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 $\braket{\varphi|\text{B}_{ij}} \leq \frac{1}{\sqrt{2}}$