偽熵是一個取決於初始態和最終態，推廣糾纏熵的新量度。基於已知結果表
明量子纏繞熵和偽熵均具有面積定律特性，我們對一個（2 + 1）維度的宇宙膨
脹度規進行了測試，以確定面積定律的有效性以及時間因素對偽熵值的影響。
此外，我們引入了量子猝滅的概念，以進一步探索可能存在的有趣現象。

Pseudo entropy is a novel quantity that generalizes the concept of entanglement
entropy, depending on both an initial state and a final state. Based on the wellknown
results indicating that both entanglement entropy and pseudo entropy
exhibit the area law property, we conducted tests on a (2+1)-dimensional cosmological
expansion metric to determine the validity of the area law and how the time
factor affects the value of pseudo entropy. Additionally, we analysis the concept
of quantum quench to further explore any potential interesting phenomena.

Contents

Contents ii
List of Figures iii
1 Introduction 1

2 Theoretical Background 3

2.1 Entropy and Area Law . . . . . . . . . . . . . . . . . . . . 3
2.1.1 Entanglement Entropy for 2-Coupled Oscillators . . . . . . 5
2.1.2 Entanglement Entropy for N-Coupled Oscillators . . . . . . 6
2.1.3 Entanglement Entropy for Massless Field . . . . . . . . . .7
2.2 Correlator Method for Entanglement Entropy . . . . . . . . . 8
2.2.1 Correlation Function and Quantum State . . . . . . . . . . 9
2.2.2 Correlation Function and Coupling Matrix . . . . . . . . . 11
2.2.3 Fitted Curve for Entanglement Entropy . . . . . . . . . . .12
2.3 Pseudo Entropy for Free Field Theories . . . . . . . . . . . 14
2.3.1 Pseudo Entropy for 2-Coupled Oscillators . . . . . . . . . 15
2.3.2 Correlator Method for Pseudo Entropy . . . . . . . . . . . 17
2.3.3 Operator Method for Pseudo Entropy . . . . . . . . . . . . 20
2.4 Pseudo Entropy for (1+1) Scalar Field . . . . . . . . . . . .25

3 (2+1)D Pseudo Entropy 28

3.1 (2+1)D Pseudo entropy for cosmological expansion . . . . . . 28
3.1.1 Massive and Almost Massless Limit . . . . . . . . . . . . .32

4 Quantum Quenches 34

4.1 Global Quench for (2+1)D Pseudo Entropy . . . . . . . . . . .34

5 Conclusions and Discussions 38

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