费米子高斯幺正算子的交换子
The commutant of fermionic Gaussian unitaries
AI总结：
研究费米子高斯幺正算子及其粒子守恒子群的t阶交换子，揭示其在随机协议和不变理论中的作用，通过Howe对偶性推导出闭式公式和构造性Gelfand-Tsetlin过程。
摘要：
本文研究了费米子高斯幺正算子及其粒子守恒子群在n个费米子模式上的t阶交换子。这些交换子控制相应群的Haar平均，因此在费米子随机协议、不变理论和资源量化中起核心作用。利用Howe对偶性，我们证明粒子守恒交换子由广义复制跳转算子生成，而一般高斯交换子由广义二次Majorana双线性以及奇偶性生成。我们推导了两种交换子的维度闭式公式，并发展了构造性Gelfand-Tsetlin过程以获得显式正交基，给出了低t值的详细例子。本框架还阐明了复制费米子态的结构，并自然连接到费米子相关性度量、广义Plücker型约束和费米子高斯态的稳定子熵。这些结果为费米子高斯动力学的高阶不变量提供了统一的代数描述。
英文摘要：
In this work, we characterize the $t$-th order commutants of fermionic Gaussian unitaries and of their particle-preserving subgroup acting on $n$ fermionic modes. These commutants govern Haar averages over the corresponding groups and therefore play a central role in fermionic randomized protocols, invariant theory, and resource quantification. Using Howe dualities, we show that the particle-preserving commutant is generated by generalized copy-hopping operators, while that for general Gaussian commutant is generated by generalized quadratic Majorana bilinears together with parity. We then derive closed formulas for the dimensions of both commutants as functions of $t$ and $n$, and develop constructive Gelfand--Tsetlin procedures to obtain explicit orthonormal bases, with detailed low-$t$ examples. Our framework also clarifies the structure of replicated fermionic states and connects naturally to measures of fermionic correlations, generalized Plücker-type constraints, and the stabilizer entropy of fermionic Gaussian states. These results provide a unified algebraic description of higher-order invariants for fermionic Gaussian dynamics.
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