Correlated Entropic Uncertainty as a Signature of Exceptional Points

TL;DR

The paper addresses why biorthogonality and the Petermann factor arise in non-Hermitian systems by introducing an entropic uncertainty framework between phase entropy and its Fourier representation. It develops circular-statistics tools and phase-map diagnostics to quantify phase locking and delocalization, showing that near exceptional points both phase and Fourier entropies peak while the Petermann factor diverges. The authors demonstrate this universal behavior in open elliptic cavities, revealing EPs as information-theoretic organizing centers that couple spectral suppression, entropy growth, and non-orthogonality. This framework provides a testable principle for non-Hermitian physics with potential applications across photonics, quantum sensing, and open-wave systems.

Abstract

Non-Hermitian physics has become a fundamental framework for understanding open systems where gain and loss play essential roles, with impact across photonics, quantum science, and condensed matter. While the role of complex eigenvalues is well established, the nature of the corresponding eigenfunctions has remained a long-standing problem. Here we show that it arises from a fundamental entropic uncertainty trade-off between phase entropy and its Fourier representation. This trade-off enforces a correlated behavior of phase and Fourier entropies near avoided crossings and exceptional points, precisely where the Petermann factor diverges and phase rigidity collapses. Our results establish biorthogonality is not as an anomaly but an intrinsic property of eigenfunctions, arising universal manifestation of uncertainty relation in non-Hermitian systems. Beyond resolving this foundational question, our framework provides a unifying and testable principle that advances the fundamentals of non-Hermitian physics and can be directly verified with existing interferometric techniques.

Figures (7)

• Figure 1: (a) Real parts of the eigenvalues $\operatorname{Re}(k)$ for the closed (integrable, Hermitian) elliptic billiard. The two indicated levels intersect in a simple crossing near $\epsilon\!\approx\!0.16$; representative eigenmodes A--F (right) show the standing-wave real-amplitude patterns on each branch and confirm the absence of hybridization. (b) The corresponding open (non-Hermitian) elliptic cavity: $\operatorname{Re}(k)$ now displays an avoided crossing (A.C.) near the same $\epsilon$ value. The central inset magnifies the avoided-crossing region to show the characteristic spectral repulsion in $\operatorname{Re}(k)$, while the side inset plots the imaginary parts $\operatorname{Im}(k)$ and reveals a true crossing of the decay rates (complementary motion in the complex eigenplane). Eigenmodes G--L (right) illustrate the structural exchange across the A.C.: modal intensity is transferred between branches as the pair hybridizes.
• Figure 2: Phase maps shown as 3D point clouds in $(x,y,\phi/\pi)$; color encodes $\phi/\pi$ (viridis), duplicating the vertical coordinate for visual emphasis. (a)--(f) Closed (Hermitian) elliptic cavity: the phase collapses to the planes $\phi/\pi\in\{0,\pm1\}$, consistent with standing-wave parity and strong phase locking. (g)--(l) Corresponding open (non-Hermitian) modes: the phase values diffuse away from the locked planes at $0$ and $\pm\pi$, indicating partial delocalization that weakens phase locking and reduces phase rigidity relative to the closed case. Panels (a)--(l) follow the same mode ordering as Fig. 1 (A--F for the closed case and G--L for the open case).
• Figure 3: (a) $R_1$ and $R_2$ for the closed-ellipse (Hermitian) modes (mode1: $k\approx5.44$, mode2: $k\approx5.92$). (b) $R_1$ and $R_2$ for the open-ellipse (non-Hermitian) modes (mode4: $k\approx5.49$, mode3: $k\approx4.96$). $R_1$ and $R_2$ are the first and second resultant lengths computed from phase distributions weighted by $|\psi|^2$ (see Methods). Data are shown as a function of the deformation parameter $\varepsilon\in[0.10,0.23]$. . The avoided crossing (A.C.) near $\varepsilon\approx0.16$ in the open system produces the characteristic dip in $R_2$ and an accompanying response in $R_1$.
• Figure 4: Phase entropy $S_{\phi}$ across an avoided crossing. (a) Hermitian (closed) modes (1,2): solid/dashed curves are the unfolded estimator, dotted/dash--dot curves are the folded estimator (histogrammed in $\theta=2\phi$). The inset shows the same traces computed without the $\mu_2$-alignment, illustrating edge-splitting artefacts. (b) Non-Hermitian (open) modes (3,4): both modes display a sharp maximum of $S_{\phi}$ at the A.C. center ($\varepsilon\approx0.1666$), signalling strong phase delocalization that coincides with the Petermann-factor spike discussed in the text. Entropies were computed from intensity-weighted histograms; values are reported in nats.
• Figure 5: (a) Avoided crossing of modes 5 and 6 very near an exceptional point; panels A--F show representative interior phase maps (A: pre--A.C., F: post--A.C.; exact $\varepsilon$ values are indicated on the panels). (b) Petermann factor $K$ (solid lines, left axis) and phase entropy $S_{\phi}$ (dashed lines, right axis) for the same modes; both quantities peak at the A.C.