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Constructible reality condition of pseudo entropy via pseudo-Hermiticity

Wu-zhong Guo, Song He, Yu-Xuan Zhang

TL;DR

This work addresses when pseudo entropy remains real by leveraging pseudo-Hermiticity of transition matrices. It proves that finite-dimensional mixed-state transitions are pseudo-Hermitian and that reduced matrices are pseudo-Hermitian under precise decompositions, then constructs classes of transitions with non-negative pseudo-Rényi entropies by choosing definite $\eta_A$ and $\eta_{\bar A}$. The authors apply the framework to quantum field theories, showing that in 2D rational CFTs certain transitions are parity-pseudo-Hermitian and that in general Minkowski QFTs the Tomita–Takesaki modular structure guarantees positive spectra for half-space transitions, thereby producing real or non-negative pseudo entropies. The results establish a practical method to generate real pseudo entropy and reveal deep connections to modular theory, with potential implications for holography and quantum information.

Abstract

As a generalization of entanglement entropy, pseudo entropy is not always real. The real-valued pseudo entropy has promising applications in holography and quantum phase transition. We apply the notion of pseudo-Hermticity to formulate the reality condition of pseudo entropy. We find the general form of the transition matrix for which the eigenvalues of the reduced transition matrix possess real or complex pairs of eigenvalues. Further, we construct a class of transition matrices for which the pseudo (Rényi) entropies are non-negative. Some known examples which give real pseudo entropy in quantum field theories can be explained in our framework. Our results offer a novel method to generate the transition matrix with real pseudo entropy. Finally, we show the reality condition for pseudo entropy is related to the Tomita-Takesaki modular theory for quantum field theory.

Constructible reality condition of pseudo entropy via pseudo-Hermiticity

TL;DR

This work addresses when pseudo entropy remains real by leveraging pseudo-Hermiticity of transition matrices. It proves that finite-dimensional mixed-state transitions are pseudo-Hermitian and that reduced matrices are pseudo-Hermitian under precise decompositions, then constructs classes of transitions with non-negative pseudo-Rényi entropies by choosing definite and . The authors apply the framework to quantum field theories, showing that in 2D rational CFTs certain transitions are parity-pseudo-Hermitian and that in general Minkowski QFTs the Tomita–Takesaki modular structure guarantees positive spectra for half-space transitions, thereby producing real or non-negative pseudo entropies. The results establish a practical method to generate real pseudo entropy and reveal deep connections to modular theory, with potential implications for holography and quantum information.

Abstract

As a generalization of entanglement entropy, pseudo entropy is not always real. The real-valued pseudo entropy has promising applications in holography and quantum phase transition. We apply the notion of pseudo-Hermticity to formulate the reality condition of pseudo entropy. We find the general form of the transition matrix for which the eigenvalues of the reduced transition matrix possess real or complex pairs of eigenvalues. Further, we construct a class of transition matrices for which the pseudo (Rényi) entropies are non-negative. Some known examples which give real pseudo entropy in quantum field theories can be explained in our framework. Our results offer a novel method to generate the transition matrix with real pseudo entropy. Finally, we show the reality condition for pseudo entropy is related to the Tomita-Takesaki modular theory for quantum field theory.
Paper Structure (25 sections, 9 theorems, 95 equations, 5 figures)

This paper contains 25 sections, 9 theorems, 95 equations, 5 figures.

Table of Contents

  1. Introduction
  2. General form of transition matrix
  3. Construction of the transition matrix
  4. Finite-dimensional examples
  5. Examples in QFTs
  6. 2-dimensional rational CFTs
  7. General QFTs in Minkowski space
  8. Comment on an exceptional case in the 2-dimensional free scalar theory
  9. Discussion
  10. Non-pseudo-Hermitian $X_A$ with real $\text{tr}[(X_A)^n]$
  11. A proof of Theorem \ref{['corol3']}
  12. Case $\mathfrak{a}$ :
  13. Case $\mathfrak{b}$ :
  14. Case $\mathfrak{c}$:
  15. Case $\mathfrak{d}$:
  16. ...and 10 more sections

Key Result

Theorem 1

An operator $M$ with a complete biorthonormal eigenbasis and a discrete spectrum is pseudo-Hermitian if and only if one of the following conditions hold:

Figures (5)

  • Figure 1: The excess of the 2nd pseudo Rényi entropy $\Delta S^{(2)}_A$ ($\Delta S^{(2)}_A\equiv S^{(2)}_A-S^{(2)}_{A;vac}$, where $S^{(2)}_{A;vac}$ denotes the 2nd Rényi entropy of $A$ when the total system is in the vacuum) of the transition matrix $\mathcal{T}_A\equiv\text{tr}_{\bar{A}}\frac{\mathcal{O}(x,t)|0\rangle\langle0|\mathcal{O}(-x,-t)}{\langle0|\mathcal{O}(-x,-t)\mathcal{O}(x,t)|0\rangle}$ in the minimal models $\mathcal{M}(p,p')$. We study the case of $\mathcal{O}=\phi_{(2,2)}$ (dot-dashed line) and $\mathcal{O}=\phi_{(2,1)}$ (solid line), respectively. One novel feature is that the 2nd pseudo entropy is real and time-independent.
  • Figure 2: Illustration of the operators $\mathcal{O}_A$ and $\mathcal{O}_{\bar{A}}$.
  • Figure 3: The plot of $S^{(n)}(\mathcal{T}_A)$.
  • Figure 4: The plot of $S^{(n)}(\mathcal{T}_A)$ and $\text{tr}[(\mathcal{T}_A)^n]$. The upper left plot shows the imaginary part of $S^{(n)}(\mathcal{T}_A)$, which are vanishing. The upper right plot shows $S^{(n)}(\mathcal{T}_A)$. The lower plot shows $\text{tr}[(\mathcal{T}_A)^n]$, which are in the region $(0,1)$.
  • Figure 5: The plot of $S^{(n)}(\mathcal{T}_A)$ and $\text{tr}[(\mathcal{T}_A)^n]$. The upper left plot shows the imaginary part of $S^{(n)}(\mathcal{T}_A)$, which are vanishing. The upper right plot shows $S^{(n)}(\mathcal{T}_A)$, which are negative. The lower plot shows $\text{tr}[(\mathcal{T}_A)^n]$.

Theorems & Definitions (9)

  • Theorem 1
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Corollary 1
  • Corollary 2