Optimal and secure measurement protocols for quantum sensor networks

Studies of quantum metrology have shown that the use of many-body entangled states can lead to an enhancement in sensitivity when compared with unentangled states. In this paper, we quantify the metrological advantage of entanglement in a setting where the measured quantity is a linear function of parameters individually coupled to each qubit. We first generalize the Heisenberg limit to the measurement of nonlocal observables in a quantum network, deriving a bound based on the multiparameter quantum Fisher information. We then propose measurement protocols that can make use of Greenberger–Horne–Zeilinger (GHZ) states or spin-squeezed states and show that in the case of GHZ states the protocol is optimal, i.e., it saturates our bound. We also identify nanoscale magnetic resonance imaging as a promising setting for this technology.

Figure 1

(a) An illustration of the network setup in a nanoscale NMR setting. Nodes, located at different points relative to a large molecule, share an entangled state; at each node there is both an unknown parameter ${\theta }_i$ and a known relative weight ${\alpha }_i$. We are concerned with estimating $\mathbf{\alpha }·\mathbf{\theta }$. (b) Illustration of the partial time evolution protocol for three qubits. Solid green segments of the timeline represent periods when a qubit is evolving due to coupling to the local parameter ${\theta }_i$, while dashed red segments represent periods after the qubit stops evolving. The switches occur at times corresponding to the qubits' weights in the final linear combination. The weight of the last qubit is ${\alpha }_3=1$.

Figure 2

Numerical optimization of ${\mathbf{\alpha }}^T{F}_Q^{-1}\mathbf{\alpha }$ for two qubits with ${\alpha }_1=1$ compared with the bound predicted by our analytic result. Each point is generated by running a gradient descent algorithm until convergence; the control parameters begin at small random values. The dashed (dotted) line is the analytic bound derived from the first (second) qubit. As ${\alpha }_2$ increases, the second qubit becomes the source of the relevant bound.