Sub-shot noise sensitivities without entanglement

It is commonly maintained that entanglement is necessary to beat the shot noise limit in the sensitivity with which certain parameters can be measured in interferometric experiments. Here we show that, with a fluctuating number of two-mode bosons, the shot-noise limit can be beaten by non-entangled bosonic states with all bosons in one mode. For a given finite maximum number of bosons, we calculate the optimal one- and two-mode bosonic states, and show that in the absence of losses, NOON states are the optimal two-mode bosonic states.