Hilbert–Schmidt speed as an efficient figure of merit for quantum estimation of phase encoded into the initial state of open n-qubit systems

Abstract

Hilbert–Schmidt speed (HSS) is a special type of quantum statistical speed which is easily computable, since it does not require diagonalization of the system state. We find that, when both HSS and quantum Fisher information (QFI) are calculated with respect to the phase parameter encoded into the initial state of an n-qubit register, the zeros of the HSS dynamics are actually equal to those of the QFI dynamics. Moreover, the signs of the time-derivatives of both HSS and QFI exactly coincide. These findings, obtained via a thorough investigation of several paradigmatic open quantum systems, show that HSS and QFI exhibit the same qualitative time evolution. Therefore, HSS reveals itself as a powerful figure of merit for enhancing quantum phase estimation in an open quantum system made of n qubits. Our results also provide strong evidence for both contractivity of the HSS under memoryless dynamics and its sensitivity to system-environment information backflows to detect the non-Markovianity in high-dimensional systems, as suggested in previous studies.

Estimating energy levels of a three-level atom in single and multi-parameter metrological schemes

“…It can be easily computed without diagonalization of dρ(j)/dj. In [71], the HSS has been introduced as a powerful figure of merit for enhancing quantum phase estimation in an open quantum system made of n qubits. More generally, because both QFI and HSS are quantum statistical speeds associated, respectively, with the Bures and HilbertSchmidt distances, it is reasonable to explore how they can be related to each other.…”

Monitoring variations of refractive index via Hilbert–Schmidt speed and applying this phenomenon to improve quantum metrology

“…The field of quantum metrology [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] has provided valuable tools, such as the classical Fisher information and the quantum Fisher information (QFI) [19][20][21][22][23][24][25][26][27], for estimating the sensitivity of measurement devices and unknown parameters. Such analyses are usually focused on finding optimal measurement strategies, with respect to environmental parameters as well as initial conditions, to achieve measurement sensitivities better than the standard quantum limit [28].…”

“…Therefore, researchers have explored alternative ways to achieve negative refraction, such as using materials with only one negative parameter (either ϵ or õ) [32][33][34][35][36][37], or using composite structures that combine different materials with positive and negative parameters. In this paper, we investigate how QFI and Hilbert-Schmidt speed (HSS), two important quantum metrological tools [10], can be used to detect and characterize these phenomena in atomic media with nonlinear optical responses.…”

Memory Effects in High-Dimensional Systems Faithfully Identified by Hilbert–Schmidt Speed-Based Witness

“…Recently, Hilbert-Schmidt speed (HSS) [60], a measure of quantum statistical speed which has the advantage of avoiding diagonalization of the evolved density matrix, has been proposed and employed as a faithful witness of non-Markovianity in Hermitian systems [61][62][63][64] and an efficient tool in quantum metrology [65,66]. These studies are so far especially limited to low-dimensional systems, while high-dimensional ones have not been investigated in detail.…”