Fragility of Optimal Measurements due to Noise in Probe States for Quantum Sensing

Abstract

For a given quantum state used in sensing, the quantum Cramér-Rao bound (QCRB) sets a fundamental limit on the precision achievable by an unbiased estimator of an unknown parameter, determined by the inverse of the quantum Fisher information (QFI). The QFI serves as an upper bound on the classical Fisher information (CFI), representing the maximum extractable information about the unknown parameter from measurements on a physical system. Thus, a central goal in quantum parameter estimation is to find a measurement, described by a POVM, that saturates the QFI (achieves maximum CFI), and thereby achieves the QCRB. In the idealization that one uses pure states and unitary encodings for sensing, discontinuities can appear in the CFI but not the QFI. In this article, we demonstrate that these discontinuities are important features, quantifying how much Fisher information is lost in the presence of noise. We refer to this as the Fisher information "fragility". We present a simple framework for understanding how discontinuities increase fragility through Jensen's inequality, and demonstrate how one can use this framework to design more robust POVMs for quantum advantage in metrology.

Figures (9)

• Figure 1: Classical Fisher information (CFI) for (a) a pure initial first excited Dicke state, with $J=16$, Eq. (\ref{['eq:initial_state']}) as a function of $\beta$, the angle of the measurement basis given by the eigenstates of $\exp\small(-i\beta\hat{J}_y\small)\hat{J}_z \exp\small(i\beta\hat{J}_y\small)$. The CFI in (a) is independent of $\beta$ except on a set of measure zero corresponding POVMs where the probability of some outcomes is zero (see insert). At these values of $\beta$ the CFI discontinuously jumps to a lower value. (b) The CFI as a function of $\beta$ after the probe state is put through a collective depolarizing channel, Eq. (\ref{['eq:collective_lindbladian']}), for different amounts of noise $\gamma t$. The dashed line represents the quantum Fisher information (QFI) which is the maximum achievable Fisher information over all possible measurements. For the noisy probe state, the CFI is continuous, dipping near the points of discontinuity in (a). Measurement bases near these values of $\beta$, while optimal for pure states, are "fragile" in presence of noise.
• Figure 2: Fisher information at $\theta=0$ of a first excited Dicke state $\ket{J,J-1}$ for a $J=16$ spin system with respect to sensing rotations around the $\hat{J}_y$ axis parameterized by $\theta$. The Fisher information is calculated for measurements in the eigenbasis of $e^{-i\beta \hat{J}_y}\hat{J}_ze^{i\beta \hat{J}_y}$. The noise is determined by the jump operator defined implicitly in Eqs. (\ref{['eq:pathological_condition']}) and (\ref{['eq:disc_special_case']}) (constructed explicitly in App. \ref{['sec:explicit_construction']}), applied at rate $\gamma$ for time $\gamma t=10^{-4}$, and is chosen such that one single discontinuity does not impact the Fisher information for values of $\beta$ surrounding it. In (a), the jump operator $\hat{L}_{M=2}$ is constructed such that $M=2$ in Eq. (\ref{['eq:disc_special_case']}), and thus the Fisher information surrounding the discontinuity at $\beta_{M=2}$ is unaffected by presence of the discontinuity. (b) and (c) show the same, with $\hat{L}_{M=8}$ and $\hat{L}_{M=14}$ respectively.
• Figure 3: Top row (a-c): Classical Fisher information (blue) of the probe state corresponding a first excited Dicke state with $J=4$, after being subjected the noise channel, Eq. (\ref{['eq:collective_lindbladian']}). The probe state is rotated about the $\hat{J}_y$ axis (after encoding of $\theta$) by angle $-\beta$ before being measured along the $\hat{J}_z$ axis. The dashed line is the Jensen's upper bound, Eq. (\ref{['eq:avg_fc']}), determined from the pure state decomposition in Eq. (\ref{['eq:bad_decomp']}). For weak noise, the bound is tight. Bottom row (d-f): traces of $\sum_iq_{i,\beta} |\phi_\theta^{(i)}\rangle\langle\phi_\theta^{(i)}|$ (orange) and $\hat{\Omega}_{\theta,\beta}$ (blue) of the decompositions in (a-c).
• Figure 4: Classical Fisher information for a spin coherent state (SCS) pointing along $(\theta,\phi)$ (a--b) and for a first excited Dicke state (with spin $J=4$) pointing along $(\theta,\phi)$ (c--d). In both cases the brightest yellow represents regions where the classical Fisher information is equal to the quantum Fisher information of the noiseless case, and the darkest blue is CFI=0. The CFI is measured along the $\hat{J}_z$ axis, and the generator of $\theta$ is $\hat{J}_y$. The noise considered here is mixing with the identity, where the noisy state $\hat{\rho}$ is related to the pure state $\ket\psi$ by $\rho=(1-\varepsilon)\ketbra{\psi}+\varepsilon\mathds1/d$ where $\varepsilon=0.01$ and $d$ is the dimension of the Hilbert space.
• Figure 5: Top row (a-c): Classical Fisher information (blue) of the probe state corresponding a first excited Dicke state with $J=4$, after being subjected the noise channel, Eq. (\ref{['eq:loc_noise']}). The probe state is rotated about the $\hat{J}_y$ axis (after encoding of $\theta$) by angle $-\beta$ before being measured along the $\hat{J}_z$ axis. The dashed line is the Jensen's upper bound, Eq. (\ref{['eq:avg_fc']}), determined from the pure state decomposition in Eq. (\ref{['eq:bad_decomp']}). For weak noise, the bound is tight. Bottom row (d-f): traces of $\sum_iq_{i,\beta} |\phi_\theta^{(i)}\rangle\langle\phi_\theta^{(i)}|$ (orange) and $\hat{\Omega}_{\theta,\beta}$ (blue) of the decompositions in (a-c).