标题： 拉格朗日乘数法在LCA群和无限量子自旋系统中的应用 显示英文标题

标题： The Hudson theorem in LCA groups and infinite quantum spin systems

摘要： 著名的Hudson定理指出，在$\mathbb{R}^d$中，高斯函数是唯一其Wigner分布处处为正的函数。 受量子信息理论的启发，D. Gross在阿贝尔群$\mathbb{Z}_d^n$上证明了一个类似的结果，当$d$为奇数时——对应于一个由$n$个qudit组成的系统——表明只有所谓的稳定化态的Wigner分布是非负的。 将这一结果扩展到有限维系统的热力学极限，自然地引导我们考虑具有紧开子群的一般$2$-正则LCA群，其中Wigner分布的非负性问题目前仍是一个开放问题。 我们通过证明如果映射$x\mapsto 2x$是保测的，则Wigner分布非负的函数恰好是二次的子特征，除了平移和乘以常数之外。 相反，如果上述映射不是保测的，Wigner分布总是会取负值。 我们详细讨论了离散群的无限和以及紧群的无限积的特殊情况，这恰好对应于无限量子自旋系统。 进一步的例子包括$n$-进系统，其中$n\geq 2$是一个任意整数（不一定是素数），以及挠群。 显示英文摘要

摘要： The celebrated Hudson theorem states that the Gaussian functions in $\mathbb{R}^d$ are the only functions whose Wigner distribution is everywhere positive. Motivated by quantum information theory, D. Gross proved an analogous result on the Abelian group $\mathbb{Z}_d^n$, for $d$ odd - corresponding to a system of $n$ qudits - showing that the Wigner distribution is nonnegative only for the so-called stabilizer states. Extending this result to the thermodynamic limit of finite-dimensional systems naturally leads us to consider general $2$-regular LCA groups that possess a compact open subgroup, where the issue of the positivity of the Wigner distribution is currently an open problem. We provide a complete solution to this question by showing that if the map $x\mapsto 2x$ is measure-preserving, the functions whose Wigner distribution is nonnegative are exactly the subcharacters of second degree, up to translation and multiplication by a constant. Instead, if the above map is not measure-preserving, the Wigner distribution always takes negative values. We discuss in detail the particular case of infinite sums of discrete groups and infinite products of compact groups, which correspond precisely to infinite quantum spin systems. Further examples include $n$-adic systems, where $n\geq 2$ is an arbitrary integer (not necessarily a prime), as well as solenoid groups.