Phase precision in optimal 2-channel quantum interferometry is studied in the limit of large photon number N 1, for losses occurring in either one or both channels. For losses in one channel an optimal state undergoes an intriguing sequence of local bifurcations as the losses or the number of photons increase. We further show that fixing the loss paramater determines a scale for quantum metrology -a crossover value of the photon number Nc beyond which the supra-classical precision is progressively lost. For large losses the optimal state also has a different structure from those considered previously. 42.50.St,06.20.Dk It has been recognized that using quantum states of light may increase the resolution of interferometric measurements [1][2][3]. Particular states of N photons achieve the Heisenberg limit of phase resolution for standard error on the phase estimate ∆ϕ = 1/N , an improvement over the classical (or shotnoise) limit ∆ϕ = 1/ √ N that is obtainable when N photons enter the interferometer one at a time. These bounds are derived by an application of the Cramer-Rao inequality [2] for the standard error of an unbiased estimator, ∆ϕ ≥ (νF) −1/2 , where F is the quantum Fisher information (QFI) [4] and ν is the number of repeated independent trials. Assuming any instrument is composed of three components: quantum input state, dynamics and measurement; the functional F depends only on the first two -it assumes an optimal measurement choice. For pure states in a single mode F/4 = ∆ 2n ≡ n 2 − n 2 (wheren is the number operator) and a familiar uncertainty relation is recovered: ∆n∆ϕ ≥ 1/2. Thus, for a lossless two-mode interferometer QFI and precision are greatest for the maximum variance state, or 'NOON state'; it saturates the Heisenberg limit. Unfortunately, it is also highly susceptible to noise, especially dissipation [5].To mitigate this problem various two-component states were proposed [6][7][8], where the loss of a number of photons in the first mode does not destroy the superposition. The precision performance under dissipation of various Gaussian states, e.g. squeezed, coherent and thermal states, has also been considered recently [9]. In all cases, the precision was found to be supra-classical for certain range of losses and N .In the lossy case the pure input state of two oscillator modes maximizing QFImust balance supra-classical precision against robustness to photon loss. In this notation the NOON state has two non-zero components, φ 0 = φ N = 1/ √ 2. For a lossy interferometer light propagates in each arm as a damped harmonic oscillator, with frequencies ω (1) , ω (2) and dissipation γ (1) , γ (2) . Equivalently, losses can be introduced by beam-splitters in each mode with reflectivity R (1,2) = 1 − exp{−γ (1,2) t}. Those lost photons siphoned out of the modes are then traced over.In the simpler case of losses in only one of the two modes, R (1) = R > 0, R (2) = 0, as might occur when that mode is directed through a partially transparent test sample, the state |φ decays into a mixtureρ = k ...

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Quantum metrology aims to realise new sensors operating at the ultimate limit of precision measurement. However, optical loss, the complexity of proposed metrology schemes and interferometric instability each prevent the realisation of practical quantumenhanced sensors. To obtain a quantum advantage in interferometry using these capabilities, new schemes are required that tolerate realistic device loss and sample absorption. We show that loss-tolerant quantum metrology is achievable with photoncounting measurements of the generalised multi-photon singlet state, which is readily generated from spontaneous parametric downconversion without any further state engineering. The power of this scheme comes from coherent superpositions, which give rise to rapidly oscillating interference fringes that persist in realistic levels of loss. We have demonstrated the key enabling principles through the four-photon coincidence detection of outcomes that are dominated by the four-photon singlet term of the four-mode downconversion state. Combining state-of-the-art quantum photonics will enable a quantum advantage to be achieved without using post-selection and without any further changes to the approach studied here.