We present an optimal strategy having finite outcomes for estimating a single parameter of the displacement operator on an arbitrary finite dimensional system using a finite number of identical samples. Assuming the uniform {\\it a priori} distribution for the displacement parameter, an optimal strategy can be constructed by making the {\\it square root measurement} based on uniformly distributed sample points. This type of measurement automatically ensures the global maximality of the figure of merit, that is, the so called average score or fidelity. Quantum circuit implementations for the optimal strategies are provided in the case of a two dimensional system.
We present an optimal strategy having finite outcomes for estimating a single parameter of the displacement operator on an arbitrary finite-dimensional system using a finite number of identical samples. Assuming the uniform a priori distribution for the displacement parameter, an optimal strategy can be constructed by making the square root measurement based on uniformly distributed sample points. This type of measurement automatically ensures the global maximality of the figure of merit, that is, the so-called average score or fidelity. Quantum circuit implementations for the optimal strategies are provided in the case of a two-dimensional system.
Indefinite causal order is known to offer enhancements in quantum metrology, notably including an unbounded advantage in the measurement of a geometric phase of a harmonic oscillator. This advantage, however, is specific to the infinite dimensional setting, and its finite dimensional analogue remains elusive, with recent findings suggesting that advantages in finite dimensions may be fundamentally limited to bounded constant factors. Here we show that, in fact, arbitrarily large advantages arise for finite dimensional systems in the finite sample regime. Specifically, we establish an unbounded separation between definite and indefinite causal order in the estimation of a geometric phase associated to two sets of
N displacements generated by discrete position and momentum operators on a
d-dimensional quantum system: for any given constant
R, there exist values of
N and
d=Ω(N2) such that a strategy with indefinite order uses an initial probe with
R times less energy than the probe required by every strategy with definite order achieving the same mean squared error, whenever the number of measurement shots
ν is bounded as
ν=O(exp(πd/16)/poly(d)). In other words, indefinite order offers an energy saving that grows arbitrarily large with the parameters of the problem. To prove this result, we establish an approximate Weyl relation for discrete Gaussian wavepackets, which is of independent technical interest.
Phase estimation in quantum interferometry is a major scenario where the quantum advantage is significantly revealed. Recently, the optimal finite-dimensional probe states (OFPSs) for phase estimation in two-mode quantum interferometry have been provided with the absence of noise [J.-F. Qin et al., Phys. Rev. A 112, 052428 (2025)]. However, the noise is inevitable in practice and the previously obtained OFPSs may cease to be optimal anymore. Hence, the forms of the true OFPSs in the existence of various noises are still open questions. Hereby, the noise of particle loss is studied and the true OFPSs under this noise have been investigated with the numerical algorithm named constrained optimization by linear approximation. Furthermore, a two-step measurement strategy is proposed to realize the ultimate precision limit in practice. The validity of this strategy is confirmed by the numerical simulation of practical experiments.