费米子高斯态的逼近定理

中文摘要 AI 辅助

多体物理中一个反复出现的原理是，充分分布式的相互作用或关联会导致有效的平均场描述。对于费米子高斯态，我们直接在马约拉纳协方差矩阵层面上建立了这一原理的定量版本。主要观察结果是，费米子协方差矩阵的可容许性条件对两点关联施加了定量的单配性约束：一个固定区域只有有限的协方差预算可分配给多个不相交区域。我们将这一原理用于三个方向。首先，对于高次有限简单图上的二次费米子哈密顿量，我们获得了基态能量和有限温度自由能的产品态逼近，对于$D$-正则图，误差阶为$D^{-1/2}$。其次，我们证明了有限费米子德菲内蒂定理：在分级费米子副本置换下$n$-可交换的$k$-副本高斯态，在迹范数下与其单副本边际的乘积$O(k/n)$-接近。特别地，在高斯类内的无限可交换性意味着精确的产品结构。第三，对于具有指数聚类的互信息空间费米子高斯态，我们证明了跨越分离缓冲区的条件互信息以$O(r^{-1})$衰减，因此，通过可恢复性，该态允许通过恢复态进行$O(r^{-1/2})$迹范数逼近。这些结果表明，费米子高斯系统的产品态和德菲内蒂逼近以及可恢复性估计都可以直接从协方差级结构推导出来。

英文摘要

A recurring principle in many-body physics is that sufficiently distributed interactions or correlations lead to an effectively mean-field description. For fermionic Gaussian states, we establish a quantitative version of this principle directly at the level of Majorana covariance matrices. The main observation is that the admissibility condition for fermionic covariance matrices imposes a quantitative monogamy constraint on two-point correlations: a fixed region has only a bounded covariance budget to distribute among many disjoint regions. We use this principle in three directions. First, for quadratic fermionic Hamiltonians on finite simple graphs of high degree, we obtain product-state approximations to both the ground-state energy and the finite-temperature free energy, with error of order \(D^{-1/2}\) for \(D\)-regular graphs. Second, we prove a finite fermionic Gaussian de Finetti theorem: a \(k\)-copy Gaussian state that is \(n\)-exchangeable under graded fermionic copy permutations is \(O(k/n)\)-close in trace norm to the product of its one-copy marginal. In particular, infinite exchangeability within the Gaussian class implies exact product structure. Third, for spatial fermionic Gaussian states with exponentially clustering mutual information, we prove that the conditional mutual information across a separating buffer decays as \(O(r^{-1})\), and hence, by recoverability, the state admits an \(O(r^{-1/2})\) trace-norm approximation by a recovered state. These results show that product and de Finetti approximations and recoverability estimates for fermionic Gaussian systems can all be derived directly from covariance-level structure.

发表机构

• Perimeter Institute for Theoretical Physics（Perimeter理论物理研究所）
• Institute for Quantum Computing, University of Waterloo（滑铁卢大学量子计算研究所）
• Dahlem Center for Complex Quantum Systems, Freie Universität Berlin（柏林自由大学达勒姆复杂量子系统中心）
• Department of Physics, University of Helsinki（赫尔辛基大学物理系）
• HUN-REN Wigner Research Centre for Physics（匈牙利研究与创新网络维格纳物理研究中心）
• Algorithmiq Ltd（Algorithmiq有限公司）
• Center for Theoretical Physics – a Leinweber Institute, Massachusetts Institute of Technology（麻省理工学院莱因韦伯理论物理中心）
• Leinweber Institute for Theoretical Physics, Stanford University（斯坦福大学莱因韦伯理论物理研究所）
• Google DeepMind（谷歌DeepMind）
• Helmholtz-Zentrum Berlin für Materialien und Energie（柏林亥姆霍兹材料与能源中心）

机构由 AI 辅助整理，请以论文原文为准。