Holographic Pseudo Entropy

TL;DR

The paper introduces pseudo entropy, a generalization of entanglement entropy obtained via post-selection, and develops its formalism across quantum mechanics, quantum field theory, and holography. Using a replica approach, it derives both pseudo Rényi and von Neumann entropies for transition matrices, and shows a holographic dual where the pseudo entropy equals the area of a minimal surface in Euclidean time-dependent AdS, interpretable as a weak value of the area operator. In qubit systems, it classifies transition matrices, proves a two-qubit monotonicity under certain conditions, and explores interpretations in terms of Bell pairs; it also demonstrates subadditivity violations in higher dimensions. The work further analyzes pseudo entropy in a two-dimensional free CFT, in locally excited holographic CFTs, and in Janus AdS/CFT, revealing systematic reductions near subsystem boundaries and a consistent gravity-CFT correspondence, including a linearity property for holographic states and a mixed-state generalization via pseudo reflected entropy.

Abstract

We introduce a quantity, called pseudo entropy, as a generalization of entanglement entropy via post-selection. In the AdS/CFT correspondence, this quantity is dual to areas of minimal area surfaces in time-dependent Euclidean spaces which are asymptotically AdS. We study its basic properties and classifications in qubit systems. In specific examples, we provide a quantum information theoretic meaning of this new quantity as an averaged number of Bell pairs when the post-selection is performed. We also present properties of the pseudo entropy for random states. We then calculate the pseudo entropy in the presence of local operator excitations for both the two dimensional free massless scalar CFT and two dimensional holographic CFTs. We find a general property in CFTs that the pseudo entropy is highly reduced when the local operators get closer to the boundary of the subsystem. We also compute the holographic pseudo entropy for a Janus solution, dual to an exactly marginal perturbation of a two dimensional CFT and find its agreement with a perturbative calculation in the dual CFT. We show the linearity property holds for holographic states, where the holographic pseudo entropy coincides with a weak value of the area operator. Finally, we propose a mixed state generalization of pseudo entropy and give its gravity dual.

Figures (25)

• Figure 1: The calculation of holographic pseudo entropy. At the time specified as the dotted circle, the bra state and ket state are different. Accordingly, the asymptotically AdS Euclidean geometry is time-dependent. The dots are excitations by inserting external sources or operators to CFTs.
• Figure 2: The replica method calculation of pseudo entropy in two dimensional CFTs for locally excited states.
• Figure 3: The Venn diagram of the classification of 2-qubit transition matrices. For each class of states, we use $(m,n)$ to represent that there are $m$ independent variables for ${\mathcal{T}}^{\psi|{\varphi}}$ and $n$ independent variables for ${\mathcal{T}}^{\psi|{\varphi}}_A$.
• Figure 4: We plot the pseudo entropy for the two qubit system as a function $\theta_1$ (horizontal axis) and $\theta_2$ (depth axis) in the left graph. The right one shows the pseudo entropy minus the averaged entanglement entropy i.e. (\ref{['difave']}). We took the range $0\leq \theta_{1,2}\leq\pi$. The region where no graph is shown gives complex valued pseudo entropy.
• Figure 5: The region of $\alpha\in[0,\pi/4]$ and $\theta\in[0,\pi/2]$. The colored part shows the region in which all eigenvalues of all reduced transition matrices are real and nonnegative. $S({\mathcal{T}}^{\psi|{\varphi}}_A) + S({\mathcal{T}}^{\psi|{\varphi}}_B) - S({\mathcal{T}}^{\psi|{\varphi}}_{AB}) < 0$ in the blue region and $S({\mathcal{T}}^{\psi|{\varphi}}_A) + S({\mathcal{T}}^{\psi|{\varphi}}_B) - S({\mathcal{T}}^{\psi|{\varphi}}_{AB}) \geq 0$ in the yellow region.

Theorems & Definitions (5)

• Example 1
• Example 2
• Example 3
• Example 4
• Example 5