Optimal distributed sensing in noisy environments

We consider distributed sensing of nonlocal quantities. We introduce quantum enhanced protocols to directly measure any (scalar) field with a specific spatial dependence by placing sensors at appropriate positions and preparing a spatially distributed entangled quantum state. Our scheme has optimal Heisenberg scaling and filters out noise with different spatial dependence than the signal. We provide states, spatial sensor configurations, and protocols to achieve optimal scaling. We explicitly demonstrate how to measure coefficients of spatial Taylor and Fourier series, and show that our approach can offer an exponential advantage as compared to strategies that do not make use of entanglement between different sites.

Figure 1

Sketch of a distributed sensing setup. Several sensors $P_{\ell }$ are located at different spatial positions. Different signals ${\alpha }_k$ have different spatial configurations of the field.

Figure 2

Examples of a few scenarios with sensors distributed on a line, discussed in the text. Noise is depicted in blue dashed and signal in red solid. (a) Short wavelength signal and local correlated noise. (b) Taylor series expansion. (c) Fourier series expansion.

Figure 3

Strength of the signal $S$ should be determined via three sensors $P_j$. The two noise sources $N_1$ and $N_2$ are equidistant from the sensor $P_1$ as well as $P_2$.

Figure 4

Polytope defined by $(\pm 1,\pm 1,0)$ describes the decoherence free subspace if there is only a single qubit present at each sensor. The two states $\mathbf{s}=\left|1,1,0\right.〉$ and $-\mathbf{s}$ lead a maximal projection onto $f_{\perp }$ of states within the polytope.