  The concepts of operator size and computational complexity play important
roles in the study of quantum chaos and holographic duality because they help
characterize the structure of time-evolving Heisenberg operators. It is
particularly important to understand how these microscopically defined measures
of complexity are related to notions of complexity defined in terms of a dual
holographic geometry, such as complexity-volume (CV) duality. Here we study
partially entangled thermal states in the Sachdev-Ye-Kitaev (SYK) model and
their dual description in terms of operators inserted in the interior of a
black hole in Jackiw-Teitelboim (JT) gravity. We compare a microscopic
definition of complexity in the SYK model known as K-complexity to calculations
using CV duality in JT gravity and find that both quantities show an
exponential-to-linear growth behavior. We also calculate the growth of operator
size under time evolution and find connections between size and complexity.
While the notion of operator size saturates at the scrambling time, our study
suggests that complexity, which is well defined in both quantum systems and
gravity theories, can serve as a useful measure of operator evolution at both
early and late times.
