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Like my other qubit cheat sheet, but specifically for phase-space-related things.

Phase point operators

These can be used to construct the Wigner function \(W\) given a density matrix \(\rho\). They satisfy

\[\begin{equation} W_{ij} (\rho) = \frac{1}{2} \text{Tr}(\rho A_{ij}) \end{equation}\]

See Wootters 1987 for more on these.

\[\begin{equation} A_{00} = \frac{1}{2}\left( \begin{array}{cc} 2 && 1-i \\ 1+i && 0 \\ \end{array} \right) \end{equation}\] \[\begin{equation} A_{01} = \frac{1}{2}\left( \begin{array}{cc} 2 && -1+i \\ -1-i && 0 \\ \end{array} \right) \end{equation}\] \[\begin{equation} A_{10} = \frac{1}{2}\left( \begin{array}{cc} 0 && 1+i \\ 1-i && 2 \\ \end{array} \right) \end{equation}\] \[\begin{equation} A_{11} = \frac{1}{2}\left( \begin{array}{cc} 0 && -1-i \\ -1+i && 2 \\ \end{array} \right) \end{equation}\]

Wigner function for a qubit

Now apply this to a general qubit of the form

\[\begin{equation} |\psi\rangle = \left( \begin{array}{c} \cos (\frac{\theta}{2}) \\ e^{i\phi}\sin (\frac{\theta}{2}) \\ \end{array} \right) \end{equation}\]

Find that

\[\begin{equation} W_{00} = \frac{1}{4}\left[1 + \cos\theta + \sin\theta(\cos\phi + \sin\phi) \right] \end{equation}\] \[\begin{equation} W_{01} = \frac{1}{4}\left[1 + \cos\theta - \sin\theta(\cos\phi + \sin\phi) \right] \end{equation}\] \[\begin{equation} W_{10} = \frac{1}{4}\left[1 - \cos\theta + \sin\theta(\cos\phi - \sin\phi) \right] \end{equation}\] \[\begin{equation} W_{11} = \frac{1}{4}\left[1 - \cos\theta - \sin\theta(\cos\phi - \sin\phi) \right] \end{equation}\]

Qs

I’m not sure if these have a default notation, but I call them \(Q_i\). I mean the quantities that the rows, columns and diagonals of the Wigner function add up to. Maybe an image would help:

Phase space
\[\begin{equation} Q_i = \frac{1}{2} - \frac{1}{2}\langle\psi | \sigma_i | \psi \rangle . \end{equation}\]

For a general qubit these are:

\[\begin{equation} Q_z = \frac{1}{2} - \frac{1}{2}\cos\theta. \end{equation}\] \[\begin{equation} Q_x = \frac{1}{2} - \frac{1}{2}\sin\theta\cos\phi. \end{equation}\] \[\begin{equation} Q_y = \frac{1}{2} - \frac{1}{2}\sin\theta\sin\phi. \end{equation}\]

Examples

Here are some examples of Wigner functions, written in the form

\[\begin{equation} W = \left[ \begin{array}{c|c} W_{01} & W_{11} \\ \hline W_{00} & W_{10} \\ \end{array} \right] \end{equation}\]

(as in the Wootters paper).

Pauli matrix eigenstates

+1 eigenstate of \(\sigma_z\):

\[\begin{equation} \left[ \begin{array}{c|c} \frac{1}{2} & 0 \\ \hline \frac{1}{2} & 0 \\ \end{array} \right] \end{equation}\]

-1 eigenstate of \(\sigma_z\):

\[\begin{equation} \left[ \begin{array}{c|c} 0 & \frac{1}{2} \\ \hline 0 & \frac{1}{2} \\ \end{array} \right] \end{equation}\]

+1 eigenstate of \(\sigma_x\):

\[\begin{equation} \left[ \begin{array}{c|c} 0 & 0 \\ \hline \frac{1}{2} & \frac{1}{2} \\ \end{array} \right] \end{equation}\]

-1 eigenstate of \(\sigma_x\):

\[\begin{equation} \left[ \begin{array}{c|c} \frac{1}{2} & \frac{1}{2} \\ \hline 0 & 0 \\ \end{array} \right] \end{equation}\]

+1 eigenstate of \(\sigma_y\):

\[\begin{equation} \left[ \begin{array}{c|c} 0 & \frac{1}{2} \\ \hline \frac{1}{2} & 0 \\ \end{array} \right] \end{equation}\]

-1 eigenstate of \(\sigma_y\):

\[\begin{equation} \left[ \begin{array}{c|c} \frac{1}{2} & 0 \\ \hline 0 & \frac{1}{2} \\ \end{array} \right] \end{equation}\]

Examples with negative probabilities

+1 eigenstate of \(\frac{1}{\sqrt{2}} (\sigma_z + \sigma_x)\):

\[\begin{equation} \left[ \begin{array}{c|c} \frac{1}{4} & \frac{1 - \sqrt{2}}{4} \\ \hline \frac{1 + \sqrt{2}}{4} & \frac{1}{4} \\ \end{array} \right] \end{equation}\]

One with the most negative possible value. Think this is an eigenstate of \(\frac{1}{\sqrt{3}} (\sigma_z + \sigma_x + \sigma_y)\) or something, but need to check:

\[\begin{equation} \left[ \begin{array}{c|c} \frac{1 + \sqrt{3}}{4} & \frac{1}{4} \\ \hline \frac{1 - \sqrt{3}}{4} & \frac{1}{4} \\ \end{array} \right] \end{equation}\]