On the Noether charge and the gravity duals of quantum complexity

Abstract

The physical relevance of the thermodynamic volumes of AdS black holes to the gravity duals of quantum complexity was recently argued by Couch et al. In this paper, by generalizing the Wald-Iyer formalism, we derive a geometric expression for the thermodynamic volume and relate its product with the thermodynamic pressure to the non-derivative part of the gravitational action evaluated on the Wheeler-DeWitt patch. We propose that this action provides an alternative gravity dual of the quantum complexity of the boundary theory. We refer this to "complexity=action 2.0" (CA-2) duality. It is significantly different from the original "complexity=action" (CA) duality as well as the "complexity=volume 2.0" (CV-2) duality proposed by Couch et al. The latter postulates that the complexity is dual to the spacetime volume of the Wheeler-DeWitt patch. To distinguish our new conjecture from the various dualities in literature, we study a number of black holes in Einstein-Maxwell-Dilation theories. We find that for all these black holes, the CA duality generally does not respect the Lloyd bound whereas the CV-2 duality always does. For the CA-2 duality, although in many cases it is consistent with the Lloyd bound, we also find a counter example for which it violates the bound as well.

Complexity in the presence of a boundary

“…As a last comment, we would like to point out that other proposals for the holographic dual of the circuit complexity are present in the literature. Two of them are easily extracted from the computation of CA (other refined proposals can be found in [39,40]). The first proposal, called CV2.0, states that the complexity can be calculated as the product of the pressure, given in terms of the cosmological constant (p = −Λ/(8πG N )), times the volume of the WDW patch [11].…”

Complexity of mixed states in QFT and holography

“…That is, let σ be the Cauchy slice on which a pure state is defined, and divide this surface into a subregion A and its complement B. Then the corresponding holographic 73 An update to the complexity = spacetime volume conjecture, denoted 'CA2.0', was proposed in [69]. However, for Einstein-Hilbert gravity with minimally coupled matter, this approach simply reduces to the CV2.0 proposal.…”

“…Their approach uses the relation C ∼ P V WDW where P = − Λ 8πG N ∼ 1/(G N L 2 ) is the 'bulk' pressure [70]. Note that the application of these arguments is not straightforward for solutions with nontrivial scalar hair [69].…”

Time dependence of complexity for Lovelock black holes

“…However, there are some drawbacks for the two proposals as well. This motivates people to search new proposals for complexity [47,48,49]. In addition, the active research in holography promotes studies of complexity for quantum field theories [50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66] as well as in condensed matter physics [67].…”