Example 1-Qubit Processes

Alexei Gilchrist

4 Quantum Operations

For some example noisy 1-qubit channels we give their representation in terms of Kraus Operators, the dynamical matrix and the process matrix.

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1 Phase Damping

Phase damping removes the coherences between \(|0\rangle\) and \(|1\rangle\) reducing a quantum superposition to a statistical mixture. It can be implemented by the following Kraus operators

\[\begin{equation} \label{eq:phasedamping} K_1 = \begin{pmatrix} 1 & 0 \\ 0 & \sqrt{1-\gamma} \end{pmatrix} \qquad K_2 =\begin{pmatrix} 0 & 0 \\ 0 & \sqrt{\gamma} \end{pmatrix}\end{equation}\]

where the quantum operation is given in the usual way

\[\begin{equation*}\mathcal{E}_\mathrm{pd}(\rho) = \sum_j K_j \rho K_j^\dagger.\end{equation*}\]

It’s easy to verify that \(\sum_j K_j^\dagger K_j = I\) as required.

A problem with the Kraus, or operator-sum, representation is that the decomposition is not unique. It’s possible to have very different Kraus operators that seem to imply quite different physical processes but that really describe the same effect. In this regard the dynamical matrix or the process matrix are better alternatives as they are unique to the process. The dynamical matrix can be calculated from the Kraus operators as,

\[\begin{equation}\label{eq:phasedampingD} D_\mathrm{pd} = K_1^* \otimes K_1 + K_2^* \otimes K_2 = \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & \sqrt{1-\gamma } & 0 & 0 \\ 0 & 0 & \sqrt{1-\gamma } & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} \right)\end{equation}\]

and it acts on a vectorised form of the density matrix: \(|\mathcal{E}_\mathrm{pd}(\rho)\rangle=D_\mathrm{pd}|\rho\rangle\).

To illustrate the ambiguity, we could also form a map that applies a phase-flip, \(Z=\left(\begin{smallmatrix}1&0\\0&-1\end{smallmatrix}\right)\), with probability \(p\) otherwise leaves the state alone, so that the process is

\[\begin{equation*}\mathcal{E}_\mathrm{pd}(\rho) = p\; Z \rho Z^\dagger + (1-p)\; \rho .\end{equation*}\]

This map looks quite different from \eqref{eq:phasedamping} but it will implement the same physical process, which can be seen immediately with the dynamical matrix:

\[\begin{equation*}D_\mathrm{pd} = (1-p)\; I \otimes I + p \;Z^* \otimes Z = \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 2 p-1 & 0 & 0 \\ 0 & 0 & 2 p-1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} \right)\end{equation*}\]

Up to a re-parametrisation with \(p = (1 + \sqrt{1 - \gamma})/2\), it’s exactly the same map.

Yet another map that implements phase damping but is an interesting model on its own, is inefficient measurement. If we were to measure the qubit in the computational basis but discard the result, the effect would be the map

\[\begin{equation*}\mathcal{E}_\mathrm{m}(\rho) = |0 \rangle\langle 0|\rho |0 \rangle\langle 0| + |1 \rangle\langle 1|\rho|1 \rangle\langle 1|\end{equation*}\]

Now, imagine the measurement is inefficient and is only implemented with probability \(p\), so the Kraus operators are:

\[\begin{equation*}K_1 = \sqrt{p}\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} = |0 \rangle\langle 0|\qquad K_2 = \sqrt{p}\begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} = |1 \rangle\langle 1|\qquad K_3 = \sqrt{1-p}\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = I,\end{equation*}\]

which forms the dynamical matrix,

\[\begin{equation*}D_\mathrm{pd} = (1-p) I\otimes I +p (|00 \rangle\langle 00|+|11 \rangle\langle 11|) = \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1-p & 0 & 0 \\ 0 & 0 & 1-p & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} \right).\end{equation*}\]

This is a phase damping map with just another parametrisation.

The final form we are going to represent the process as is the process matrix, \(\rho_{\mathcal{E}_\mathrm{pd}}\). This is a matrix, equivalent to a quantum state, that is given by the Jamiolkowski isomorphism. It is also unique to the process and doesn’t have the freedom the Kraus representation has. It’s most easily obtained by reshuffling the dynamical matrix:

\[\begin{equation*}|\rho_\mathcal{E}\rangle = I\otimes \mathrm{SWAP}\otimes I |D\rangle\end{equation*}\]

The result for phase damping is (after transforming the vector \(|\rho_\mathcal{E}\rangle\) back to a matrix \(\rho_\mathcal{E}\)),

\[\begin{equation}\label{eq:phasedampingrho} \rho_{\mathcal{E}_\mathrm{pd}} = \left( \begin{array}{cccc} 1 & 0 & 0 & \sqrt{1-\gamma } \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \sqrt{1-\gamma } & 0 & 0 & 1 \\ \end{array} \right)\end{equation}\]

where the parametrisation in \eqref{eq:phasedampingD} was used.

Example

Say the mixture \(\rho_0 = p|0 \rangle\langle 0|+(1-p)|1 \rangle\langle 1|\) goes through a phase-damping channel. We expect there to be no change since there are no coherences in the state.

