Continuity of the quantum Fisher information

In estimating an unknown parameter of a quantum state the quantum Fisher information (QFI) is a pivotal quantity, which depends on the state and its derivate with respect to the unknown parameter. We prove the continuity property for the QFI in the sense that two close states with close first derivatives have close QFIs. This property is completely general and irrespective of dynamics or how states acquire their parameter dependence and also the form of parameter dependence—indeed this continuity is basically a feature of the classical Fisher information that in the case of the QFI naturally carries over from the manifold of probability distributions onto the manifold of density matrices. We demonstrate that, in the special case where the dependence of the states on the unknown parameter comes from one dynamical map (quantum channel), the continuity holds in its reduced form with respect to the initial states. In addition, we show that, when one initial state evolves through two different quantum channels, the continuity relation applies in its general form. A situation in which such a scenario can occur is an open-system metrology where one of the maps represents the ideal dynamics, whereas the other map represents the real (noisy) dynamics. In the making of our main result, we also introduce a regularized representation for the symmetric logarithmic derivative which works for general states even with incomplete rank, and it features continuity similar to the QFI.

Figure 1

Schematic of a general estimation scenario for dynamics. (Top) Ideal case, where the prepared state ${\varrho }_0$ and the parameter-dependent map ${\mathcal{E}}_x$ (as well as the measurement operation) are assumed ideal (noiseless). (Bottom) Real case, where some noise has affected the scenario and changed the initial state ${\varrho }_0\to {\varrho }_0^{\prime }$ and the dynamics ${\mathcal{E}}_x\to {\mathcal{E}}_x^{\prime }$. Note that ${\mathcal{E}}_x$ and ${\mathcal{E}}_x^{\prime }$ can be by construction general open-system dynamics, described by completely positive, trace-preserving quantum maps or channels [22]. We, however, call ${\mathcal{E}}_x$ “ideal” or “noiseless” in the sense that this is what we designed originally and ${\mathcal{E}}_x^{\prime }$ “real” or “noisy” in the sense that some extra uncontrollable or unaccounted-for source of noise has changed the designed ${\mathcal{E}}_x$ to ${\mathcal{E}}_x^{\prime }$.

Figure 2

$|{\mathcal{F}}^{\left(\mathrm{C}\right)}\left(\mathbf{p}\right)-{\mathcal{F}}^{\left(\mathrm{C}\right)}\left(\mathbf{q}\right)|$ vs $x$ for the example discussed in remark (v).

Figure 3

(Top) Difference of the QFIs for two density matrices given in Sec. 4a vs their distance, for $x\in [0,\pi /2]$. It is evident, from the multivaluedness of $|{\mathcal{F}}^{\left(\mathrm{Q}\right)}\left(\mathit{\varrho }\right)-{\mathcal{F}}^{\left(\mathrm{Q}\right)}\left(\mathit{\sigma }\right)|$ at the point where ${\parallel \varrho -\sigma \parallel }_1=0$, that at $x=\pi /4,\pi /2$ the QFI exhibits violation of the reduced continuity (while satisfying the continuity in the general sense of Theorem t1). (Bottom) Comparison of the exact value of $|{\mathcal{F}}^{\left(\mathrm{Q}\right)}\left(\mathit{\varrho }\right)-{\mathcal{F}}^{\left(\mathrm{Q}\right)}\left(\mathit{\sigma }\right)|$ and our upper bound (28).

Figure 4

Schematic of a metrological scenario with general parameter-dependent dynamics represented by a linear map ${\mathcal{E}}_x$. Note that this is an example in line with the general scheme of Fig. 1 (top).

Figure 5

Schematic of a metrological scenario with general parameter-dependent dynamics ${\mathcal{E}}_x$. (Top) The noiseless scenario. (Bottom) The noisy scenario. Here the noise operations ${\mathcal{N}}_{{\delta }_1}$ and ${\mathcal{N}}_{{\delta }_2}$ affect, respectively, the preparation and the dynamics.

Figure 6

Example 4d. (a) Exact value of $|{\mathcal{F}}^{\left(\mathrm{Q}\right)}\left(\mathit{\varrho }\right)-{\mathcal{F}}^{\left(\mathrm{Q}\right)}\left(\mathit{\sigma }\right)|$ vs $x$. (b) Bound (28) vs $x$. (c) The two quantities in the same plot. (d) The two quantities in the same plot where our bound is rescaled with a factor of $\approx 0.02$. The good agreement here indicates that our bound captures the behavior of the difference of the exact QFIs faithfully.

Figure 7

Example 4d. $|{\mathcal{F}}^{\left(\mathrm{Q}\right)}\left(\mathit{\varrho }\right)-{\mathcal{F}}^{\left(\mathrm{Q}\right)}\left(\mathit{\sigma }\right)|$ vs $\parallel {\partial }_x\varrho -{\partial }_x{\sigma \parallel }_1$. This monotonic behavior indicates that the very nonvanishing of the difference of the QFIs is rooted at the nonvanishing property of $\parallel {\partial }_x\varrho -{\partial }_x{\sigma \parallel }_1$.

Figure 8

Counterclockwise integration contour in the form of a strip of width $2\varepsilon$, which encloses all nonvanishing eigenvalues of $\varrho$ and $\sigma$.

Figure 9

Graphical proof that ${max}_{\varepsilon \in [0,{\lambda }_{min}\left(\tilde{\varrho }\right)]}min\{\varepsilon ,{\lambda }_{min}\left(\tilde{\varrho }\right)-\varepsilon \}={\lambda }_{min}\left(\tilde{\varrho }\right)/2$.