Optimal Generators for Quantum Sensing

We propose a computationally efficient method to derive the unitary evolution that a quantum state is most sensitive to. This allows one to determine the optimal use of an entangled state for quantum sensing, even in complex systems where intuition from canonical squeezing examples breaks down. In this paper we show that the maximal obtainable sensitivity using a given quantum state is determined by the largest eigenvalue of the quantum Fisher information matrix (QFIM) and the corresponding evolution is uniquely determined by the coinciding eigenvector. Since we optimize the process of parameter encoding rather than focusing on state preparation protocols, our scheme is relevant for any quantum sensor. This procedure naturally optimizes multiparameter estimation by determining, through the eigenvectors of the QFIM, the maximal set of commuting observables with optimal sensitivity.

Figure 1

One-axis twisting with $N=20$. (a) and (b) Collective Bloch sphere at $t=0$ and $t=1/(\chi N^{\frac{2}{3}})$, respectively. The color represents $|{⟨\theta ,\varphi |\psi (t)⟩|}^2$ at each point. (c) The three eigenvalues ${\lambda }_i$ of $\mathcal{F}$. Also plotted as a black dotted-dashed line is ${\mathcal{F}}_{\mathrm{OAT}}$ from Eq. (8). (d) Location of the optimal generator during the squeezing process. The color represents the QFI for the given generator. The time axis for $t\lesssim \pi /(2\chi )-2/(\chi \sqrt{N})$ is shown with a black arrow, while the discontinuous jump at $t\sim \pi /(2\chi )-2/(\chi \sqrt{N})$ is shown with a purple arrow. The eigenvalue then grows at this final axis for the remainder of the process.

Figure 2

Three-axis twisting of a SU(4) system with $N=20$. (a) The largest eight eigenvalues ${\lambda }_i$ of $\mathcal{F}$. Also shown as a black dashed line is the largest QFI from an operator in the $\mathfrak{J}$, $\mathfrak{K}$, and $\mathfrak{E}$ subgroups. (b) Coefficients ${\mathcal{O}}^{\mu }$ and corresponding basis operator ${\widehat{G}}_{\mu }$ of the optimal generator $\widehat{\mathcal{G}}={\mathcal{O}}^{\mu }{\widehat{G}}_{\mu }$.