Phase super-sensitivity Oleg Żero General Question: Given the power measurement, how precisely are we allowed to determine the phase-shitf that caused it? My Question: What is are the best quantum states to ensure maximum information when the system is lossy? Δφ

Phase super-sensitivity Oleg Żero ħω P(φ) φ Classical light’s sensitivity is shot-noise limited. It is called classical limit. Quantum light is limited by the number of photons. It is called quantum limit, or super-sensitivity.

Phase super-sensitivity Oleg Żero Speaking ”quantum”, classical light is a special superposition of Fock states. However, in quantum world, the state can be anything. ħω P(φ) φ The goal is to find states, such that:

Phase super-sensitivity Then some detectors will click... ...and some others 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

Phase super-sensitivity Oleg Żero We need to define an estimator... 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.

Phase super-sensitivity Oleg Żero This is getting abstract... I know. This is so-called Cramer-Rao bound. But we aslo have quantum Cramer Rao bound. It is expressed through some abstract operator.

Phase super-sensitivity Oleg Żero Why not maximising the quantum Fisher information? We need. Neither of these possibilities exists for us... 1. possibility... A canonical measurement where 2. possibility... A magical mechanism for monitoring of losses. φ

Phase super-sensitivity Oleg Żero So we are left with classical 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.

Phase super-sensitivity Oleg Żero We build the estimator that maximizes Fisher information. From Bayes theorem we know that: So performing the measurement again and again...

Phase super-sensitivity Oleg Żero We build the estimator that maximizes Fisher information. From Bayes theorem we know that: So performing the measurement again and again...

Phase super-sensitivity Oleg Żero We build the estimator that maximizes 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.

Phase super-sensitivity Oleg Żero The results. Fisher information [a.u.] Transmittance 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.

Phase super-sensitivity Oleg Żero The results. Fisher information [a.u.] Transmittance 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!