Agnostic Parameter Estimation with Large Spins

TL;DR

This work generalizes agnostic quantum metrology to large-spin probes, showing that maximal quantum Fisher information can be achieved for an unknown rotation axis by entangling a large-spin probe with an ancilla and using postselection. For orthogonal ancilla states, the success probability scales as $p=2/m$ while the maximal FI remains $\mathcal{F}_{\max}=4s^2$, highlighting a trade-off between probe dimension and probabilistic preparation of the optimal encoding. The analysis extends to non-orthogonal ancilla states, where the postselection probability depends on the ancilla-state overlap and the entangled-state Schmidt coefficients, yet still enables $H$-agnostic estimation with finite success probability. The results connect to spin-1/2 protocols, recover known limits, and suggest experimental routes in optics to realize agnostic metrology with large spins.

Abstract

The quantum Fisher information of a quantum state with respect to a certain parameter quantifies the sensitivity of the quantum state to changes in that parameter. Maximizing the quantum Fisher information is essential for achieving the optimal estimation precision of quantum sensors. A typical quantum sensor involves a qubit(e.g. a spin-1/2) probe undergoing an unknown rotation, here the unknown rotation angle is the parameter to be estimated. A well known limitation is that if the rotation axis is unknown, the maximal quantum Fisher information is impossible to attain. This limitation has been lifted recently by leveraging entanglement between the probe qubit and an ancilla qubit. Namely, through measurement of the ancilla after the axis is revealed, one can prepare the probe that is optimal for any unknown rotation axis. This proposal, however, works only for a spin-1/2. Considering large spin probes can achieve a larger quantum Fisher information, offering enhanced metrological advantage, we here utilize the entanglement between a large spin probe and an ancilla to achieve optimal quantum Fisher information for estimating the rotation angle, without prior knowledge of the rotation axis. Different from the previous spin-1/2 case, achieving the optimal precision with large spins generally requires post-selection, resulting in a success probability dependent on the dimension of the Hilbert space. Furthermore, we extend the encoding state from the maximally entangled case to general entangled states, showing that optimal metrology can still be achieved with a certain success probability.

Figures (5)

• Figure 1: Circuit diagram for optimal parameter estimation without prior knowledge of the rotation axis. At $t_0$, the probe-ancilla pair are in an entangled state, while the axis $\bm n$ is unknown. The unitary $U_{\bm{n}}(\beta) = e^{-i\beta H}$ acts on the probe at $t_1$. At $t_2$, $\bm n$ is revealed. A measurement on $B$ effectively teleports to the past an optimal encoding state for the probe $A$ via postselection.
• Figure 2: The blue curve represents the probability of successfully collapsing to the metrologically optimal state for the probe $A$ as a function of the Hilbert space dimension $m$. The red curve represents the maximum quantum Fisher information under the optimal state as a function of $m$.
• Figure 3: The probability of successfully collapsing to the metrologically optimal state $U_A|n_-\rangle_A$ for the probe $A$ as a function of $\theta$ and $\xi_1^2$ under several different $\xi_2^2$. (a) $\xi_2^2=0.2$. (b) $\xi_2^2=0.4$. (c) $\xi_2^2=0.6$. (d) $\xi_2^2=0.7$. Due to $\xi_1^2+\xi_2^2+\xi_3^2=1$, when $\xi_1$ and $\xi_2$ are fixed, $\xi_3$ will naturally be fixed as well. The white dotted lines correspond to the case that $\xi_1=\xi_3$.
• Figure 4: Contour plot of total successful probability $P$ over the parameter space $\xi_2^2-\theta$. $\xi_1=\xi_3=\sqrt{(1-\xi_2^2)/2}$. The dotted line corresponds to the maximally entangled case $\xi_1=\xi_2=\xi_3=\frac{1}{\sqrt{3}}$, for which $|\psi_1\rangle=|n_+\rangle$, $|\psi_2\rangle=|E_2\rangle$, $|\psi_3\rangle=|n_-\rangle$ and satisfies $_B\langle\psi_i|\psi_j\rangle_B = \delta_{ij}$.
• Figure 5: The probability of successfully collapsing to the metrologically optimal state $U_A|n_-\rangle_A$ for the probe $A$ as a function of $\xi_3$. The dotted line corresponds to the case that $\xi_2=0, \theta=0$. The solid line corresponds to the case that $\xi_1=0, \theta=\pi/2$.