Entanglement, non-hermiticity, and duality

Abstract

Usually duality process keeps energy spectrum invariant. In this paper, we provide a duality, which keeps entanglement spectrum invariant, in order to diagnose quantum entanglement of non-Hermitian non-interacting fermionic systems. We limit our attention to non-Hermitian systems with a complete set of biorthonormal eigenvectors and an entirely real energy spectrum. The original system has a reduced density matrix \rho_\mathrm{o}ρo and the real space is partitioned via a projecting operator \mathcal{R}_{\mathrm o}ℛo. After dualization, we obtain a new reduced density matrix \rho_{\mathrm{d}}ρd and a new real space projector \mathcal{R}_{\mathrm d}ℛd. Remarkably, entanglement spectrum and entanglement entropy keep invariant. Inspired by the duality, we defined two types of non-Hermitian models, upon \mathcal R_{\mathrm{o}}ℛo is given. In type-I exemplified by the "non-reciprocal model'', there exists at least one duality such that \rho_{\mathrm{d}}ρd is Hermitian. In other words, entanglement information of type-I non-Hermitian models with a given \mathcal{R}_{\mathrm{o}}ℛo is entirely controlled by Hermitian models with \mathcal{R}_{\mathrm{d}}ℛd. As a result, we are allowed to apply known results of Hermitian systems to efficiently obtain entanglement properties of type-I models. On the other hand, the duals of type-II models, exemplified by "non-Hermitian Su-Schrieffer-Heeger model’’, are always non-Hermitian. For the practical purpose, the duality provides a potentially computation route to entanglement of non-Hermitian systems. Via connecting different models, the duality also sheds lights on either trivial or nontrivial role of non-Hermiticity played in quantum entanglement, paving the way to potentially systematic classification and characterization of non-Hermitian systems from the entanglement perspective.