Precision limit under weak-coupling with ancillary qubit

TL;DR

The paper addresses achieving Heisenberg-limited phase estimation in quantum metrology without relying on entangled GHZ states or squeezing Hamiltonians. It introduces a measurement-based protocol where a spin-ensemble probe weakly couples to an ancillary qubit via a general XXZ interaction and uses an unconditional qubit measurement to create a two-path evolution, generating a GHZ-like superposition and yielding $F_Q \sim N^2$ for large $N$. The authors derive optimal conditions on coupling strengths, detuning, and evolution times under which exact or asymptotic Heisenberg scaling is achieved, and show robustness to encoding-direction and coupling imperfections. Phase sensitivity is analyzed under parity detection on either the ancilla or the probe, with both approaches approaching the Heisenberg limit at suitable operating points. The work offers an economical route to surpass the standard quantum limit in metrology using unconditional qubit measurements as a resource, applicable to platforms like NV centers and quantum dots.

Abstract

We propose a measurement-based quantum metrology protocol in a composite model, where the probe system (a spin ensemble) is coupled to an ancillary two-level system (qubit) with a general Heisenberg XXZ interaction. With an optimized and weak probe-ancilla coupling strength and a proper duration of joint evolution, the two parallel evolution paths of the probe system induced by the unconditional measurement on qubit can transform an eigenstate of the collective angular momentum operator of spin ensemble to be a two-component state with a large distance in eigenspace. The quantum Fisher information about the phase encoded in the probe system of polarized states or their superposition, that could be relaxed to mixed states, can therefore manifest an exact or asymptotic quadratic scaling with respect to the probe size (spin number) $N$. The quadratic scaling behavior is found to be insensitive to the imperfect encoding operator and coupling strength. By virtue of the parity detection on the ancillary qubit or the probe system, the phase sensitivity can approach the Heisenberg limit. We suggest that the unconditional measurement on qubit could become an efficient resource to replace Greenberger-Horne-Zeilinger-like states and squeezing Hamiltonian for exceeding the standard quantum limit in metrology precision.

Figures (6)

• Figure 1: A circuit model of our measurement-based metrology. The composite system prepared in a separable state $|\psi\rangle\otimes|\varphi\rangle$ experiences two stages of free joint unitary evolutions $U(t_1)$ and $U(t_2)$. In between them, a to-be-estimated phase parameter $\theta$ is encoded into the large-spin-probe system via a unitary rotation $R_{\alpha}(\theta)$ on the equatorial plane and meanwhile an unconditional measurement $M_{\pm}=|\pm\rangle\langle\pm|$ in the basis of $\sigma_x$ is performed on the ancillary qubit. The output state is determined by the parity detections on the probe system or the ancillary qubit.
• Figure 2: QFI as a function of $N$ for a thermal state $\rho_P^{\rm th}$ with various inverse temperatures. The black-dashed line and the black dot-dashed line indicate the Heisenberg and shot-noise scalings, respectively. Here $\rho_A=|\varphi\rangle\langle\varphi|=|+\rangle\langle+|$, $t_1=(N+1)\pi/(4\Delta_A)$, $\omega_P/\Delta_A=40/(N+1)$, $g_z/\Delta_A\approx6/N$, and $g/\Delta_A\approx4\sqrt{2}/N$.
• Figure 3: Renormalized QFI $F_Q/N^2$ as a function of the deviation $\delta$ about encoding direction for the probe system prepared as (a) and (b) a polarized state $|\psi\rangle=|j,-j\rangle_{\rm opt}$ or (c) and (d) a thermal state $\rho^{\rm th}_P$ with $\beta=1$. The solid lines in (a) and (b) indicate the approximated analytical result in Eq. (\ref{['QFIapp']}). The probe eigenfrequency $\omega_P$ is optimized by Eq. (\ref{['optimalFrequency']}) with (a) and (c) an even number $n_P=10$ and (b) and (d) an odd number $n_P=11$. The probe size is $N=100$. $t_1=(N+1)\pi/(4\Delta_A)$, $g_z/\Delta_A\approx6/N$, and $g/\Delta_A\approx4\sqrt{2}/N$.
• Figure 4: Renormalized QFI $F_Q/N^2$ in the parametric space of $g/\Delta_A$ and $g_z/\Delta_A$ for the probe system prepared as the polarized state $|\psi\rangle=|j,-j\rangle_{\rm opt}$ with the probe spin number $N=100$. The white dot-dashed line presents the analytical result under the condition in Eq. (\ref{['optimal g gz']}). Here $t_1=t_{1, {\rm opt}}(n_1=0)=(N+1)\pi/(4\Delta_A)$.
• Figure 5: Renormalized phase sensitivity $N|\Delta\theta|$ under the parity detection on the ancillary system. The black-dashed line represents HL. The probe is initialized as $|\psi\rangle=|j,-j\rangle_{\rm opt}$ and the evolution time of Stage 2 is $t_2=3t_{1,{\rm opt}}(n_1=0)$. The other parameters are the same as Fig. \ref{['QFI shift']}.