Extended convexity of quantum Fisher information in quantum metrology

We prove an extended convexity for quantum Fisher information of a mixed state with a given convex decomposition. This convexity introduces a bound which has two parts: (i) The classical part associated with the Fisher information of the probability distribution of the states contributing to the decomposition, and (ii) the quantum part given by the average quantum Fisher information of the states in this decomposition. Next we use a non-Hermitian extension of a symmetric logarithmic derivative in order to obtain another upper bound on quantum Fisher information, which helps to derive a closed form for the bound in evolutions having the semigroup property. We enhance the extended convexity with this concept of a non-Hermitian symmetric logarithmic derivative (which we show is computable) to lay out a general metrology framework where the dynamics is described by a quantum channel and derive the ultimate precision limit for open-system quantum metrology. We illustrate our results and their applications through two examples where we also demonstrate how the extended convexity allows identifying a crossover between quantum and classical behaviors for metrology.

Figure 1

The optimal lower bound of the error $1/\sqrt{\mathcal{F}{}_{\mathrm{conv}}^{\left(\mathrm{Q}\right)}}$ (solid black) and its approximations $c_2^{-1/2}/N$ (dashed-red line) and $c_1^{-1/2}/\sqrt{N}$ (dashed-blue line) for $N\ll N^{☆}$ and $N\gg N^{☆}$, respectively. Here $q=0.995,\tau =1$, and $x=1$, which yield $N^{☆}\approx 50$, which also agrees with the plot (the light blue area). The inset compares our bound $1/\sqrt{\mathcal{F}{}_{\mathrm{conv}}^{\left(\mathrm{Q}\right)}}$ (dashed-red line) with the exact value of $1/\sqrt{\mathcal{F}{}^{\left(\mathrm{Q}\right)}}$ (solid-black line), which again both agree well up to $N\approx 65$.