Subregion Action and Complexity

TL;DR

This work analyzes the finite part of the on-shell Einstein-Hilbert action for spacetime subregions bounded by null surfaces in the Complexity=Action framework, focusing on a $(d+2)$-dimensional AdS black brane that models a thermofield double state. It carefully assembles the diffeomorphism-invariant action $I=I^{(0)}+I^{\rm amb}$, including null-boundary and joint terms, and computes explicit expressions for regions behind the horizon (WDW, past interior, future interior) and outside the horizon (triangular and square exterior subregions). A key contribution is the introduction of mutual complexity ${\cal C}(A|B)$ and the demonstration of subadditivity for exterior subregions, along with a detailed analysis of how joint points, horizon intersections, and the ambiguous null normalization influence the finite parts of the action. The results illuminate how subregion CA complexity behaves under time evolution, clarify the separate roles of interior versus exterior regions in driving growth, and propose a framework for relating CA subregion quantities to purified or mixed-state complexity in the dual field theory. Altogether, the paper advances a consistent, holographically computable notion of subregion complexity and its mutual/relative variants, with implications for how complexity bounds and information-theoretic correlations are encoded in gravitational physics.

Abstract

We evaluate finite part of the on-shell action for black brane solutions of Einstein gravity on different subregions of spacetime enclosed by null boundaries. These subregions include the intersection of WDW patch with past/future interior and left/right exterior for a two sided black brane. Identifying the on-shell action on the exterior regions with subregion complexity one finds that it obeys subadditivity condition. This gives an insight to define a new quantity named mutual complexity. We will also consider certain subregion that is a part of spacetime which could be causally connected to an operator localized behind/outside the horizon. Taking into account all terms needed to have a diffeomorphism invariant action with a well-defined variational principle, one observes that the main contribution that results to a nontrivial behavior of the on-shell action comes from joint points where two lightlike boundaries (including horizon) intersect. A spacelike boundary gives rise to a linear time growth, while we have a classical contribution due to a timelike boundary that is given by the free energy.

Figures (5)

• Figure 1: Penrose diagram of the WDW patch of an eternal AdS black hole assuming $t_R=t_L$. Left: WDW patch on which the on-shell action is computed to find the complexity. Right: Past patch corresponding to the WDW patch. The past patch may be identified as a part that is casually connected to an operator localized at $r=r_m$ behind the horizon.
• Figure 2: Left: Intersection of WDW patch with the past interior. Right: Intersection of WDW patch with the future interior.
• Figure 3: Spacetime region corresponding to evaluation of holographic uncomplexity. The regions shown by blue color compute uncomplexity given by the equation \ref{['UC']}. This is not equal to difference between maximum complexity and the complexity of the state at a given time (see equation \ref{['UU']}).
• Figure 4: Left: A localized operator at $P$. The colored region is the part that is involved in the construction of the operator localized at $r=r_p$. Right: The orange region is the intersection of WDW patch and entanglement wedge at time slice $t_R=0$ for half of an eternal black hole.
• Figure 5: Left: Intersection of WDW patch and entanglement wedge for large entangling region at time slice $t_R=0$ for half of an eternal black hole. Right: Two subregions denoted by $\ell_1$ and $\ell_2$.