Introduction to Bell states
Bell states are a set of four maximally entangled two-qubit quantum states. For two photons with distinct external states \nu_1 and \nu_2, and each photon’s polarization in a two-dimensional space spanned by |\mathbf x\rangle and |\mathbf y\rangle, the Bell states are defined as follows:
\begin{aligned} |\boldsymbol \Psi^+(\nu_1, \nu_2)\rangle &= \frac{1}{\sqrt{2}} \left( |\mathbf x_1, \mathbf y_2\rangle + |\mathbf y_1,\mathbf x_2\rangle \right) \\ |\boldsymbol \Psi^-(\nu_1, \nu_2)\rangle &= \frac{1}{\sqrt{2}} \left( |\mathbf x_1, \mathbf y_2\rangle - |\mathbf y_1, \mathbf x_2\rangle \right) \\ |\boldsymbol \Phi^+(\nu_1, \nu_2)\rangle &= \frac{1}{\sqrt{2}} \left( |\mathbf x_1, \mathbf x_2\rangle + |\mathbf y_1, \mathbf y_2\rangle \right) \\ |\boldsymbol \Phi^-(\nu_1, \nu_2)\rangle &= \frac{1}{\sqrt{2}} \left( |\mathbf x_1, \mathbf x_2\rangle - |\mathbf y_1, \mathbf y_2\rangle \right) \end{aligned}
To ensure these states form a valid basis in quantum mechanics, it is necessary to verify that they are normalized and mutually orthogonal. Let’s demonstrate this for the Bell state |\boldsymbol \Psi^+(\nu_1, \nu_2)\rangle.
First, let’s check the normalization of |\boldsymbol \Psi^+(\nu_1, \nu_2)\rangle:
\begin{aligned} \langle \boldsymbol \Psi^+(\nu_1, \nu_2) | \boldsymbol \Psi^+(\nu_1, \nu_2) \rangle =& \frac{1}{2} \left( \langle \mathbf x_1, \mathbf y_2| + \langle \mathbf y_1, \mathbf x_2| \right) \left( |\mathbf x_1, \mathbf y_2\rangle + |\mathbf y_1, \mathbf x_2\rangle \right) \\ =& \frac{1}{2} \left( \langle \mathbf x_1| \mathbf x_1\rangle \langle \mathbf y_2| \mathbf y_2\rangle + \langle \mathbf x_1| \mathbf y_1\rangle \langle \mathbf y_2| \mathbf x_2\rangle \right. \\ & \left. + \langle \mathbf y_1| \mathbf x_1\rangle \langle \mathbf x_2| \mathbf y_2\rangle + \langle \mathbf y_1| \mathbf y_1\rangle \langle \mathbf x_2| \mathbf x_2\rangle \right) \\ =& \frac{1}{2} \left( 1 \cdot 1 + 0 \cdot 0 + 0 \cdot 0 + 1 \cdot 1 \right) = 1 \end{aligned}
This calculation shows that |\boldsymbol \Psi^+(\nu_1, \nu_2)\rangle is indeed normalized.
Next, let’s verify the orthogonality of |\boldsymbol \Psi^+(\nu_1, \nu_2)\rangle with another Bell state, for instance |\boldsymbol \Psi^-(\nu_1, \nu_2)\rangle:
\begin{aligned} \langle \boldsymbol \Psi^+(\nu_1, \nu_2) | \boldsymbol \Psi^-(\nu_1, \nu_2) \rangle =& \frac{1}{2} \left( \langle \mathbf x_1, \mathbf y_2| + \langle \mathbf y_1, \mathbf x_2| \right) \left( |\mathbf x_1, \mathbf y_2\rangle - |\mathbf y_1, \mathbf x_2\rangle \right) \\ =& \frac{1}{2} \left( \langle \mathbf x_1| \mathbf x_1\rangle \langle \mathbf y_2| \mathbf y_2\rangle - \langle \mathbf x_1| \mathbf y_1\rangle \langle \mathbf y_2| \mathbf x_2\rangle \right. \\ & \left. + \langle \mathbf y_1| \mathbf x_1\rangle \langle \mathbf x_2| \mathbf y_2\rangle - \langle \mathbf y_1| \mathbf y_1\rangle \langle \mathbf x_2| \mathbf x_2\rangle \right) \\ =& \frac{1}{2} \left( 1 \cdot 1 - 0 \cdot 0 + 0 \cdot 0 - 1 \cdot 1 \right) = 0 \end{aligned}
This result confirms that |\boldsymbol \Psi^+(\nu_1, \nu_2)\rangle and |\boldsymbol \Psi^-(\nu_1, \nu_2)\rangle are orthogonal. Similar calculations can be performed to verify the normalization and mutual orthogonality of all four Bell states, confirming that they form a complete orthonormal basis.
For more insights into this topic, you can find the details here.