Achieving the Quantum Fisher Information Bound in Pseudo-Hermitian Sensors

TL;DR

The paper tackles whether pseudo-Hermitian sensors can surpass Hermitian limits in quantum parameter estimation by formulating a covariant quantum Fisher information (CQFI) on the η-deformed Hilbert space and introducing a covariant derivative that preserves state normalization. It proves a duality with the ordinary QFI of an associated Hermitian system, establishing an upper bound F_max and showing that optimal measurements saturate the CQFI bound via the covariant SLD eigenbasis, with projections Π_i=|ν_i⟩⟨ν_i|_η. The results imply that CQFI is bounded by the Hermitian maximum, yet under certain parameterizations, especially near EPs, pseudo-Hermitian systems can exhibit enhanced sensitivity due to nonlinear mappings in the Hermitian frame. The work provides a practical, dimension-preserving framework for achieving ultimate precision in pseudo-Hermitian sensing, including strategies for metric-dependent measurements and exemplars with nonreciprocal and PT-symmetric Hamiltonians.

Abstract

Non-Hermitian systems have attracted considerable interest over the last few decades due to their unique spectral and dynamical properties not encountered in Hermitian counterparts. An intensely debated question is whether non-Hermitian systems, described by pseudo-Hermitian Hamiltonians with real spectra, can offer enhanced sensitivity for parameter estimation when they are operated at or close to exceptional points. However, much of the current analysis and conclusions are based on mathematical formalism developed for Hermitian quantum systems, which is questionable when applied to pseudo-Hermitian Hamiltonians, whose Hilbert space metric is intrinsically parameter dependent. Here, we develop a covariant formulation of quantum Fisher information (QFI) defined on the deformed Hilbert space of pseudo-Hermitian Hamiltonians. This covariant framework ensures the preservation of the state norm and enables a consistent treatment of parameter sensitivity. We further show that the covariant QFI of pseudo-Hermitian systems is dual to the ordinary QFI of corresponding Hermitian systems. Importantly, this correspondence naturally imposes an upper bound on the covariant QFI and allows one to identify optimal projections which saturate the corresponding classical Fisher information to this ultimate limit. The developed framework also enables to set the criteria under which pseudo-Hermitian sensors can exhibit an advantage over their Hermitian counterparts of the same dimensionality.

Figures (4)

• Figure 1: Covariant classical Fisher information obtained from different measurement projectors in the nonreciprocal system of Example 1.b. The parameter $\gamma$ controls the deviation of the measurement basis $|\chi(\gamma)\rangle$ from the optimal axis defined by the eigenbasis of the covariant SLD operator (see the main text for details). The upper bound of the covariant quantum Fisher information for each case is shown by a red dotted curve. The vertical dash gray line corresponds to the value of $\theta=\theta_{\rm EP}=-0.5$ [arb. units], when the pseudo-Hermitian sensor is at the exceptional point.
• Figure 2: Covariant classical Fisher information in the $\cal PT$-symmetric system, obtained from different measurement projectors: (a) when the reference point $x_0$ coincides with the parameter point of the probe state $|\xi_t\rangle$, and (b) when $x_0$ differs from the probe-state parameter point. The parameter $\gamma$ controls the deviation of the measurement basis $|\chi(\gamma)\rangle$ from the optimal axis defined by the eigenbasis of the covariant SLD operator. The upper bound of the covariant quantum Fisher information for each case is shown by a red dotted curve. In both panels, the vertical dash gray line corresponds to the value of $\theta=\theta_{\rm EP}$, when the corresponding sensor is at the exceptional point.
• Figure S1: QFI for the state $|0_t\rangle$ in Eq. (\ref{['0t']}), calculated using the SQFI (blue dashed curves) and CQFI (green solid curves) formalisms. The dynamics is governed by the pseudo-Hermitian Hamiltonian in Eq. (\ref{['Hex']}). The system parameters are: (a) $\delta=0.5$ [arb. units], and (b) $\delta=0.2$ [arb. units]. The time is set to $t=\pi$ [arb. units]. The EPs are defined by $\theta_{\rm EP}=-0.5$ [arb. units] on panel (a), and $\theta_{\rm EP}=-0.2$ [arb. units] on panel (b). The locations of the EPs $\theta_{\rm EP}$ coincide with the positions of the $y$-axis on both panels.
• Figure S2: This figure is a complement to Fig. 1(a) in the main text for the case when the reference point $x_0$, chosen for the projectors, does not coincide with that of the system's state $|\psi_t\rangle$. The chosen system parameters are: (a) $\delta_0=\delta=1/2$, $\theta_0=-1/4$, $t_0=t=1$.