Quantum Fisher information on two manifolds of two-mode Gaussian states

We investigate two special classes of two-mode Gaussian states of light that are important from both the experimental and theoretical points of view: the mode-mixed thermal states and the squeezed thermal ones. Aiming to a parallel study, we write the Uhlmann fidelity between pairs of states belonging to each class in terms of their defining parameters. The quantum Fisher information matrices on the corresponding four-dimensional manifolds are diagonal and allow insightful parameter estimation. The scalar curvatures of the Bures metric on both Riemannian manifolds of special two-mode Gaussian states are evaluated and discussed. They are functions of two variables, namely, the mean numbers of photons in the incident thermal modes. Our comparative analysis opens the door to further investigation of the interplay between geometry and statistics for Gaussian states produced in simple optical devices.

Figure 1

The scalar curvature $R_{\mathrm{MT}}\left({\overline{n}}_1,{\overline{n}}_2\right)$, Eq. (7.1), of the two-mode MTSs. This is a convex surface whose general aspect is that of a symmetric valley that descends and widens continuously. Its talweg belongs to the symmetry plane ${\overline{n}}_1={\overline{n}}_2$ and is drawn in Fig. 2.

Figure 2

(a) The vertical intersection (7.3) of the surface (7.1) and its symmetry plane ${\overline{n}}_1={\overline{n}}_2$. This is the talweg of the surface (7.1) and is made up of the symmetric two-mode TSs. (b) The vertical intersection (7.4) of the surface (7.1) and the plane ${\overline{n}}_1+{\overline{n}}_2=1$. The vertical sections (a) and (b) are orthogonal and meet at their unique zero, ${\overline{n}}_1=\frac{1}{2}$.

Figure 3

The scalar curvature $R_{\mathrm{ST}}\left({\overline{n}}_1,{\overline{n}}_2\right)$, Eq. (7.6), of the two-mode STSs. This surface consists of two opposite symmetric valleys separated by a watershed: a narrow, steeply ascending valley and a wider one which is slowly descending. It possesses a unique saddle point $S$ at the intersection of the talweg and the watershed. The talweg is the normal section (7.8) in the symmetry plane ${\overline{n}}_1={\overline{n}}_2$, while the watershed is the normal section (7.9) in the plane ${\overline{n}}_1+{\overline{n}}_2=2{\overline{n}}_s$. These normal sections, which are vertical and orthogonal, are represented in Figs. 4 and 4.

Figure 4

(a) The vertical intersection (7.8) of the surface (7.6) and its symmetry plane ${\overline{n}}_1={\overline{n}}_2$. This is the talweg of the surface (7.6) and is made up of the symmetric two-mode STSs. The function (7.8) has a steep rise from its lowest value $R_{\mathrm{ST}}(0,0)=-16$, reached for any two-mode SVS, to a maximum $R_{\mathrm{ST}}\left({\overline{n}}_s,{\overline{n}}_s\right)=-\frac{143}{14}\approxeq -10.2143,$ reached at the point ${\overline{n}}_s=-0.5+\sqrt{1.15}\approxeq 0.5724$. Then it has a moderate fall toward the asymptotic value ${lim}_{\overline{n}\to \infty }{R}_{\mathrm{ST}}\left(\overline{n},\overline{n}\right)=-12$. The function (7.8) changes concavity at the unique inflection point ${\overline{n}}_i\approxeq 0.9565$, where it has the value $R_{\mathrm{ST}}\left({\overline{n}}_i,{\overline{n}}_i\right)\approxeq -10.5140$. (b) The vertical intersection (7.9) of the surface (7.6) and the plane ${\overline{n}}_1+{\overline{n}}_2=2{\overline{n}}_s$. This is the watershed on the surface (7.6), which is orthogonal to the talweg and touches it at their common extremum point, ${\overline{n}}_1={\overline{n}}_s$.

Figure 5

The scalar curvatures (7.5) and (7.10) of the two-mode MTSs and STSs at the physicality edge. They are figured as the marginal vertical curves on the surfaces (7.1) and (7.6), respectively, which meet the plane ${\overline{n}}_2=0$ along them. The former is decreasing and convex, while the latter is increasing and concave. Their starting values are, respectively, $R_{\mathrm{MT}}(0,0)=+\infty$ for the vacuum state and $R_{\mathrm{ST}}(0,0)=-16$ for any two-mode SVS. Nevertheless, they have a common asymptotic limit: $R_{\mathrm{MT}}(+\infty ,0)=R_{\mathrm{ST}}(+\infty ,0)=2.$