Krylov complexity of density matrix operators

Pawel Caputa; Hyun-Sik Jeong; Sinong Liu; Juan F. Pedraza; Le-Chen Qu

doi:10.1007/JHEP05(2024)337

paper

Krylov complexity of density matrix operators

arXiv:2402.09522 · doi:10.1007/JHEP05(2024)337

Abstract

Quantifying complexity in quantum systems has witnessed a surge of interest in recent years, with Krylov-based measures such as Krylov complexity ( $C_{K}$ ) and Spread complexity ( $C_{S}$ ) gaining prominence. In this study, we investigate their interplay by considering the complexity of states represented by density matrix operators. After setting up the problem, we analyze a handful of analytical and numerical examples spanning generic two-dimensional Hilbert spaces, qubit states, quantum harmonic oscillators, and random matrix theories, uncovering insightful relationships. For generic pure states, our analysis reveals two key findings: (I) a correspondence between moment-generating functions (of Lanczos coefficients) and survival amplitudes, and (II) an early-time equivalence between $C_{K}$ and $2 C_{S}$ . Furthermore, for maximally entangled pure states, we find that the moment-generating function of $C_{K}$ becomes the Spectral Form Factor and, at late-times, $C_{K}$ is simply related to $N C_{S}$ for $N \geq 2$ within the $N$ -dimensional Hilbert space. Notably, we confirm that $C_{K} = 2 C_{S}$ holds across all times when $N = 2$ . Through the lens of random matrix theories, we also discuss deviations between complexities at intermediate times and highlight subtleties in the averaging approach at the level of the survival amplitude.

v1: 41 pages, 10 figures; v2: references added; v3: matching the published version

Krylov complexity of density matrix operators

Abstract

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