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Unattainable & attainable bounds for quantum sensors

In quantum metrology, it is widely believed that the quantum Cramer-Rao bound is attainable bound while it is not true. In order to clarify this point, we explain why the quantum Cramer-Rao bound cannot be attained geometrically. In this manuscript, we investigate noiseless channel estimation under energy constraint for states, using a physically reasonable error function, and present the optimal state and the attainable bound. We propose the experimental generation of the optimal states for enhanced metrology using squeezing transformations. This makes the estimation of unitary channels physically implementable, while existing unitary estimation protocols do not work