Quantum channel estimation problem

In a paradigmatic quantum metrological problem, the goal is to estimate the value of a single parameter \(\theta\) encoded in some physical process described by a quantum channel (formally a Completely Positive Trace-Preserving map – CPTP):

\[\Lambda_\theta: \mathcal{L}(\mathcal{I}) \mapsto \mathcal{L}(\mathcal{O}),\]

where \(\mathcal{I}\), \(\mathcal{O}\) are Hilbert spaces of input and output quantum systems and \(\mathcal{L}(\mathcal{H})\) is a set of linear operators on \(\mathcal{H}\), that is a superset of density matrices on \(\mathcal{H}\).

The channel \(\Lambda_\theta\) can be probed by an arbitrary \(\theta\)-independent input state

\[\rho_0 \in \mathcal{L}(\mathcal{I} \otimes \mathcal{A}),\]

where \(\mathcal{A}\) is an auxiliary system called an ancilla; \(\Lambda_\theta\) does not act on \(\mathcal{A}\), but the entanglement between \(\mathcal{I}\) and \(\mathcal{A}\) may sometimes be used to enhance the estimation precision in the presence of noise [22, 23, 28, 29]. The resulting output state is

\[\rho_\theta := \left( \Lambda_\theta \otimes \mathbb{I}_{\mathcal{A}} \right)(\rho_0) \in \mathcal{L}(\mathcal{O} \otimes \mathcal{A}),\]

where \(\mathbb{I}_{\mathcal{A}}\) represents the identity map on \(\mathcal{L}(\mathcal{A})\)—note the notational distinction from \(\mathbb{1}_{\mathcal{A}}\), which denotes identity operator on the \(\mathcal{A}\) space.

The state \(\rho_\theta\) is measured by a generalized measurement \(\Pi=\{\Pi_i\}\) resulting in the probability of outcome \(i\) equal to \(p_\theta(i) := \mathrm{Tr}\!\left(\rho_\theta \Pi_i\right)\).

The parameter \(\theta\) is then estimated using an estimator \(\tilde\theta(i)\) and the goal is to find the protocol producing results as close as possible to the true value of \(\theta\). In case of the simplest single-channel estimation strategy, as depicted in Fig. 1 (A), the optimization of the protocol is equivalent to optimizing the input state \(\rho_0\) and the measurement \(\{\Pi_i\}\). However, more complex protocols may additionally involve optimization over control operations \(\textrm{C}_i\), as in scenarios (C) and (D) in Fig. 1.

In general, the exact form of the optimal protocol depends on the actual estimation framework chosen, be it frequentist or Bayesian [30, 31, 32], as well as the choice of the cost function/figure of merit to be minimized/maximized. In this work, we adopt the frequentist approach, and focus on optimizing the QFI as the figure of merit to be maximized.