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Matteo G. A. Paris is a theoretical physicist working on quantum information, quantum optics and quantum measurement. His article Quantum estimation for quantum technology by Matteo G. A. Paris explains how to estimate quantities encoded in a quantum state when a direct measurement of the desired parameter is unavailable.

The treatment uses the symmetric logarithmic derivative and Quantum Fisher information to bound attainable precision and compare measurement strategies. It connects the choice of measurement with statistical estimation, rather than assuming that repeated observations automatically achieve the best possible accuracy.^[ref]

Institutional profile: www.unimi.it/it/ugov/person/matteo-paris.

The Quantum Fisher information measures the local sensitivity of a parametrized density matrix ${\rho }_{\theta }$. The symmetric logarithmic derivative $L_{\theta }$ satisfies ${\partial }_{\theta }\rho =(L_{\theta }\rho +\rho L_{\theta })/2$, and $F_Q=\mathrm{Tr}(\rho L_{\theta }^2)$.

For $n$ independent preparations, a locally unbiased estimator obeys $\mathrm{Var}(\hat{\theta })\ge 1/(n{F}_Q)$ under the usual regularity assumptions. For a pure family $|{\psi }_{\theta }⟩=e^{-i\theta G}|{\psi }_0⟩$, $F_Q=4\mathrm{Var}(G)$. A state with definite $G$ acquires only a global phase and has zero information about $\theta$; a superposition can have observable relative-phase sensitivity.^[ref]

For a differentiable family of density matrices ${\rho }_{\theta }$, the symmetric logarithmic derivative is a Hermitian operator $L_{\theta }$ satisfyingIf ${\rho }_{\theta }={\sum }_j{p}_j|j⟩⟨j|$, its determined matrix entries are $(L_{\theta }{)}_{jk}=2⟨j|{\partial }_{\theta }{\rho }_{\theta }|k⟩/(p_j+p_k)$ whenever $p_j+p_k>0$. Entries wholly in the kernel are undetermined when a solution exists and do not affect $\mathrm{Tr}({\rho }_{\theta }{L}_{\theta }^2)$. Rank-changing boundary points need extra care.

For ${\rho }_{\theta }=\mathrm{diag}(\theta ,1-\theta )$, $0<\theta <1$, one gets $L_{\theta }=\mathrm{diag}(1/\theta ,-1/(1-\theta ))$ and Quantum Fisher information $F_Q=1/[\theta (1-\theta )]$. This recovers ordinary Fisher information for a binary observation. In noncommuting families the symmetrized equation supplies the operator version needed to quantify local sensitivity.^[ref]