The information basis is given by the vectors \(|0 \rangle\) and \(|1 \rangle\). If a qubit is in one of the two information basis vectors, then the measurement outcomes have all-or-nothing probability:
\begin{align*}
\langle 0 | 0 \rangle &= 1, & \langle 1 | 0 \rangle &= 0, & \langle 0 | 1 \rangle &= 0, & \langle 1 | 1 \rangle &= 1
\end{align*}
The two basis vectors in the (fair) Bell basis are given by
\begin{align*}
|{+}\rangle &= \frac{1}{\sqrt{2}} \left( |0 \rangle + |1 \rangle \right) & |{-}\rangle &= \frac{1}{\sqrt{2}} \left( |0 \rangle - |1 \rangle \right)
\end{align*}
If a qubit is in one of the two (fair) Bell basis vectors, measurement outcomes are equally likely:
\begin{align*}
\langle 0 | {+} \rangle &= \frac{1}{\sqrt{2}}, & \langle 1 | {+} \rangle &= \frac{1}{\sqrt{2}}, & \langle 0 | {-} \rangle &= \frac{1}{\sqrt{2}}, & \langle 1 | {-} \rangle &= -\frac{1}{\sqrt{2}}
\end{align*}
The Hadamard matrix,
\begin{equation*}
H = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}
\end{equation*}
takes one information basis vector and turns it into a Bell basis vector:
\begin{align*}
H |0 \rangle &= |{+} \rangle & H |1 \rangle &= |{-} \rangle
\end{align*}
Since each information basis measurement of a Bell basis vector is equally likely, you can think of this basis as a fair coin.
So far, this Hadamard matrix has appear greatly in the quantum information MOOC. You can also have unfair coins, corresponding to a biased Bell basis. Here is one such biased basis:
\begin{align*}
|{+}_{\epsilon} \rangle &= \sqrt{\frac{1 - \epsilon}{2}} |0\rangle + \sqrt{\frac{1 + \epsilon}{2}} |1\rangle & |{-}_{\epsilon} \rangle &= \sqrt{\frac{1 + \epsilon}{2}} |0\rangle - \sqrt{\frac{1 - \epsilon}{2}} |1\rangle
\end{align*}
Here we must require
\begin{equation*}
{-1} \leq \epsilon \leq 1
\end{equation*}
in order for the probability to be real and normalized. Note that the biased basis is ortho-normal:
\begin{align*}
\langle {+}_{\epsilon} | {+}_{\epsilon} \rangle &= 1, & \langle {-}_{\epsilon} | {+}_{\epsilon} \rangle &= 0, & \langle {+}_{\epsilon} | {-}_{\epsilon} \rangle &= 0, & \langle {-}_{\epsilon} | {-}_{\epsilon} \rangle &= 1
\end{align*}
However, information basis measurements are now not equally likely:
\begin{align*}
\langle 0 | {+}_{\epsilon} \rangle &= \sqrt{\frac{1 - \epsilon}{2}}, & \langle 1 | {+}_{\epsilon} \rangle &= \sqrt{\frac{1 + \epsilon}{2}}, & \langle 0 | {-}_{\epsilon} \rangle &= \sqrt{\frac{1 + \epsilon}{2}}, & \langle 1 | {-}_{\epsilon} \rangle &= -\sqrt{\frac{1 - \epsilon}{2}}
\end{align*}
Note that \(\epsilon = 0\) reduces to the fair Bell basis.
The matrix that produces the biased Bell basis is
\begin{equation*}
H_{\epsilon} = \frac{1}{\sqrt{2}} \begin{pmatrix} \sqrt{1 - \epsilon} & \sqrt{1 + \epsilon} \\ \sqrt{1 + \epsilon} & -\sqrt{1 - \epsilon} \end{pmatrix}
\end{equation*}
I suppose that the fairness of the usual Bell basis is important for the algorithms where this gate is used. But can you do anything useful with a biased Bell basis?
