]
- created by using [[hadamard-gate]] on the control bit of a [[controlled-not]]: $
\left(\operatorname{CNOT}_{1 \to 2}\right) \left(\mathbf{H} \otimes \mathbf{1}\right)$
- $
\ket{\Phi^+} = \frac{1}{\sqrt{2}} \left[\ket{00} + \ket{11}\right]$ from $\ket{00}$
- $
\ket{\Psi^+} = \frac{1}{\sqrt{2}} \left[\ket{01} + \ket{10}\right]$ from $\ket{01}$
- $
\ket{\Phi^-} = \frac{1}{\sqrt{2}} \left[\ket{00} - \ket{11}\right]$ from $\ket{10}$
- $
\ket{\Psi^-} = \frac{1}{\sqrt{2}} \left[\ket{01} - \ket{10}\right]$ from $\ket{11}$
- the four bell states form an [[orthonormal]] [[basis]]
- for an unentangled 2-[[qubit]] state $
\ket{\varphi} = \left(a\ket{0}+b\ket{1}\right)\otimes\left(c\ket{0}+d\ket{1}\right)$, the [[inner-product]] with any bell state is bounded $\braket{\varphi|\text{B}_{ij}} \leq \frac{1}{\sqrt{2}}$