[Paper Review] Notes on time entanglement and pseudo-entropy

K. Narayan, Hitesh Kumar Saini|arXiv (Cornell University)|Mar 2, 2023

Advanced Thermodynamics and Statistical Mechanics8 citations

TL;DR

The paper treats the time evolution operator as a density operator to explore timelike entanglement and its relation to pseudo-entropy, presenting explicit finite-system examples and connections to imaginary-temperature entanglement.

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.

Motivation & Objective

• Probe entanglement-like structures produced by the time evolution operator for timelike separations.
• Clarify relationships between time evolution entanglement and pseudo-entropy concepts.
• Illustrate with concrete finite-dimensional systems (qubits, qutrits, oscillators) and with projections.
• Explore thermofield-double type states and projections as special cases.
• Discuss implications for 2-d CFT timelike intervals and time-dependent interactions.

Proposed method

• Define the normalized time-evolution density operator ρt(t) = U(t)/Tr U(t) and its partial trace ρt^A to obtain entropies.
• Relate ρt to the reduced transition matrix from pseudo-entropy and show how a time-evolution without projection mirrors finite temperature entanglement with β = it.
• Introduce the time-evolution operator with a projection onto an initial state |I⟩ and study the resulting reduced density matrices ρt^{|I⟩,A}.
• Examine thermofield-double type states and show how diagonal/structured initial states map to known entanglement patterns.
• Analyze special normalizations, such as ρt^0(t) (normalized at t = 0) and ρt^{0,|i⟩}, to compare with ordinary entanglement and pseudo-entropy.
• Explore 2-d CFTs with timelike intervals to connect with timelike entanglement discussions.

Experimental results

Research questions

• RQ1 What entanglement-like structures arise when the time-evolution operator is treated as a density operator for timelike separations?
• RQ2 How does projecting the time-evolved state onto an initial state modify the entanglement properties and relate to pseudo-entropy?
• RQ3 In finite systems (qubits, qutrits, oscillators), under what conditions do the resulting entropies become real or remain complex?
• RQ4 How do thermofield-double type states influence the time-entanglement structure and its real-valued subfamilies?
• RQ5 What is the behavior of timelike entanglement in 2-d CFTs for timelike intervals, and how does this relate to imaginary-temperature interpretations?

Key findings

• The time evolution operator, viewed as a density matrix, yields entanglement-like quantities that are generally complex-valued for timelike separations, with real subfamilies under special conditions.
• Time-entanglement structures reproduce ordinary finite-temperature entanglement patterns with imaginary temperature β = it, up to analytic continuations.
• For simple 2-qubit and thermofield-double-type initial states, the reduced entropies exhibit real, oscillatory behavior controlled by energy differences, including cases with a single nontrivial phase.
• Introducing projections onto initial states leads to reduced transition matrices that align with pseudo-entropy concepts and reveal how initial-state choices shape the entanglement structure.
• Normalization choices (e.g., at t = 0) significantly alter the resulting entanglement patterns, sometimes yielding real, purely thermodynamic-like entropy forms.
• In 2-d CFTs, timelike intervals allow a framework for time-entanglement entropy that parallels timelike holographic interpretations and connects to prior de Sitter space discussions.