Optimal parameter estimation with a fixed rate of abstention

The problems of optimally estimating a phase, a direction, and the orientation of a Cartesian frame (or trihedron) with general pure states are addressed. Special emphasis is put on estimation schemes that allow for inconclusive answers or abstention. It is shown that such schemes enable drastic improvements, up to the extent of attaining the Heisenberg limit in some cases, and the required amount of abstention is quantified. A general mathematical framework to deal with the asymptotic limit of many qubits or large angular momentum is introduced and used to obtain analytical results for all the relevant cases under consideration. Parameter estimation with abstention is also formulated as a semidefinite programming problem, for which very efficient numerical optimization techniques exist.

Figure 1

Profile of the components ${\xi }_j$ of the transformed fiducial state $|{\tilde{\Psi }}_0\rangle$ in the asymptotic limit. The solid line is the limiting function $\phi \left(t\right)$ for $Q=0.17$ ($\lambda =1.1$). The points represent the actual components, as obtained by numerical optimization for $n=30$ (empty circles) and $n=220$ (filled circles). The dotted line is the solution for unrestricted abstention, Eq. (38), whereas the dashed horizontal line represents the components $\psi \left(t\right)=1=\sqrt{n}{c}_j$ of the initial fiducial state $|{\Psi }_0\rangle$ scaled by $\lambda$.

Figure 2

Plot of $n{S}_{\min }=2N(1-F)$ vs $Q$ (solid line) for an asymptotically large number, $N=n$, of parallel spins on the equator of the Bloch sphere. The dots have been obtained by numerical optimization with $n=100$.

Figure 3

Profile of the transformed fiducial state $|{\tilde{\Psi }}_0\rangle$ for $Q=0.56$ ($\lambda =1.5$). The thin (thick) lines correspond to $n=20$ ($n=80$). The circles are obtained by numerical optimization. The dashed lines represent the constraint $\lambda \psi (t-1/2)$, where $\psi \left(\tau \right)$ is given in Eq. (52).

Figure 4

Plot of $n^2{S}_{\min }[=(1/2)N^2(1-F)]$ versus $Q$. The solid lines are the analytical expressions in (94), whereas the circles are numerical results. In order to approach the asymptotic limit, higher values of $n$ are needed for smaller $Q$. Accordingly, two different values of $n$ have been used; $n=50$ (filled circles) and $n=120$ (empty circles).

Figure 5

Plot of $n{S}_{\min }$[$=N(1-F)$] vs $Q$ (solid line) for a signal state consisting of an asymptotically large number, $N=2n$, of antiparallel spins with null total magnetic number. The dots have been obtained by numerical optimization with $n=100$.

Figure 6

Plot of $n{S}_{\min }[=N(1-F)]$ vs $Q$. The solid black line is the leading asymptotic expression in Eq. (126). The dashed line is the curve $(Q,n{S}_{\min })$ given by Eqs. (124) and (125) for $n=20$. The blue empty (red filled) circles are numerical results for $n=20$ ($n=90$). The empty diamond is also a numerical result for $Q=1$ and $n=1000$.