Figures of Merit for Channels

The current package has built-in capability for computing the following figures of merit. The choice of the figure of merit can be specified using the fig_of_merit= parameter to QPtomographer.bistates.run() or QPtomographer.channelspace.run().

The diamond norm distance

The diamond norm distance between two channels \(\mathcal{E}_{X\to Y}\) and \(\mathcal{F}_{X\to Y}\) is defined as

\[\frac{1}{2} \left\lVert \mathcal{E} - \mathcal{F} \right\rVert_{\diamond} = \sup_{\sigma_{XR}} \frac{1}{2} \left\lVert \mathcal{E}(\sigma_{XR}) - \mathcal{F}(\sigma_{XR}) \right\lVert_{1}\ ,\]

where the supremum is taken over all bipartite states on \(X\) and a reference system \(R\simeq X\). In the right hand side, the channels act on the system \(X\) and act as the identity channel on \(R\). The optimization may be restricted to pure states without loss of generality.

The diamond norm distance can be formulated as the following semidefinite program, in terms of a positive semidefinite variable \(Z_{YR} \geqslant 0\) and a real variable \(\alpha\):[1]

\[\begin{split}\begin{array}{rc} \frac{1}{2} \left\lVert \Delta_{X\to Y} \right\rVert_{\diamond} \quad = \quad \mbox{minimize:} \quad & \alpha \qquad\ , \\ \mbox{subject to:}\quad & \mbox{tr}_Y(Z_{YR}) \leqslant \alpha\mathbb{I}_{R} \\ & Z_{YR} \geqslant \Delta_{X\to Y}(\tilde{\Phi}_{X:R}) \end{array}\end{split}\]

where we have defined

\[{|\tilde\Phi\rangle}_{X:R} = \sum_k {|k\rangle}_X\otimes{|k\rangle}_R\ .\]

The entanglement fidelity for channels

An alternative quantity which is relevant to quantify the distance of a channel \(\mathcal{E}_{X\to X'}\) (where \(X'\simeq X\)) to the identity channel \(\mbox{id}_{X\to X'}\) is the so-called entanglement fidelity

\[F_{e}(\mathcal{E}_{X\to X'}) = F^2(\mathcal{E}_{X\to X'}(\varphi_{XR}), \varphi_{XR}) = {\langle\varphi|}_{XR} \mathcal{E}_{X\to X'}({|\varphi\rangle}{\langle\varphi|}_{XR}) {|\varphi\rangle}_{XR}\ ,\]

where we have defined the maximally entangled state \(|\varphi\rangle_{XR}\) (using a reference system \(R\simeq X\)) and the (square) fidelity between states \(F^2(\rho,\sigma)\), as:

\[|\varphi\rangle_{XR} = \frac{1}{\sqrt{\mbox{dim}(X)}} \, \sum_k {|k\rangle}_X\otimes{|k\rangle}_{R}\ ;\]
\[F^2(\rho,\sigma) = \left\lVert \sqrt{\rho} \sqrt{\sigma} \right\rVert_{1}^{2} \ .\]

The worst-case entanglement fidelity

Finally, one may optimize the entanglement fidelity over all possible input states and seek out the worst case, instead of computing the entanglement fidelity only for the maximally entangled state. Hence, the worst-case entanglement fidelity of a channel \(\mathcal{E}_{X\to X'}\) (where \(X'\simeq X\)) is defined as

\[F_{\mbox{worst}}(\mathcal{E}_{X\to X'}) = \inf_{\sigma_{XR}} F^2(\mathcal{E}_{X\to X'}(\sigma_{XR}), \sigma_{XR})\ ,\]

where the optimization ranges over all bipartite quantum states defined over the input and a reference system \(R\simeq X\), and where \(F(\rho,\sigma)\) is defined as above.

The optimization may be restricted to pure states without loss of generality, because the minimum of a concave function is over a convex set is reached at the boundary of the set.

The worst-case entanglement fidelity may be computed by evaluating the following semidefinite program (as described in arXiv:1808.00358), in terms of the real variable \(\mu\) and positive semidefinite variable \(\rho_X \geqslant 0\):

\[\begin{split}\begin{array}{rcl} F_{\mbox{worst}}(\mathcal{E}_{X\to X'})\quad = \quad\mbox{minimize:}\quad & \mu \qquad &\ , \\ \mbox{subject to:}\quad & \mbox{tr}(\rho_X) = 1 &\\ & \left[\begin{array}{c|c} \mathbb{I}\vphantom{\Bigg[]} & M_{X'R}^\dagger \mbox{vec}(\rho_X) \\ \hline \mbox{vec}(\rho_X)^\dagger M_{X'R} & \mu \end{array}\right] \geqslant 0 & \end{array}\end{split}\]

where \(M_{X'R}\) is a factorization of the Choi matrix of the process, satisfying:

\[M_{X'R} \, M_{X'R}^\dagger = \mathcal{E}_{X\to X'}(\tilde\Phi_{XR})\ ,\]

with \({|\tilde\Phi\rangle}_{XR}\) as defined above. The factorization can be obtained using a Cholseky or LDLT factorization, for instance; or more generally by computing any matrix square root. The unitary freedom of the matrix square root is irrelevant here.

In the above, the operation \(\mbox{vec}(\rho)\) “vectorizes” the given operator by stacking its coefficients into one large column; it is more precisely defined as

\[\mbox{vec}(\rho) = (\mathbb{I}\otimes\rho) {|\tilde{\Phi}\rangle}_{XR}\ .\]

References

[1]Watrous, J. (2009). Semidefinite Programs for Completely Bounded Norms. Theory of Computing, 5(11), 217–238; arXiv:0901.4709