Krylov complexity of density matrix operators

• URL: http://arxiv.org/abs/2402.09522v3
• Date: Thu, 6 Jun 2024 19:32:49 GMT
• Title: Krylov complexity of density matrix operators
• Authors: Pawel Caputa, Hyun-Sik Jeong, Sinong Liu, Juan F. Pedraza, Le-Chen Qu,
• Abstract summary: Krylov-based measures such as Krylov complexity ($C_K$) and Spread complexity ($C_S$) are gaining prominence. We investigate their interplay by considering the complexity of states represented by density matrix operators.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $N\geq2$ 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.

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