Abstract
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.
Highlights
Precise versions of these notions in two models: the Sachdev-Ye-Kitaev (SYK) model and 2d Jackiw-Teitelboim (JT) gravity
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
From the geometric interpretation of effective theory [12, 56], we find the following relation between the size in the SYK model and the complexity in JT gravity at the limit q 1, N βJ 1 and the early time, π2 ∆n[ψβ1(θ + iu)] = −2 − 2θ tan θ + π sec θ cosh t βJ
Summary
This section reviews K-complexity associated with the Krylov basis in the SYK model. Unlike an a priori basis, the Krylov basis is uniquely determined by the evolution Hamiltonian and the initial state. ∂tφn(t) = bnφn−1(t) − bn+1φn+1(t), φ0(0) = 1 This Schrödinger equation (2.4) effectively describes a quantum particle moving in one dimensional chain in which each lattice site corresponds to an element of the Krylov basis. We note that this mapping from operators to quantum particles shares similar ideas with the mapping from unitary operators to points in a complexity geometry [23, 53]. [29] showed that the Lanczos coefficients are bounded by a linear function when n N , where N is the system size, implying that the average K-complexity grows at most exponentially up to the scrambling time, αt ∼ log N. We show that the Lanczos coefficients are bounded by a constant when n N , and that the average K-complexity can grow no faster than linearly in time at late times
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