  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
R\'enyi 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 R\'enyi 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/R\'enyi entropy and the non-unitary CFT prediction.
Furthermore, we apply the generic entanglement/R\'enyi entropy to
symmetry-protected topological phases with non-Hermitian perturbations. We find
the generic $n$-th R\'enyi entropy captures the expected entanglement property,
whereas the traditional R\'enyi entropy can exhibit unnatural singularities due
to its improper definition.
