Krylov complexity of density matrix operators

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 2C_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 NC_S for N ≥ 2 within the N-dimensional Hilbert space. Notably, we confirm that C_K = 2C_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.