The concept of complexity has become pivotal in multiple disciplines, including quantum information, where it serves as an alternative metric for gauging the chaotic evolution of a quantum state. This paper focuses on Krylov complexity, a specialized form of quantum complexity that offers an unambiguous and intrinsically meaningful assessment of the spread of a quantum state over all possible orthogonal bases. This study is situated in the context of Gaussian quantum states, which are fundamental to both Bosonic and Fermionic systems and can be fully described by a covariance matrix. While the covariance matrix is essential, it is insufficient alone for calculating Krylov complexity due to its lack of relative phase information is shown. The relative covariance matrix can provide an upper bound for Krylov complexity for Gaussian quantum states is suggested. The implications of Krylov complexity for theories proposing complexity as a candidate for holographic duality by computing Krylov complexity for the thermofield double States (TFD) and Dirac field are also explored.

中文翻译：

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费米子和玻色子高斯态的克雷洛夫复杂度

的概念复杂已经成为包括量子信息在内的多个学科的关键，它可以作为衡量量子态混沌演化的替代指标。本文重点讨论克雷洛夫复杂度，量子复杂性的一种特殊形式，它提供了对量子态在所有可能的正交基上的传播的明确且本质上有意义的评估。这项研究以高斯量子态为背景，高斯量子态是玻色子和费米子系统的基础，可以通过协方差矩阵来充分描述。虽然协方差矩阵是必不可少的，但由于缺乏相对相位信息，仅用它来计算 Krylov 复杂度是不够的。建议相对协方差矩阵可以提供高斯量子态 Krylov 复杂度的上限。还探讨了通过计算热场双态（TFD）和狄拉克场的克雷洛夫复杂度来提出复杂度作为全息对偶性候选者的理论的克雷洛夫复杂度的含义。