  Entanglement entropy is a powerful tool in characterizing universal features
in quantum many-body systems. In quantum chaotic Hermitian systems, typical
eigenstates have near maximal entanglement with very small fluctuations. Here,
we show that for Hamiltonians displaying non-Hermitian many-body quantum chaos,
modeled by the Ginibre ensemble, the entanglement entropy of typical
eigenstates is greatly suppressed. The entropy does not grow with the Hilbert
space dimension for sufficiently large systems and the fluctuations are of
equal order. We derive the novel entanglement spectrum that has infinite
support in the complex plane and strong energy dependence. We provide evidence
of universality and similar behavior is found in the non-Hermitian
Sachdev-Ye-Kitaev (nSYK) model, indicating the general applicability of the
Ginibre ensemble to dissipative many-body quantum chaos.
