Complex-valued Holographic Pseudo Entropy via Real-time AdS/CFT Correspondence

TL;DR

Problem: pseudo entropy generalizes entanglement entropy by involving initial and final states, but Euclidean holographic realizations yield real values despite non-Hermiticity. Approach: introduce a real-time, Lorentzian path-integral prescription and formulate the Schwinger-Keldysh framework to compute pseudo entropy in QFT and its holographic dual. Findings: holographic pseudo entropy is generally complex and dual to a time-dependent extremal codimension-2 surface with possible imaginary contributions from regularized extrinsic curvature, extending the covariant holographic entanglement entropy. Impact: provides a framework for complex entanglement measures in dynamical AdS/CFT and informs the role of non-Hermitian structures in quantum gravity.

Abstract

The pseudo entropy is a promising recent generalization of the entanglement entropy to the situations in which both the initial and final state are involved, with the density matrix promoted to the transition matrix. However, in contrast to the non-Hermiticity of the generic transition matrix, the holographic pseudo entropy formulated via the Euclidean AdS/CFT turns out to be always real-valued, which potentially conceals the crucial natures of this novel quantity. In this note, we make the first attempt to formulate a real-time prescription for computations to incorporate naturally the pseudo entropy, as a generally complex-valued entanglement measure, into the AdS/CFT context. It is then conjectured that the holographic pseudo entropy is dual to the extremal codimension-2 area surface in the generally time-dependent Lorentzian asymptotically AdS spacetime, but may also receive imaginary contribution from the regularized extrinsic curvature term of the area surface, which is not included in the covariant holographic entanglement entropy. In this real-time prescription, the holographic pseudo entropy can be considered as a generalization of the covariant holographic entanglement entropy, as well.

Figures (4)

• Figure 1: Complex-time contour for $\langle \Psi_2\vert \Psi_1\rangle$ with the direction of time flow shown explicitly and spatial coordinates expressed. For $\langle \Psi_1\vert \Psi_2\rangle$, one inverses the flow of time in the Lorentzian segment and translates the Euclidean segments vertically. For our purpose, the Hamiltonian in the two Euclidean segments are set to be time-independent, and different from each other so that the initial and final state thus produced differ, which finally ends up with a non-trivial time-dependence in the Lorentzian segment violating the time-translation and -reflection symmetry.
• Figure 2: The path integral representation of (a) $\tau^{1\vert 2}_{\mathcal{A}}$ and (b) $\tau^{2\vert1}_{\mathcal{A}}$, related to the forward spacetime $\mathcal{B}^{1\vert 2}$ and backward spacetime $\mathcal{B}^{2\vert1}$, respectively, with the Euclidean segments omitted. There is a cut along the subregion $\mathcal{A}$ on the Cauchy slice $\Sigma_0=\mathcal{A}\cup\mathcal{A}_c$, with the entangling surface $\partial \mathcal{A}$ explicitly shown. The cuts in different spacetime are shown in different colors, blue or red. The arrows colored in orange indicate the direction of the time flow.
• Figure 3: The time-folded contour of (a) $\tau^{1\vert 2}_\mathcal{A}$ and (b) $\tau^{2\vert1}_{\mathcal{A}}$, which is of the Schwinger-Keldysh type. The path integrals has been glued up on ${\Sigma}_T$ as discussed in the main text. The cut along subregion $\mathcal{A}$ related to $\tau^{1\vert 2}_\mathcal{A}$ or $\tau^{2\vert 1}_\mathcal{A}$ is now arranged into a single branch of the Schwinger-Keldysh contour.
• Figure 4: Schwinger-Keldysh construction for $\rho_{\mathcal{A}}(t)$ in the case of the covariant holographic entanglement entropy. $\rho_{\mathcal{A}}(t)$ is the generic time-dependent reduced density matrix. The cut along the subregion $\mathcal{A}$ on the Cauchy slice $\Sigma_0$ of interest lies at the boundary between the forward and backward branch of the Schwinger-Keldysh contour.