Biased Bell Basis | M.E. Irizarry-Gelpí

Mon 05 March 2018
Physics
#8.370.2x

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?