Noisy Metrology beyond the Standard Quantum Limit

Parameter estimation is of fundamental importance in areas from atomic spectroscopy and atomic clocks to gravitational wave detection. Entangled probes provide a significant precision gain over classical strategies in the absence of noise. However, recent results seem to indicate that any small amount of realistic noise restricts the advantage of quantum strategies to an improvement by at most a multiplicative constant. Here, we identify a relevant scenario in which one can overcome this restriction and attain superclassical precision scaling even in the presence of uncorrelated noise. We show that precision can be significantly enhanced when the noise is concentrated along some spatial direction, while the Hamiltonian governing the evolution which depends on the parameter to be estimated can be engineered to point along a different direction. In the case of perpendicular orientation, we find superclassical scaling and identify a state which achieves the optimum.

Figure 1

General metrology protocol where a known probe state evolves according to a physical evolution depending on an unknown parameter $\omega$. After a sufficient amount of data are collected, an estimate for the parameter is constructed.

Figure 2

Cut through the space of all valid qubit maps. The identity channel $\mathcal{I}[\varrho ]=\varrho$ is at the top, and (extremal) unitary channels ${\mathcal{U}}_{\omega t}$ are at the circumference. One can think of phase $\omega t$ as increasing along the angular direction and decoherence $\gamma t$ along the radial direction. The spirals represent trajectories of channels for parallel ${\mathcal{E}}_{\omega }^z(t)$ and transversal ${\mathcal{E}}_{\omega }^x(t)$ noise, approaching the completely mixing channel at the disk center. Arrows are separated by fixed time steps. We see that ${\mathcal{E}}_{\omega }^x(t)$ loses coherence slower, being always closer to the boundary, and also has larger angular “speed” $|d{\mathcal{E}}_{\omega }^x/d\omega |$. The classical simulation method [15] provides a geometric bound on estimation precision based on the distance to the boundary: $\delta \omega \sqrt{T}\ge \sqrt{[ϵ(t{)}^{2}t]/N}$. For asymptotic $N$, the bound is determined by the behavior for small $t$ at the start of the spirals. We get ${ϵ}_z(t{)}^2=2\gamma /t+O(1)$ and ${ϵ}_x(t{)}^2=({\gamma }^2{t}^2)/12+O(t^4)$.

Figure 3

(a) Precision scaling for bounds and GHZ strategy. The bound for parallel noise with optimization of the single-round duration is shown (dotted line), as well as the bounds for transversal noise without (dashed line) and with (solid lines) optimization. Without optimization (taking $t$ optimal for a single qubit), the asymptotic scaling is still SQL-like, but with optimization, superclassical scaling is maintained. In the latter case, the bound is saturated by the GHZ-state strategy (dots). The thin line shows $1/N^{5/6}$ scaling for reference, and the borders of the upper and lower shaded regions show SQL-like and Heisenberg scaling, respectively. (b) Bound (solid lines) and GHZ-state strategy (dashed line) for noise with a dominant transversal and a small parallel component ${\alpha }_x/{\alpha }_z=99$. The scaling is superclassical for moderate $N$, while the $z$ component determines the asymptotic scaling, which is hence SQL-like. The CE bound approaches $\sqrt{2\gamma ϵ/N}$ (lower thin dotted line) and the GHZ-state $\sqrt{2\gamma ϵe/N}$ (middle thin dotted line). The finite-$N$ CE bound for parallel noise with the same strength is shown for reference (upper thin dotted line) as well as for transversal noise (thin solid line).