Resolving unattainable Cramer-Rao bounds for quantum sensors

In quantum metrology, it is widely believed that the quantum Cramer-Rao bound is an attainable bound. However, this is not always true. In order to clarify this point, we explain why the quantum Cramer-Rao bound cannot be attained. Here, we investigate noiseless channel estimation under energy constraint for states, using a physically reasonable error function, and present a state achieving the optimal rate as well as the attainable bound. We propose an experimental generation of the optimal states for enhanced metrology using squeezing transformations. This makes phase/unitary estimation physically implementable.