Nonlinear quantum metrology

This dissertation reviews the theory of information geometry as applied to quantum theory, and proposes some new methods for high-precision quantum metrology. Information geometry defines the statistical distance for quantum states. It also solves the problem of optimizing measurements for quantum parameter estimation at a theoretical level. This is summarized by the quantum Cramer-Rao bound, a bound on the mean squared error achievable. The second part of the dissertation reviews and extends some topics in linear quantum metrology. Linear quantum metrology typically considers the problem of estimating an unknown coupling parameter acting with identical single-system couplings for the individual constituents of a quantum probe. We review the relation entangled and sequential protocols. In essence, the entangled protocol can be seen as a formal quantum parallelization of the sequential protocol. We study the case of clock synchronization protocols and show that, perhaps surprisingly, both protocols behave in the same way under general uncorrelated decoherence. We present some applications of the sequential estimation protocol for mixed-state quantum computation. The third part defines nonlinear quantum metrology, where the coupling interaction can have multi-body couplings running over all subsets of constituents of the quantum probe with a given size. We obtain an scaling for parameter estimation that surpasses optimal linear quantum metrology, even with separable initial states of the probe and separable measurements. Interestingly, initial entanglement results in a universal improvement equal to the square root of the size of the probe. We first study with particular detail the case of a coupling between the probe and the unknown parameter of the form J2z , where Jz is an angular momentum operator. Finally, we propose an engineered interaction in a two-mode BEC for which entanglement is not created at any stage. Due to some lucky cancelation between the scattering lengths of two modes of 87Rb atoms we obtain an effective evolution of the form nJz, valid for short-times. This interaction clearly shows that entanglement is not a fundamental component of nonlinear quantum metrology, and explains the resilience of the nonlinear metrology scaling under decoherence.