Krylov complexity of density matrix operators
Quantifying complexity in quantum systems has witnessed a surge of interest in recent years, with Krylov-based measures such as Krylov complexity (C<inf>K</inf>) and Spread complexity (C<inf>S</inf>) gaining prominence. In this study, we investigate their interplay by conside...
| Autores: | , , , , |
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| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2024 |
| País: | España |
| Institución: | Consejo Superior de Investigaciones Científicas (CSIC) |
| Repositorio: | DIGITAL.CSIC. Repositorio Institucional del CSIC |
| OAI Identifier: | oai:digital.csic.es:10261/413953 |
| Acceso en línea: | http://hdl.handle.net/10261/413953 https://www.scopus.com/inward/record.uri?eid=2-s2.0-85195315103&doi=10.1007%2FJHEP05%282024%29337&partnerID=40&md5=e5f546170d361920286202fc0e907633 |
| Access Level: | acceso abierto |
| Palabra clave: | AdS-CFT Correspondence Field Theories in Lower Dimensions Integrable Field Theories Random Systems |
| Sumario: | Quantifying complexity in quantum systems has witnessed a surge of interest in recent years, with Krylov-based measures such as Krylov complexity (C<inf>K</inf>) and Spread complexity (C<inf>S</inf>) 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<inf>K</inf> and 2C<inf>S</inf>. Furthermore, for maximally entangled pure states, we find that the moment-generating function of C<inf>K</inf> becomes the Spectral Form Factor and, at late-times, C<inf>K</inf> is simply related to NC<inf>S</inf> for N ≥ 2 within the N-dimensional Hilbert space. Notably, we confirm that C<inf>K</inf> = 2C<inf>S</inf> 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. © The Author(s) 2024. |
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