Uncertainty inequalities in a non-Hermitian scenario: the problem of the metric

TL;DR

The paper tackles the challenge of defining physical observables and uncertainties for non-Hermitian quantum dynamics by constructing regime-appropriate metric operators within a Krein-space framework. It derives a generalized Heisenberg–Robertson uncertainty relation that remains valid across unbroken $\ ext{PT}$ symmetry, broken-symmetry, and exceptional-point regimes, and applies the formalism to a two-level $\mathcal{PT}$-symmetric model. In the unbroken phase, uncertainties exhibit oscillations, while in the broken phase and at exceptional points the system relaxes toward a minimum-uncertainty steady state; the eigenfunctions become non-orthogonal in the Krein-space sense in the broken regime. The work further shows agreement with Lindblad master-equation descriptions in the steady state, providing a robust bridge between non-Hermitian effective dynamics and open-system formalisms. Overall, the metric-centric approach is shown to be essential for meaningful physical predictions in non-Hermitian quantum mechanics beyond the exact symmetry regime.

Abstract

We investigate uncertainty relations for quantum observables evolving under non-Hermitian Hamiltonians, with particular emphasis on the role of metric operators. By constructing appropriate metrics in each dynamical regime, namely the unbroken-symmetry phase, the broken-symmetry phase, and at exceptional points, we provide a consistent definition of expectation values, variances, and time evolution within a Krein-space framework. Within this approach, we derive a generalized Heisenberg-Robertson uncertainty inequality which is valid across all spectral regimes. As an application, we analyze a two level model with parity-time reversal symmetry and show that, while the uncertainty measure exhibits oscillatory behavior in the unbroken phase, it evolves toward a minimum-uncertainty steady state in the broken symmetry phase and at exceptional points. We further compare our metric-based description with a Lindblad master-equation approach and show their agreement in the steady state. Our results highlight the necessity of incorporating appropriate metric structures to extract physically meaningful predictions from non-Hermitian quantum dynamics.

Figures (12)

• Figure 1: Dynamical phases of the model of Eq. \ref{['hamieq1']} in terms of the values of $d=(s/r)^2-\sin^2\theta$ in the plane $(\theta,s/r)$. The region for which $d>0$ corresponds to the $\mathcal{PT}$-symmetry phase of the model, while the region with $d<0$ corresponds to the broken symmetry phase. The white border between both regions corresponds to the localisation of the EPs, where $d=0$.
• Figure 2: Contour plots of $UR(\sigma_x,\sigma_y)$, Eq. \ref{['eq:URsgxsgy']}, in the $(\phi,p)$-plane at $t=0$ for different values of $\eta$ within the $\mathcal{PT}$-symmetry region ($\eta^2<1$).
• Figure 3: Contour plots of $UR(\sigma_x,\sigma_y)$, Eq. \ref{['eq:URsxsynoPT']}, in the $(\phi,p)$-plane at $t=0$ for different values of $\eta$ within the broken symmetry region ($\eta^2>1$).
• Figure 4: Overlap $|\langle + |- \rangle _S|$ between the states with energies $E_+$ and $E_-$, as a function of $\eta$, Eq. (\ref{['eq:over']}).
• Figure 5: Contour plots of $UR(\sigma_x,\sigma_y)$, Eq. \ref{['eq:URsxsyEP']}, in the $(\phi,p)$-plane at $t=0$ at the exceptional points $(\eta=\pm 1)$.