Notes on time entanglement and pseudo-entropy

TL;DR

The paper investigates timelike entanglement by treating the time-evolution operator ${\cal U}(t)=e^{-iHt}$ as a density operator and studying entanglement-like measures obtained from partial traces.It clarifies the relationship between time evolution entanglement and pseudo-entropy, showing that many constructions map to imaginary-temperature entanglement, with complex-valued entropies in general.Through explicit finite-system examples (2-qubit, 2-qutrit, coupled/uncoupled oscillators) and a CFT treatment of timelike intervals, it reveals rich structure including real subfamilies and oscillatory behavior, as well as special cases with real entropy under symmetry.The work also analyzes time-evolution with projections onto initial states (and thermofield-double type preparations), time-normalization at $t=0$, and time-dependent interactions, underscoring both parallels and differences with pseudo-entropy and holographic contexts.

Abstract

Following arXiv:2210.12963 [hep-th], we investigate aspects of the time evolution operator regarded as a density operator and associated entanglement-like structures in various quantum systems. These involve timelike separations and generically lead to complex-valued entropy, although there are interesting real subfamilies. There are many parallels and close relations with reduced transition matrices and pseudo-entropy, which we discuss and clarify. For instance, a related quantity involves the time evolution operator along with a projection onto some initial state, which amounts to analysing pseudo-entropy for the initial state and its time-evolved final state.