Estimation of Phase and Diffusion: Combining Quantum Statistics and Classical Noise

Quantum Artificial Intelligence Laboratory (QuAIL), NASA Ames Research Center, Moffett Field, California 94035, USA(Dated: January 10, 2014)Coherent ensembles of Nqubits present an advantage in quantum phase estimation over separable mixtures,but coherence decay due to classical phase diffusion reduces overall precision. In some contexts, the strengthof diffusion may be the parameter of interest. We examine estimation of both phase and diffusion in large spinsystems using a novel mathematical formulation. For the first time, we show a closed form expression for thequantum Fisher information for estimation of a unitary parameter in a noisy environment. The optimal probestate has a non-Gaussian profile and differs also from the canonical phase state; it saturates a new tight precisionbound. For noise below a critical threshold, entanglement always leads to enhanced precision, but the shot-noiselimit is beaten only by a constant factor, independent of N. We provide upper and lower bounds to this factor,valid in low and high noise regimes. Unlike other noise types, it is shown for N˛1 that phase and diffusioncan be measured simultaneously and optimally by canonical phase measurements.