Quantum Parameter Estimation in the Laboratory

In this thesis, I report on three projects which I have completed during my PhD. All of these projects are related to quantum parameter estimation.When simultaneously measuring all the parameters describing a unitary transformation, a decision needs to be made regarding the relative importance of these different parameters. In our first project, we show that the geometry of the group of unitaries in question can guide us toward a natural way of weighting these dif- ferent parameters. This allows us to transform the matrix Cram ́er-Rao inequality into a simple parameter-independent scalar inequality. We use this new inequality to calculate the best possible performances for a measurement of the parameters that describe a general SU(d) unitary and we find an optimal scalable family of quantum states which achieves this performance for SU(3) unitaries. In our second project, we experimentally create and characterize the tetrahedron state in the polarization of four photons. This entangled state is the optimal 4-photon state for measuring the three parameters describing a SU(2) polarization rotation. Interestingly, it possesses the same rotational symmetries as a tetrahedron, giving the state its name. We perform state tomography on our experimental state and measure a quantum state fidelity of F = (0.46 ± 0.02) with the ideal state. Our state nevertheless shares many qualitative features with the latter. Finally, in our last project, we propose a technique named SPLICE which allows us to beat “Rayleigh’s curse”—the inability to measure the separation between two incoherent point sources of light when said separation is below the imaging apparatus’ Rayleigh’s criterion. We test this technique in a proof of principle experiment in the laboratory and demonstrate its advantage over traditional methods.Ph.D