Limited-control metrology approaching the Heisenberg limit without entanglement preparation

Current metrological bounds typically assume full control over all particles that are involved in the protocol. Relaxing this assumption, we study metrological performance when only limited control is available. As an example, we measure a static magnetic field when a fully controlled quantum sensor is supplemented by particles over which only global control is possible. We show that even for a noisy quantum sensor, a protocol that maps the magnetic field to a precession frequency can achieve transient super-Heisenberg scaling in measurement time and Heisenberg scaling in the particle number. This leads to an estimation uncertainty that approaches that achievable under full control to within a factor independent of the particle number for a given total time. Applications to hybrid sensing devices and the crucial role of the quantum character of the sensor are discussed.

Figure 1

The proposed measurement scheme uses a control sequence on the sensor spin (blue) to weakly measure the auxiliary spins (black) without the need of further control after initialization in either the pure state $|+++\cdots \rangle$ as shown here or the completely mixed state. The weak measurement is realized by initializing a sensor spin (blue) into $|0\rangle$, applying $\pi /2$ pulses, and a dynamical decoupling (DD) sequence to weakly entangle the sensor spin with the auxiliary spins and finally measuring the sensor spin, applying an effective CPTP map onto the auxiliary spins. In between these measurements that are synchronized with an external local oscillator, the auxiliary spins acquire a phase $\varphi \propto {\mu }_{n}B$, leading to a total phase $n\varphi$ after $n$ cycles.

Figure 2

Upper graph: Numerical Fisher information scaling (red, initial product state; orange, initial mixed state), the analytical approximation for small $N$ Eq. (11) (red line, invalid for large $N$ due to the average density matrix calculation being invalid for $\gamma N>1$) and the asymptotic ($\gamma N\gg 1$) behavior (blue) for $\varphi =0.7$ and $M=100$ nuclear spins, each coupled with $k_0{T}_s=0.01$ to the NV center. The HL (black) for $M+1$ spins is shown for comparison. The results are averaged over 96 (32 for mixed state) runs with $N=2^{26}$ measurements each. Lower graph: Same data (red) and results for $k_0{T}_s=0.05$ (green). The Fisher information is initially larger but dominated by backaction earlier, resulting in a smaller prefactor for the asymptotic regime. In both graphs and all other simulations, the achievable Fisher information approaches the HL to within a factor $\lesssim 20$. This Fisher minimal distance is independent of the interaction strength $k_0{T}_s$ and the number of nuclei $M$; see the Appendix.

Figure 3

For the same parameters as in Fig. 2: Numerical Fisher information scaling [red, initial product state; orange, initial mixed state ($k_0{T}_s=0.01$ at $N=2^{26}$ each); green, initial product state ($k_0{T}_s=0.05$ at $N=2^{22}$)] with the number of nuclei $M$. The blue lines show the predicted scaling $\propto M^2$ from Eq. (15) and Fig. 2.

Figure 4

For $M=10$ nuclei: Fisher information after $2^{24}$ measurements for $\beta =0.01\times 2/\pi$ for different ${\gamma }_2$ averaged over 192 runs compared to a ${\gamma }^{-3}$ curve with the numerically obtained prefactor.

Figure 5

Left: Maximum of the ratio between the Fisher information of our protocol and the Heisenberg limit for different coupling strength. The results do not depend on the number of nuclei $M$, but are dominated by variations originating from the 9600 averages in the Monte Carlo simulation. Right: Curves for $M=100$ nuclei. The highest value is reached at $N\approx {\beta }^{-2}$ as expected. Artifacts of the simulation are clearly visible as for an infinite number of repetitions smooth curves are expected.

Figure 6

Logarithmic negativity for $M$ spins, each spin coupled with $\beta =0.01\times 2/\pi$ with negligible decay ${\gamma }_2=0$. The left graph shows the entanglement of one of the auxiliary spins with the remaining $M-1$ spins and the right graph shows entanglement in an equal bipartition of the auxiliary spins. The results are averaged over 2000 runs.