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Figure from:Geometric perspective on quantum parameter estimation

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FIG. 9. Networked approach to quantum sensing for a decomposition of the parameters across 5 sensors with potentially entangled states (dashed lines). The j th node is encoded by the parameter subset 4, ;1. The dashed lines represent (quantum) correlations between the nodes. This approach is well suited to field sensing, where each node is a spin system. — Figure 9 Networked approach to quantum sensing for a decomposition of the parameters across 5 sensors with potentially entangled states (dashed lines). The j th node is encoded by the parameter subset 4, ;1. The dashed lines represent (quantum) correlations between the nodes. This approach is well suited to field sensing, where each node is a spin system.

Related Figures (9)

FIG. 1. We measure an observable M to obtain an estimate Oe, for a parameter of interest 6. We can relate the error A@ in the estimate to the standard deviation AM by relating the tangent at @eg. with angle a to the ratio between AM (red vertical line) and the error A@.

FIG. 2. The Mach-Zehnder interferometer, drawn with two evanes- cently coupled 50:50 beam splitters. Here, a and 6 define two input modes. An optical path difference between the internal arms of the interferometer results in a relative phase difference. The objective is to estimate the relative phase 9 = © — ¢ by measuring the photon intensities at the output using the photodetectors on the right.

FIG. 3. Fig. 3a: the probability simplex where the measurement has three outcomes with probabilities p;, pa, and p3. Every probability distribution is represented by a point in the simplex (P,4 and Pz). Fig. 3b: 6 parametrises a path through the probability simplex.

FIG. 4. The distance between probability distributions P4 and Pg diverges when one of them (B) lies on the hull of the simplex.

FIG. 5. General channel parameter estimation scheme with the possi- bility of adaptive control, shown in parallel with the phase estimation procedure using a Mach-Zehnder interferometer. The quantum chan- nel A(@) imparts parameters @ on the input probe state. The probe is measured by an operator of the form II(x) dx and distributes estimates according to Born’s rule. By data processing the measurement out- comes, we obtain our estimate 4. The goal of quantum metrology is in finding both the optimal probe state and observable that minimises the covariance matrix of unbiased estimates.

where equality holds if and only if Im[Tr[p(6)M(x)L@]] = 0, i.e. if the vectors lie in the real space IR*. This requires the s_p to be Hermitian. Introducing (ALg)* as the variance of the sLp, we use the Schwartz inequality to obtain: Next, we maximise this quantity over all possible povms I(x). Given the complex vectors a,8 € C7, and Re[a]Re[B] < ||@B||, we develop Eq. (90) into

FIG. 6. General parallel parameter estimation strategies. The initial probe states, o(0) are represented by the circles on the left. The probe states may be entangled before passing through an evolving channel, illustrated by the grey central squares. The state passes through another potentially entangling evolution and is finally measured using local povms I];, represented by the blue polygons on the right. This setup allows for four classes of estimation, where the input states many be entangled or not (Q/C on the left), and where the input state may be subjected to entangled or separable measurement observables (Q/C on the right).

FIG. 7. System-bath interaction model where Hs, He, and Hy are the system, environment, and interaction Hamiltonians.

FIG. 8. Ancilla assisted quantum channel extension scheme, de- scribed by A(6) @ 1 and a povm measurement Ij.

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