Quantum Fisher Information

Given some probabilistic model that specifies probabilities \(p_\theta(i)\) of obtaining different measurement outcomes \(i\) depending on the value of a parameter \(\theta\), one can lower-bound the achievable mean squared error \(\Delta^2 \tilde{\theta}\) of any locally unbiased estimator \(\tilde{\theta}(i)\) via the famous classical Cramér-Rao (CCR) bound [33]:

(1)\[\Delta^2 \tilde\theta \ge \frac{1}{k F_C(p_{\theta})},\]

where \(k\) is the number of independent repetitions of an experiment and \(F_C\) is classical Fisher information (CFI):

(2)\[F_C(p_{\theta}) := \sum_i \frac{\dot p_\theta(i)^2}{p_\theta(i)},\]

where \(\dot{p}_\theta\) denotes the derivative with respect to \(\theta\). The bound is asymptotically saturable, in the limit of many repetitions of an experiment \(k \rightarrow \infty\), with the help of e.g. a maximum-likelihood estimator [33]. Hence, the larger the CFI, the better parameter sensitivity of the model.

In quantum estimation models, the outcomes are the results of the measurement \(\{\Pi_i\}\) performed on a quantum state \(\rho_\theta\), \(p_\theta(i) = \mathrm{Tr}\!\left(\rho_\theta \Pi_i \right)\). Hence, depending on the choice of the measurement, one may obtain larger or smaller CFI. The QFI corresponds to the maximal achievable CFI, when optimized over all admissible quantum measurements [24, 25, 26]:

(3)\[F_Q(\rho_{\theta}) := \max_{\{ \Pi_i \}_i} F_C(p_{\theta}) = \mathrm{Tr}\!\left(\rho_{\theta} L^2\right),\]

where \(L\) is the symmetric logarithmic derivative (SLD) of \(\rho_\theta\):

\[\dot \rho_\theta = \frac{1}{2}\left( \rho_{\theta}L + L\rho_{\theta} \right).\]

Moreover, the optimal measurement can, in particular, be chosen as the projective measurement in the eigenbasis of the SLD.

Combining the above with the CCR bound (1) leads to the quantum Cramér-Rao (QCR) bound that sets a lower bound on achievable estimation variance when estimating a parameter encoded in a quantum state \(\rho_\theta\):

\[\Delta^2 \tilde{\theta} \geq \frac{1}{k F_Q(\rho_\theta)},\]

and is saturable in the limit of \(k \rightarrow \infty\).

This makes the QFI a fundamental concept in quantum estimation theory, and makes it meaningful to formulate optimization problems in quantum metrology in the form of an optimization of the QFI of the final quantum state obtained at the output of a given metrological protocol.

The basic single-channel estimation task, depicted in Fig. 1 (A), corresponds then to the following optimization problem

(4)\[F_Q(\Lambda_\theta ) = \max_{\rho_0} F_Q\!\left[ \Lambda_\theta \otimes \mathbb{I}_\mathcal{A}(\rho_0) \right].\]

We will refer to \(F_Q(\Lambda_\theta)\) as the channel QFI, as it corresponds to the maximal achievable QFI of the output state of the channel, when the maximization over all input states is performed.

In case of protocols with multiple channel uses, \(N > 1\), one could simply probe each channel independently, using independent probes and independent measurements. In this case, by additivity of QFI, one would end up with the corresponding \(N\)-channels QFI — \(F_Q^{(N)}(\Lambda_\theta)\) equal, for this simplistic strategy, to \(N\) times the single channel QFI.

However, one may also consider more sophisticated strategies involving multiple channel uses, which may provide significant advantage over the independent probing strategy, in principle leading even to quadratic scaling of QFI — \(F_Q^{(N)}(\Lambda_\theta)\propto N^2\), referred to as the Heisenberg scaling (HS) [1].

In case of noiseless unitary estimation models \(\Lambda_\theta(\rho) = U_\theta \rho U_\theta^\dagger\) the optimal probe states that maximize the QFI are easy to identify analytically [1] and no sophisticated tools are required. In case of noisy models, however, the actual optimization over the protocol is much more involved, as in principle it requires optimization over multipartite entangled states Fig. 1 (B) or multiple control operations \(\textrm{C}_i\) Fig. 1 (C) or both Fig. 1 (D).

This package allows for optimization of all kinds of metrological strategies, employing ancillary systems, entanglement and control operations, with the only physical restriction that the operations applied have a definite causal order, and hence can be regarded as so-called quantum combs in the sense formalized in [34] and further in this text.

Regarding optimization strategies, we start with the discussion of MOP method, which is applicable to small-scale scenarios, and then move on to discuss a more versatile ISS optimization which can be combined with tensor-network formalism to deal with large-scale metrological problems.