Mutual information between thermo-field doubles and disconnected holographic boundaries

TL;DR

This work defines thermo-mutual information (TMI) as the mutual information between physical and thermo-double degrees of freedom in thermo-field doubled systems, and proves its key properties: UV finiteness, non-negativity, and an upper bound by ordinary mutual information. It develops both field-theoretic and holographic approaches to compute TMI, validating the concept through a two-spin quantum system, a 2d massless Dirac fermion, and a BTZ black hole holographic setup, with explicit replica/Euclidean constructions and twist-operator results. The holographic analysis uses the Ryu–Takayanagi covariant framework to obtain thermo-mutual information between regions on disconnected AdS boundaries, revealing how TMI behaves with temperature and region geometry, and highlighting the role of purifications and the Hawking–Page transition. The discussion outlines extensions to higher dimensions and other AdS geometries, and contrasts TMI with renormalized entanglement measures, suggesting TMI as a robust probe of thermal entanglement and inter-boundary correlations in holographic CFTs.

Abstract

We use mutual information as a measure of the entanglement between 'physical' and thermo-field double degrees of freedom in field theories at finite temperature. We compute this "thermo-mutual information" in simple toy models: a quantum mechanics two-site spin chain, a two dimensional massless fermion, and a two dimensional holographic system. In holographic systems, the thermo-mutual information is related to minimal surfaces connecting the two disconnected boundaries of an eternal black hole. We derive a number of salient features of this thermo-mutual information, including that it is UV finite, positive definite and bounded from above by the standard mutual information for the thermal ensemble. We relate the construction of the reduced density matrices used to define the thermo-mutual information to the Schwinger-Keldysh formalism, ensuring that all our objects are well defined in Euclidean and Lorentzian signature.

Figures (7)

• Figure 1: The mutual information and thermo-mutual information in the two-spin system. The solid blue line is the MI and the solid red line is the TMI. The solid black line at the top of the graph denotes $2\log 2$ which is the maximum value of the mutual information. The dashed blue and red lines denote the correlation functions $\frac{1}{2} \left(\left\langle S_{A1}^z S_{B1}^z \right\rangle_\Omega\right)^2$ and $\frac{1}{2} \left(\left\langle S_{A1}^z S_{B2}^z \right\rangle_\Omega \right)^2$ respectively.
• Figure 2: Schwinger-Keldysh time integration contours in the complex $t$ plane. The Euclidean contour is the dashed red line; the TFD contour is the solid blue line. These contours are further discussed in Appendix \ref{['app:PI']}.
• Figure 3: Depiction of the Euclidean path integral (\ref{['eq:rhoABE']}). (I) shows the placement of delta function insertions used to construct the Euclidean analogue of $\rho_{A1 \cup B1}$; (II) shows the placement for the Euclidean analogue of $\rho_{A1 \cup B2}$. The twist fields (see §\ref{['sec:CFT']}) are also depicted, note that the orientation of the cuts at $-\beta/2$ are reversed.
• Figure 4: Examples of the MI and TMI of the 2D massless Dirac fermion. Left: the MI (solid blue) and TMI (solid red) as a function of $\beta$. In this plot $L_A = L_B = 1$ and $S=1/2$. The solid black line denotes the maximum value of the MI. Right: the MI and TMI as function of separation $S$. Here $L_A = L_B = 1$ and $\beta = 5$.
• Figure 5: The Penrose-Carter diagram for a maximally extended AdS-Schwarzschild black hole. The two dark shaded regions labeled $R_\pm$ are the regions covered by two AdS-Schwarzschild coordinate patches where the killing vector $\partial_t$ is timelike, bounded by the $45^\circ$ dashed lines indicating the horizon. The red jagged line is the curvature singularity, which bends inwards for AdS-Schwarzschild black holes in $D>3$.