Non-hermitian Density Matrices from Time-like Entanglement and Wormholes

TL;DR

This paper investigates the relationship between time-like entanglement and non-hermitian density matrices in quantum systems, revealing their roles in causal influences, non-unitary evolutions, and traversable wormholes.

Contribution

It classifies non-hermitian density matrices arising from different setups and demonstrates their connection to traversable wormholes and time-like entanglement in holographic dualities.

Findings

Time-like entanglement is essential for traversable wormholes.

Non-hermitian density matrices can result from causal influences and non-unitary evolutions.

Holographic examples include traversable AdS wormholes.

Abstract

We extensively explore the connections between time-like 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 non-unitary 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 time-like 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 time-like entanglement.