The First Law of Complexity
Abstract
We investigate the variation of holographic complexity for two nearby target states. Based on Nielsen's geometric approach, we find the variation only depends on the end point of the optimal trajectory, a result which we designate the first law of complexity. As an example, we examine the complexity=action conjecture when the AdS vacuum is perturbed by a scalar field excitation, which corresponds to a coherent state. Remarkably, the gravitational contributions completely cancel and the final variation reduces to a boundary term coming entirely from the scalar field action. Hence the null boundary of Wheeler-DeWitt patch appears to act like the "end of the quantum circuit".
Cite
@article{arxiv.1903.04511,
title = {The First Law of Complexity},
author = {Alice Bernamonti and Federico Galli and Juan Hernandez and Robert C. Myers and Shan-Ming Ruan and Joan Simón},
journal= {arXiv preprint arXiv:1903.04511},
year = {2020}
}
Comments
7 pages + supplemental material, 2 figures; v2: clarifications and reference added, published version