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Resolving unattainable Cramer–Rao bounds for quantum sensors

In quantum metrology, it is widely believed that the quantum Cramér–Rao bound is an attainable bound. However, this is not always true. In order to clarify this point, we explain why the quantum Cramér–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.