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