Upper bounds

In Sec. Minimization over purifications we explained that the QFI can be computed via minimization of the norm of \(\alpha\) (6) over all Kraus representations \(\{K_{\theta, k}\}\) of the channel \(\Lambda_\theta\). It follows that in order to compute QFI for parallel or adaptive strategy we need to optimize over Kraus representations \(\{K^{(N)}_{\theta, k}\}\) of the whole channel \(\Lambda^{(N)}_\theta=\Lambda^{\otimes N}_\theta\)—a task that is exponentially hard in \(N\). However, if we only want to obtain the upper bounds, we may dramatically reduce the complexity of the optimization, and reformulate the computation of the bound in a way that only minimization over Kraus representations of the single channel \(\Lambda_\theta\) is required [12, 13, 21, 29, 45].

In case of parallel strategies this leads to an upper bound of the form [11, 12, 29]:

(13)\[F^{(N)}_Q(\Lambda_\theta) \le 4N \min_{h} \left( \|\alpha(h)\| + (N-1)\|\beta(h)\|^2 \right),\]

where \(\alpha(h)\) is defined in (6), while

\[\beta(h) = \sum_k \dot K_{\theta, k}^\dagger(h) K_{\theta, k}(h).\]

This bound allows immediate exclusion of the possibility of the Heisenberg scaling in the model—it is enough to show that there exist \(h\) for which \(\beta(h)=0\), which may be formulated as a simple algebraic condition [12, 18].

In case of the adaptive strategy, one may again employ the idea of MOP to bound the maximal increase of QFI when the adaptive strategy is extended by one additional step including one more sensing channel [21]:

\[F_{\mathrm{Q}}^{(N+1)}(\Lambda_\theta) \le F^{(N)}_{\mathrm{Q}}(\Lambda_\theta) + 4\min_{h} \left[ \|\alpha(h)\| + \sqrt{F^{(N)}_{\mathrm{Q}}(\Lambda_\theta)} \|\beta(h)\| \right].\]

Both of these bounds can be computed efficiently regardless of the value of \(N\) [21]. They are implemented in the par_bounds and ad_bounds functions respectively.

Importantly, the bounds for parallel and adaptive strategies are asymptotically (for \(N\rightarrow\infty\)) equivalent and saturable [18, 21]—there are strategies that asymptotically achieve the optimal value of QFI predicted by the bounds.

Depending on whether there exist \(h\) for which \(\beta(h)=0\) we may have two possible asymptotic behaviors, namely the standard scaling (SS):

(14)\[\lim_{N\rightarrow\infty} F^{(N)}_{Q}(\Lambda_\theta)/N = 4\min_{h,\beta(h)=0}\|\alpha(h)\|,\]

and the Heisenberg scaling (HS):

(15)\[\lim_{N\rightarrow\infty} F^{(N)}_{Q}(\Lambda_\theta)/N^2 = 4\min_h\|\beta(h)\|^2.\]

In the package, the asymptotic bound can be computed using asym_scaling_qfi, a function that automatically also determines the character of the scaling.

Finally, there are methods to compute the bounds also in the case of correlated noise models that have been developed recently in [45]. We do not provide the exact formulation of the optimization problem that one needs to solve to find the relevant bound as it is much more involved than in case of uncorrelated noise models discussed above. In the package, the correlated noise bounds may be computed using ad_bounds_correlated a finite number of channel uses, and ad_asym_bound_correlated in order to obtain the asymptotic behavior. Unlike in the uncorrelated noise models, there is no guarantee here that the bounds are saturable. Still, they may be systematically tightened at the expense of increasing the numerical complexity, see Sec. Correlated noise models for a more detailed discussion.