The eigenstate thermalization hypothesis (ETH) represents a cornerstone in the theoretical understanding of the emergence of thermal behavior in closed quantum systems. The ETH asserts that expectation values of simple observables in energy eigenstates are accurately described by smooth functions of the thermodynamic parameters, with fluctuations and off-diagonal matrix elements exponentially suppressed in the entropy. We investigate to what extent the ETH holds in non-Hermitian many-body systems and come to the surprising conclusion that the fluctuations between eigenstates are of equal order to the average, indicating no eigenstate thermalization. We support this conclusion with mathematically rigorous results in the Ginibre ensemble and numerical results in other random matrix ensembles, the non-Hermitian Sachdev-Ye-Kitaev model, and a local non-Hermitian spin chain, indicating universality in a broad class of chaotic non-Hermitian quantum systems.

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• Eigenstate thermalization
• Quantum chaos
• Non-Hermitian systems
• Random matrix theory
• Sachdev-Ye-Kitaev model