Subsystem complexity and holography

Abstract

As a probe of circuit complexity in holographic field theories, we study subsystem 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. In the action approach, we find two possible cases depending on an ambiguity in the definition of the action associated with a counterterm. In one case, there is 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. In the other case, the match is to what we call the basis complexity, the minimum number of gates required to prepare the given mixed state starting from a minimal complexity state with the same eigenvalue spectrum. One way to fix this ambiguity is to choose an action definition such that UV divergent part is positive, in which case the best match to the action result is the basis complexity. In contrast, the holographic volume does not appear to match any of our definitions of mixed-state complexity.

On subregion action complexity in AdS3 and in the BTZ black hole

“…Keeping just the terms linear in l in eq. (3.27), we find agreement with the subregion complexity C BTZ,R A computed for one side of the Kruskal diagram, see [46,47]:…”

“…A similar behaviour of subregion CA is found in the thermofield double state where the subsystems are taken as the two disconnected boundaries of spacetime. This case was investigated for asymptotically AdS black holes in D dimensions [46,47], showing that the complexity=action is subadditive when η <η D and superadditive for η >η D . The value ofη D is given by the zero of g D (η) [46]:…”

Complexity in the presence of a boundary

“…The extension of these proposals to the non-static cases can also be found in [33]. Field theory considerations about the subregion complexity appear in [55,56].…”

Complexity of mixed states in QFT and holography

“…Obviously, the purifications of a given mixed state are not unique. However, a natural definition of mixed state complexity -the so-called purification complexity [19] is defined as the minimal pure state complexity among all possible purifications of our mixed state, i.e., as usual, we are optimizing over the circuits which take the reference state to a target state |Ψ AA c , which is a purification of the desired mixed stateρ A , but we must also optimized over the possible purifications ofρ A , i.e., allows us to use the prescription of [14] for evaluating the complexity of the possible purifications, and we then minimize over the parameters of the purifications, as in eq. (1.1) above.…”