Optimal parameter estimation with a fixed rate of abstention

Abstract

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.

Probabilistic metrology or how some measurement outcomes render ultra-precise estimates

“…and subject to boundary conditions that are fixed by the probe state, the strength of the noise, and the success probability. Other equivalent variational formulations can be found in [33,34,68] for pure states and in [69] for the pointwise approach.…”

“…We use the Karush-Kuhn-Tucker optimization method to minimize equation (18) under the constraints in equation (20). For a given value of s j , the so-called complementary slackness condition [34,70] guarantees that the solution ( ) j x to equation (18) saturates the inequality in equation (20) for x in a certain region called coincidence set, while it coincides with an eigenfunction of the Hamiltonian j , defined after equation (18), for x outside this region. The continuity of ( ) j x and its derivative provide some matching conditions at the border of the coincidence set and a unique solution can be easily found.…”

“…That is, these schemes are optimized in order to provide a valid estimate for each possible measurement outcome, in such a way that the average precision is maximized. Recently it has been shown that for a fixed probe state, and in the absence of noise, the precision of some particular (favorable) outcomes can be greatly enhanced, well beyond the limits set for deterministic strategies [32][33][34][35]. The possibility to post-select, i.e., to abstain from providing an estimate some times, can even change the uncertainty from SQL to Heisenberg scaling.…”

Optimal quantum operations at zero energy cost

“…Note that both in the amplification and in the cloning example, the input and the target states are drawn from a set of states: more precisely, the problem is to transform the input state |ϕ t = e −itHsys/ |ϕ into the target state |ψ t = e −itHsys/ |ψ for an arbitrary (and possibly unknown) value of the parameter t. However, since we require our operations to be energy-preserving, we do not need to optimize them for every value of t: indeed, every energy-preserving transformation that approximates the transition |ϕ → |ψ will do equally well in approximating the transition |ϕ t → |ψ t , due to the covariance condition of Eqs. (24) and (25). This point is made clear if we measure the performance of our operations in terms of the fidelity between the target state and the actual output.…”

“…For example, we will see that a beam of N atoms, each of them prepared in the superposition |S = (|E + |G )/ √ 2 of the ground state and the first excited state, can be probabilistically transformed at no energy cost into a stronger beam of N 2− atoms in a state that is nearly identical to the state of N 2− identical copies of |S , up to an exponentially small error. The ability to efficiently approximate forbidden transformations of coherence at zero energy cost is a new twist of the postselection approach widely applied in quantum information [19][20][21][22][23][24][25][26][27][28][29][30][31][32] and complements existing results on the resource theory of coherence [33? -38].…”