Using the Kraus operators \eqref{eq:phasedamping}:

\[\begin{gather*} \sum_j K_j \rho_0 K_j^\dagger = \begin{pmatrix} 1 & 0 \\ 0 & \sqrt{1-\gamma} \end{pmatrix} \begin{pmatrix} p & 0 \\ 0 & 1-p \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & \sqrt{1-\gamma} \end{pmatrix}^\dagger +\\ \begin{pmatrix} 0 & 0 \\ 0 & \sqrt{\gamma}\end{pmatrix} \begin{pmatrix} p & 0 \\ 0 & 1-p \end{pmatrix} \begin{pmatrix} 0 & 0 \\ 0 & \sqrt{\gamma}\end{pmatrix}^\dagger = \begin{pmatrix} p & 0 \\ 0 & 1-p \end{pmatrix}\end{gather*}\]

Using the dynamical matrix \eqref{eq:phasedampingD}:

\[\begin{equation*} D|\rho_0\rangle = \left(\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1-p & 0 & 0 \\ 0 & 0 & 1-p & 0 \\ 0 & 0 & 0 & 1 \\ \end{array}\right) \begin{pmatrix} p \\ 0 \\ 0 \\ 1-p \end{pmatrix} = \begin{pmatrix} p \\ 0 \\ 0 \\ 1-p \end{pmatrix}\end{equation*}\]

And finally using the process matrix \eqref{eq:phasedampingrho}:

\[\begin{gather*} \mathrm{Tr}_A\{(\rho_0^T\otimes I)\rho_\mathcal{E}\} = \mathrm{Tr}_A\left\{ \begin{pmatrix} p & 0 \\ 0 & 1-p \end{pmatrix}^T\!\!\! \otimes I\; \begin{pmatrix} 1 & 0 & 0 & \sqrt{1-\gamma } \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \sqrt{1-\gamma } & 0 & 0 & 1 \end{pmatrix}\right\}\\ = \mathrm{Tr}_A\left\{ \begin{pmatrix} p & 0 & 0 & (1-p)\sqrt{1-\gamma } \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ p\sqrt{1-\gamma } & 0 & 0 & 1-p\end{pmatrix} \right\} = \begin{pmatrix} p & 0 \\ 0 & 1-p \end{pmatrix}\end{gather*}\]

2 Depolarising Channel

The depolarising channel which drives the qubit to the maximally mixed state \(\rho = I/2\). To implement the channel we apply with probability \(p\) either a Pauli \(I\), \(X\), \(Y\), or \(Z\) is applied; otherwise we leave it alone. That is, the Kraus operators are:

\[\begin{equation*}\left\{ \sqrt{1-3p/4}\;I,\quad \sqrt{p/4}\;X,\quad \sqrt{p/4}\;Y,\quad \sqrt{p/4}\;Z \right\}\end{equation*}\]

The dynamical matrix for a depolarising channel is

\[\begin{equation*}D_\mathrm{dep} = \begin{pmatrix} 1-\frac{p}{2} & 0 & 0 & \frac{p}{2} \\ 0 & 1-p & 0 & 0 \\ 0 & 0 & 1-p & 0 \\ \frac{p}{2} & 0 & 0 & 1-\frac{p}{2} \end{pmatrix}\end{equation*}\]

and the process matrix is,

\[\begin{equation*}\rho_{\mathcal{E}_\mathrm{dep}} = \begin{pmatrix} 1-\frac{p}{2} & 0 & 0 & 1-p \\ 0 & \frac{p}{2} & 0 & 0 \\ 0 & 0 & \frac{p}{2} & 0 \\ 1-p & 0 & 0 & 1-\frac{p}{2} \end{pmatrix}\end{equation*}\]

3 Amplitude Damping

Amplitude damping models a lossy channel that drives the qubit to its ground state \(|0\rangle\). One implementation is the set of Kraus operators

\[\begin{equation*}K_1 = \begin{pmatrix} 1 & 0 \\ 0 & \sqrt{1-\gamma} \end{pmatrix} \qquad K_2 =\begin{pmatrix} 0 & \sqrt{\gamma} \\ 0 & 0 \end{pmatrix},\end{equation*}\]

The map as a dynamical matrix is,

\[\begin{equation*}D_\mathrm{ad} = K_1^* \otimes K_1 + K_2^* \otimes K_2 = \left( \begin{array}{cccc} 1 & 0 & 0 & \gamma \\ 0 & \sqrt{1-\gamma } & 0 & 0 \\ 0 & 0 & \sqrt{1-\gamma } & 0 \\ 0 & 0 & 0 & 1-\gamma \\ \end{array} \right)\end{equation*}\]

and as a process matrix, \begin{equation*} \rho_{\mathcal{E}_\mathrm{ad}} = \begin{pmatrix} 1 & 0 & 0 & \sqrt{1-\gamma } \\ 0 & 0 & 0 & 0 \\ 0 & 0 & \gamma & 0 \\ \sqrt{1-\gamma } & 0 & 0 & 1-\gamma \end{pmatrix}. \end{equation*}