Non-Hermitian density matrices from timelike entanglement and wormholes

We extensively explore the connections between timelike entanglement and non-Hermitian density matrices in quantum many-body systems. We classify setups where we encounter non-Hermitian density matrices into two types: one is due to causal influences under unitary evolutions, and the other is due to nonunitary evolutions in non-Hermitian systems. We provide various examples of these setups including interacting harmonic oscillators, two dimensional conformal field theories, and holographic dualities. In them, we compute the timelike entanglement entropy and imagitivity, which measures how much density matrices are non-Hermitian. In both two classes, typical holographic examples are given by traversable AdS wormholes. We explain how causal influences in a wormhole dual to a pair of non-Hermitian quantum systems is possible even without interactions between them. We argue that to realize a traversable wormhole we need not only ordinary quantum entanglement but also timelike entanglement.