Quantum entanglement is one essential element to characterize many-body quantum systems. However, the entanglement measures are mostly discussed in Hermitian systems. Here, we propose a natural extension of entanglement and Rényi entropies to non-Hermitian quantum systems. There have been other proposals for the computation of these quantities, which are distinct from what is proposed in the current paper. We demonstrate the proposed entanglement quantities which are referred to as generic entanglement and Rényi entropies. These quantities capture the desired entanglement properties in non-Hermitian critical systems, where the low-energy properties are governed by the non-unitary conformal field theories (CFTs). We find excellent agreement between the numerical extrapolation of the negative central charges from the generic entanglement/Rényi entropy and the non-unitary CFT prediction. Furthermore, we apply the generic entanglement/Rényi entropy to symmetry-protected topological phases with non-Hermitian perturbations. We find the generic n-th Rényi entropy captures the expected entanglement property, whereas the traditional Rényi entropy can exhibit unnatural singularities due to its improper definition.

The fidelity susceptibility is a tool for studying quantum phase transitions in the Hermitian condensed matter systems. Recently, it has been generalized with the biorthogonal basis for the non-Hermitian quantum systems. From the general perturbation description with the constrain of parity-time (PT) symmetry, we show that the fidelity F is always real for the PT-symmetric states. For the PT-broken states, the real part of the fidelity susceptibility equals to one half of the sum of the fidelity susceptibility of the PT-broken and the PT-partner states, Re[XF ] = 1 2 (XF + XF ). The negative infinity of the fidelity susceptibility is explored by the perturbation theory when the parameter approaches the exceptional point (EP). Moreover, at the second-order EP where two eigenstates and eigenenergies coalesce, we prove that the real part of the fidelity between PTsymmetric and PT-broken states is ReF = 1 2 . We demonstrate these general properties for noninteracting and interacting systems by two examples: the two-legged non-Hermitian Su-Schrieffer-Heeger (SSH) model and the non-Hermitian XXZ spin chain.

The fidelity susceptibility is a tool for studying quantum phase transitions in the Hermitian condensed matter systems. Recently, it has been generalized with the biorthogonal basis for the non-Hermitian quantum systems. From the general perturbation description with the constraint of parity-time (PT) symmetry, we show that the fidelity F is always real for the PT-unbroken states. For the PT-broken states, the real part of the fidelity susceptibility Re[XF] is corresponding to considering both the PT partner states, and the negative infinity is explored by the perturbation theory when the parameter approaches the exceptional point (EP). Moreover, at the second-order EP, we prove that the real part of the fidelity between PT-unbroken and PT-broken states is ReF=12. Based on these general properties, we study the two-legged non-Hermitian Su-Schrieffer-Heeger (SSH) model and the non-Hermitian XXZ spin chain. We find that for both interacting and non-interacting systems, the real part of fidelity susceptibility density goes to negative infinity when the parameter approaches the EP, and verifies it is a second-order EP by ReF=12.