Krylov complexity of density matrix operators

Quantifying complexity in quantum systems has witnessed a surge of interestin recent years, with Krylov-based measures such as Krylov complexity (C_K)and Spread complexity (C_S) gaining prominence. In this study, we investigatetheir interplay by considering the complexity of states represented by densitymatrix operators. After setting up the problem, we analyze a handful ofanalytical and numerical examples spanning generic two-dimensional Hilbertspaces, qubit states, quantum harmonic oscillators, and random matrix theories,uncovering insightful relationships. For generic pure states, our analysisreveals two key findings: (I) a correspondence between moment-generatingfunctions (of Lanczos coefficients) and survival amplitudes, and (II) anearly-time equivalence between C_K and 2C_S. Furthermore, for maximallyentangled pure states, we find that the moment-generating function of C_Kbecomes the Spectral Form Factor and, at late-times, C_K is simply related toNC_S for N≥2 within the N-dimensional Hilbert space. Notably, weconfirm that C_K = 2C_S holds across all times when N=2. Through the lensof random matrix theories, we also discuss deviations between complexities atintermediate times and highlight subtleties in the averaging approach at thelevel of the survival amplitude.

Further reading#

• Access Paper in arXiv.org