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arXiv:2009.06050

Hilbert-Schmidt speed as an efficient tool in quantum metrology

Hossein Rangani Jahromi, Rosario Lo Franco

Published Sep 13, 2020• 0 citations
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Abstract

We investigate how the Hilbert-Schmidt speed (HSS), a special type of quantum statistical speed, can be exploited as a powerful and easily computable tool for quantum phase estimation in a $n$-qubit system. We find that, when both the HSS and quantum Fisher information (QFI) are computed with respect to the phase parameter encoded into the initial state of the $n$-qubit register, the zeros of the HSS dynamics are essentially the same as those of the QFI dynamics. Moreover, the positivity (negativity) of the time-derivative of the HSS exactly coincides with the positivity (negativity) of the time-derivative of the QFI. Our results also provide strong evidence for contractivity of the HSS under completely positive and trace preserving maps in high-dimensional systems, as predicted in previous studies.

DOI
10.48550/arxiv.2009.06050

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We investigate how the Hilbert-Schmidt speed (HSS), a special type of quantum statistical speed, can be exploited as a powerful and easily computable tool for quantum phase estimation in a nnn-qubit system. We find that, when both the HSS and quantum Fisher information (QFI) are computed with respect to the phase parameter encoded into the initial state of the nnn-qubit register, the zeros of the HSS dynamics are essentially the same as those of the QFI dynamics. Moreover, the positivity (negativity) of the time-derivative of the HSS exactly coincides with the positivity (negativity) of the time-derivative of the QFI. Our results also provide strong evidence for contractivity of the HSS under completely positive and trace preserving maps in high-dimensional systems, as predicted in previous studies.
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.
A witness of non-Markovianity based on the Hilbert-Schmidt speed (HSS), a special type of quantum statistical speed, has been recently introduced for low-dimensional quantum systems. Such a non-Markovianity witness is particularly useful, being easily computable since no diagonalization of the system density matrix is required. We investigate the sensitivity of this HSS-based witness to detect non-Markovianity in various high-dimensional and multipartite open quantum systems with finite Hilbert spaces. We find that the time behaviors of the HSS-based witness are always in agreement with those of quantum negativity or quantum correlation measure. These results show that the HSS-based witness is a faithful identifier of the memory effects appearing in the quantum evolution of a high-dimensional system with a finite Hilbert space.
We address the ratio between the trace speed and the Hilbert-Schmidt speed for differentiable finite-dimensional quantum states, R=∥∂φρ∥1/∥∂φρ∥2\mathcal R=\|\partial_φρ\|_1/\|\partial_φρ\|_2R=∥∂φ​ρ∥1​/∥∂φ​ρ∥2​. Since the tangent ∂φρ\partial_φρ∂φ​ρ is Hermitian and traceless, R\mathcal RR obeys stronger bounds than those for generic operators. For any nonzero tangent of rank rrr, we prove the sharp bounds 2≤R≤r\sqrt{2}\le\mathcal R\le\sqrt{r}2​≤R≤r​ for even rrr and 2≤R≤r−1/r\sqrt{2}\le\mathcal R\le\sqrt{r-1/r}2​≤R≤r−1/r​ for odd rrr, characterizing all equality cases. Nonstationary pure-state and qubit families saturate the lower bound R=2\mathcal R=\sqrt2R=2​. For odd Hilbert-space dimension ddd, we further prove the sharp global maximum R≤d−1/d\mathcal R\le\sqrt{d-1/d}R≤d−1/d​. Interpreting R2\mathcal R^2R2 as the participation ratio of the singular-value distribution yields an effective-rank picture, reff=R2r_{\mathrm{eff}}=\mathcal R^2reff​=R2. We decompose the effective rank into classical eigenvalue and quantum eigenvector contributions and obtain the bound reff≤rC+rQr_{\mathrm{eff}}\le r_C+r_Qreff​≤rC​+rQ​, with equality when either component vanishes. Linking the effective rank to the quantum Fisher information FFF gives reff≥8 TS2/Fr_{\mathrm{eff}}\ge 8\,\mathrm{TS}^2/Freff​≥8TS2/F, showing that a large effective rank is required when this lower bound substantially exceeds the universal minimum value 222. Finally, a hierarchy of quantum speed limits shows how the effective rank controls the tightness of bounds expressed through the Hilbert-Schmidt speed.
Hilbert-Schmidt speed (HSS)-based witness, a quantifier of quantum statistical speed having the advantage of avoiding the diagonalization of the evolved density matrix, has been introduced as a trustful witness of non-Markovianity in low-dimensional systems. We investigate the sensitivity of this HSS-based witness to detect non-Markovianity in high-dimensional and multipartite open quantum systems. We find that the time behaviors of the HSS-based witness are always in agreement with those of quantum negativity or quantum correlation measures. These results show that the HSS-based witness is a faithful identifier of the memory effects appearing in the quantum evolution of a high-dimensional system.
We investigate the Hilbert--Schmidt speed as a witness of non-Markovianity in finite-dimensional quantum dynamics. For a differentiable one-parameter family of quantum states, the Hilbert--Schmidt speed is defined, up to a conventional factor, as the Hilbert--Schmidt norm of the corresponding Hermitian traceless tangent operator. We show that, for unital dynamics in arbitrary finite dimension, P-divisibility implies the monotonic decrease of the Hilbert--Schmidt speed. Hence, any increase in this quantity signals the breakdown of P-divisibility, and therefore also excludes CP-divisibility. For qubit systems, we establish a stronger result: every P-divisible evolution decreases the Hilbert--Schmidt speed, independently of unitality. Thus, in dimension two, HSS growth is a valid witness of non-Markovian behaviour for arbitrary positive divisible dynamics. This conclusion is dimension dependent. In dimensions d≥3d\geq3d≥3, non-unitality can generate HSS growth even under CP-divisible dynamics. We demonstrate this by constructing an explicit non-unital CP-divisible qutrit semigroup for which the Hilbert--Schmidt speed strictly increases. Finally, for unital GKSL dynamics, we derive a generator-level dissipation identity explaining the monotonic decay of the Hilbert--Schmidt speed. These results specify the regimes in which HSS growth can be interpreted as evidence of the failure of divisibility and clarify its limitation for higher-dimensional non-unital evolutions.
In this paper, we demonstrate that the trace speed (TS), a statistical measure based on the trace norm of the derivative of the density matrix, serves as a powerful and precise tool for locating exceptional points (EPs) in quantum systems with parity-time (PT) and anti-PT symmetry. This detection of EPs is performed by benchmarking TS against the Hilbert-Schmidt speed (HSS), another efficient statistical measure. Analytical derivations and numerical simulations reveal that the dynamics of TS undergo characteristic changes when traversing EPs, enabling their exact positions to be determined unambiguously. Furthermore, TS can be more readily accessible experimentally than HSS, since it relates directly to trace-norm changes that can be estimated from feasible measurement schemes. Owing to its high sensitivity to small parameter variations and straightforward laboratory implementation, TS is a promising figure of merit for ultra-sensitive quantum sensing and for the optimization of EP-based photonic architectures. These results provide a foundation for exploring more complex non-Hermitian systems and their technological applications.