Subsystem Complexity and Holography

TL;DR

This work probes subsystem circuit complexity in holographic field theories by evaluating complexity analogues based on entanglement wedges under complexity=volume and complexity=action. It develops several mixed-state complexity definitions inspired by tensor networks and compares them to holographic results for neutral/charged eternal black holes and shock waves, finding that CA can qualitatively align with purification or basis complexity (depending on action counterterms) while CV generally fails to map cleanly. The study provides explicit CA/CV calculations for subregions, clarifies additivity properties, and offers a framework to relate subregion complexity to purified-spectrum-basis structures with potential implications for the holographic encoding of mixed states. Overall, the results strengthen the case for CA-like notions encoding mixed-state complexity, while highlighting challenges and ambiguities in CV-type proposals for subregions.

Abstract

We study circuit complexity for spatial regions in holographic field theories. We study analogues based on the entanglement wedge of the bulk quantities appearing in the "complexity = volume" and "complexity = action" conjectures. We calculate these quantities for one exterior region of an eternal static neutral or charged black hole in general dimensions, dual to a thermal state on one boundary with or without chemical potential respectively, as well as for a shock wave geometry. We then define several analogues of circuit complexity for mixed states, and use tensor networks to gain intuition about them. We find a promising qualitative match between the holographic action and what we call the purification complexity, the minimum number of gates required to prepare an arbitrary purification of the given mixed state. On the other hand, the holographic volume does not appear to match any of our definitions of mixed-state complexity.

Figures (7)

• Figure 1: Intersection of entanglement wedge ${\cal E}$ and Wheeler-DeWitt patch ${\cal W}$ for half of an eternal black hole.
• Figure 2: Separation of the WdW patch in terms of its intersection with the entanglement wedges ${\cal E}_L \cap {\cal W}$ and ${\cal E}_R \cap {\cal W}$, and with the regions behind the past and future horizons ${\cal W}_{\rm int}^{\pm}$ for an eternal black hole.
• Figure 3: Null segments on the boundary of ${\cal W}_L = {\cal E}_L \cap {\cal W}$. The four corners arise at the intersection of neighboring segments of the boundary.
• Figure 4: Penrose diagram for an eternal charged black hole. ${\cal W}_L$, the intersection of the entanglement wedge associated to the left boundary ${\cal E}_L$ and the Wheeler-DeWitt patch ${\cal W}$ associated to the full boundary of the black hole, is shown in purple.
• Figure 5: Representation of the left and right entanglement wedges ${\cal E}_L$ and ${\cal E}_R$ in red and green respectively, for an eternal black hole in the presence of a shock wave inserted on the left boundary. In blue it is represented the associated WdW patch