Unbounded quantum Fisher information in two-path interferometry with finite photon number

Abstract

The minimum error of unbiased parameter estimation is quantified by the quantum Fisher information in accordance to the Cramér-Rao bound. We indicate that only superposed NOON states by simultaneous measurements can achieve the maximum quantum Fisher information with form N 2 for a given photon number distribution by a POVM in linear two-path interferometer phase measurement. We present a series of specified superposed states with infinite quantum Fisher information but with finite average photon numbers. The advantage of this unbounded quantum Fisher information will be beneficial to many applications in quantum technology.

Using states with a large photon number variance to increase quantum Fisher information in single-mode phase estimation

“…One can consider distributions such as the Riemann-Zeta distribution, the Beta negative binomial, or the Yule-Simon distribution, all of which exhibit a diverging or an infinite variance of the photon number. Particularly, the Riemann Zeta distribution has already been considered as an interesting example showing an infinite QFI in two-mode schemes [73]. These examples seem to provide the completely precise estimation, but it turned out that it is not the case (see more detailed discussion in [31]).…”

Non-asymptotic analysis of quantum metrology protocols beyond the Cramér–Rao bound

“…The latter limitation has been addressed in the context of the maximum-likelihood strategy [11,12], and more recently with the quantum Ziv-Zakai and Weiss-Weinstein bounds [13,14], which also incorporate the effect of the prior information. Nevertheless, the previous restrictions are somestimes not taken into account, in spite of the fact that a naive use of the Fisher information can predict schemes with an apparent infinite precision [15][16][17] which are inefficient in practice [4,13,16,18,19]. Since in general it is not possible to foresee when and how the Cramér-Rao bound is going to fail in a concrete practical scenario from the asymptotic theory itself, a closer analysis of those schemes that are asymptotically optimal is needed.…”

Breaking the weak Heisenberg limit

“…The actual meaning of the strong Heisenberg limit has been much debated [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30]. However, the weak Heisenberg limit has not so extensively examined [5,23,30], although it has a more deep practical meaning as discussed above.…”