won’t. We have a statistical model that provides us tools. • Outcomes • Probability density functions • Parameterized with unknown variable • Under measurement we collect data: We need to define the so-called estimator: φ The phase. Our guess on phase. This is how far off we are. What if we loose some photons? Oleg Żero
The mean square error is a candidate. • Variance is our measure of uncertainty. • Bias is related to the systematic error. Estimator is to be called unbiased and efficient if • bias is zero, and • variance meets This is so-called Cramer-Rao bound. variance bias2 where Is known as Fisher information.
tools to solve a quantum problem. We have have mixed state. And photon counters. So we can identify the following: • The probabilities are taken from quantum coefficients. • The parameter to be known is our phase. • The photodetectors’ clicks are our outcomes.
Fisher information. This is a function of likelihood of how accurate our guess is. And it has the properties we need: • It peaks up whenever our guess is correct. • It minimizes the variance. So all we need to do is to find input Fock coefficients, whose output probabilities maximize the Fisher information.
through branch [unity] Transmittance through branch [unity] Symmetric case Asymmetric case N = 2 The N00N state. Any particular state for a given interval. Classical measurement. The overall maximum. The N00N state matches the quantum limit when no loss is present. Otherwise, it’s decoherence. When loss is present, other more exotic states do better. There is a trade-off between sensitivity and robustness.
through branch [unity] Transmittance through branch [unity] Symmetric case Asymmetric case N = 2 The N00N state. Any particular state for a given interval. Classical measurement. The overall maximum. T. W. Lee et al.: “Optimization of quantum interferometric metrological sensors in the presence of photon loss” Physical Review A 80, 063803 (2009). Others were also pursuing this problem at that time. But comparing their work for N = 2, 3 photons, shows Perfect match!