Precision assessment in non-Hermitian systems: a comparative study of three formalisms

Abstract

Quantifying measurement precision in quantum systems is vital for advancing quantum technologies such as sensing, communication, and computation. The quantum Fisher information (QFI) sets the ultimate precision bound in Hermitian systems; however, extending this concept to non-Hermitian systems, even those with real spectra, poses conceptual challenges due to their non-unitary dynamics. We compare three probability-conserving approaches for evaluating QFI in such systems: (i) simple normalization, (ii) metric formalism, and (iii) master-equation framework. Although all three ensure probability conservation, they differ in physical interpretation and in how they quantify estimation precision. Our study is particularly motivated by previous studies that have shown that the simple normalization method for non-Hermitian Hamiltonian generated dynamics may lead to misleading or even unphysical conclusions for certain quantum information theoretic tasks. We emphasize, in this article, that the metric formalism naturally enables the use of standard Hermitian metrology tools in cases where it provides a coherent and physically consistent framework for non-Hermitian systems.

Figures (8)

• Figure 1: Different possible regions in the parameter space of a specific Hamiltonian $H = -\mathrm{i} \gammagg\mathrm{i} \gamma$, illustrating the transition along the slice $g=g_0$. The system moves from $\Omega_\mathrm{H}^\mathrm{R}$ at $\gamma = 0$ to $\Omega_\mathrm{NH}^\mathrm{R}$ for $0 < \gamma < g$, and then to $\Omega_\mathrm{NH}^\mathrm{C}$ for $0 < g < \gamma$. Here, $\Omega_{\rm H}^{\rm R}$ and $\Omega_{\rm NH}^{\rm C}$ denote regions in the parameter space of $H$ corresponding to Hermitian Hamiltonians with real eigenvalues and NHHs with complex eigenvalues, respectively, while $\Omega_{\rm NH}^{\rm R}$ refers to the intermediate region of NHHs that still possess real eigenvalues.
• Figure 2: Types of dynamics and their mutual relationships (left), and correponding formalisms (right). (a) Note that the master-equation formalism corresponds to the standard quantum-trajectory approach without post-selection, i.e., allowing an arbitrary number of quantum jumps, whereas the normalization formalism corresponds to the quantum-trajectory approach post-selected on no quantum jumps. Post-selection in open systems (Liouvillian) dynamics leads to non-Hermitian effective Hamiltonians. (b) Given a non-Hermitian Hamitlonian, in an appropriately transformed reference frame, dubbed "Einstein's quantum elevator" following Ju2022, a non-Hermitian Hamiltonian is rendered Hermitian. An equivalent description of the physical system is obtained via the metric formalism.
• Figure 3: Different ways of evaluating the QFI given an initial state $\rho_\mathrm{in}(\theta)$ and a NHH $\tilde{H}$ generating the evolution. The approaches include: (1) normalizing the non-Hermitian state $\tilde{\rho}(t)$ by its trace at each time step, (2) transforming the state into a metric space with a modified inner product, (3) solving the full master equation to incorporate both coherent evolution and dissipation in open systems. The reason we compare these approaches is that they all describe probability-conserving dynamics.
• Figure 4: A set of parameters $\bm{\theta}$ is associated with non-Hermitian system $\mathcal{S}_\mathrm{NH}$, where direct estimation of $\bm{\theta}$ is challenging. To simplify the process, $\mathcal{S}_\mathrm{NH}$ is mapped, using a proper metric, to a Hermitian system $\mathcal{S}_\mathrm{H}$, which retains the same parameters but allows for easier inference. The estimation of $\bm{\theta}$ in metric space is then achieved using the conventional QFI formalism for Hermitian systems.
• Figure 5: (a)Evolution of different states in the Bloch sphere: $\rho_\mathrm{metric}(t)$ in Eq. (\ref{['eq:rho1metric']}), time normalized $\tilde{\rho}_\mathrm{norm}(t)$ in Eq. (\ref{['eq:rhoNorm']}), and $\rho_\mathrm{me}(t)$ -- the solution of the master equation Eq. (\ref{['eq:rhoME']}). The initial state corresponds to Bloch vector $(0.48,0,-0.2)$ and pertains to $\theta = 0.6$ and $x = 0.24$ in Eq. (\ref{['eq:initialstate']}). The Hamiltonian parameters used are $g = 0.5$ and $\gamma = 0.4$. (b)The evolution projected onto the Bloch sphere equator.