Entanglement, non-hermiticity, and duality
Limei Chen, Shuai A. Chen, Peng Ye
Abstract
Usually duality process keeps energy spectrum invariant. In thispaper, we provide a duality, which keeps entanglement spectruminvariant, in order to diagnose quantum entanglement of non-Hermitiannon-interacting fermionic systems. We limit our attention tonon-Hermitian systems with a complete set of biorthonormal eigenvectorsand an entirely real energy spectrum. The original system has a reduceddensity matrix \rho_\mathrm{o} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mstyle mathvariant="normal"> <mml:mi>o</mml:mi> </mml:mstyle> </mml:msub> </mml:math> and the real space is partitioned via a projecting operator \mathcal{R}_{\mathrm o} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mstyle mathvariant="script"> <mml:mi>ℛ</mml:mi> </mml:mstyle> <mml:mstyle mathvariant="normal"> <mml:mi>o</mml:mi> </mml:mstyle> </mml:msub> </mml:math> .After dualization, we obtain a new reduced density matrix \rho_{\mathrm{d}} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mstyle mathvariant="normal"> <mml:mi>d</mml:mi> </mml:mstyle> </mml:msub> </mml:math> and a new real space projector \mathcal{R}_{\mathrm d} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mstyle mathvariant="script"> <mml:mi>ℛ</mml:mi> </mml:mstyle> <mml:mstyle mathvariant="normal"> <mml:mi>d</mml:mi> </mml:mstyle> </mml:msub> </mml:math> .Remarkably, entanglement spectrum and entanglement entropy keepinvariant. Inspired by the duality, we defined two types ofnon-Hermitian models, upon \mathcal R_{\mathrm{o}} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mstyle mathvariant="script"> <mml:mi>ℛ</mml:mi> </mml:mstyle> <mml:mstyle mathvariant="normal"> <mml:mi>o</mml:mi> </mml:mstyle> </mml:msub> </mml:math> is given. In type-I exemplified by the "non-reciprocal model'', there exists at least one duality such that \rho_{\mathrm{d}} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mstyle mathvariant="normal"> <mml:mi>d</mml:mi> </mml:mstyle> </mml:msub> </mml:math> is Hermitian. In other words, entanglement information of type-Inon-Hermitian models with a given \mathcal{R}_{\mathrm{o}} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mstyle mathvariant="script"> <mml:mi>ℛ</mml:mi> </mml:mstyle> <mml:mstyle mathvariant="normal"> <mml:mi>o</mml:mi> </mml:mstyle> </mml:msub> </mml:math> is entirely controlled by Hermitian models with \mathcal{R}_{\mathrm{d}} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mstyle mathvariant="script"> <mml:mi>ℛ</mml:mi> </mml:mstyle> <mml:mstyle mathvariant="normal"> <mml:mi>d</mml:mi> </mml:mstyle> </mml:msub> </mml:math> . 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-HermitianSu-Schrieffer-Heeger model’’, are always non-Hermitian. For thepractical purpose, the duality provides a potentially computation routeto entanglement of non-Hermitian systems. Via connecting differentmodels, the duality also sheds lights on either trivial or nontrivialrole of non-Hermiticity played in quantum entanglement, paving the wayto potentially systematic classification and characterization ofnon-Hermitian systems from the entanglement perspective.