Krylov complexity of density matrix operators
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866916278276456448 |
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| author | Caputa, Pawel Jeong, Hyun-Sik Liu, Sinong Pedraza, Juan F. Qu, Le-Chen |
| author_facet | Caputa, Pawel Jeong, Hyun-Sik Liu, Sinong Pedraza, Juan F. Qu, Le-Chen |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_09522 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Krylov complexity of density matrix operators Caputa, Pawel Jeong, Hyun-Sik Liu, Sinong Pedraza, Juan F. Qu, Le-Chen High Energy Physics - Theory Statistical Mechanics Quantum Physics 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. |
| title | Krylov complexity of density matrix operators |
| topic | High Energy Physics - Theory Statistical Mechanics Quantum Physics |
| url | https://arxiv.org/abs/2402.09522 |