AI 中文总结
本文针对具有紧半单对称代数的多体量子系统，利用表示理论构建态无关不确定性通用框架，推导得到总方差的尖锐态无关界限，证实集体自旋系统总不确定性下限仅与粒子数奇偶性相关。
AI 中文摘要
不确定性关系约束不相容可观测量的涨落，但多数常见界限依赖于量子态。态无关不确定性关系则研究对任意量子态而言，不可避免的涨落程度。对于由连续对称性生成的可观测量，当对称性表示为不可约时，尖锐的态无关界限是已知的。然而，多体集体系统通常表现为对称代数的可约张量积表示，这引发了如何确定其总不确定性的问题。我们针对具有紧半单对称代数𝔤的多体量子系统解决了该问题。利用对称性结构，我们构建了基于表示理论的态无关不确定性通用框架。总方差可精确分解为不可约扇区内的本征涨落与扇区间非负色散，这给出了多体希尔伯特空间ℋ上总方差Δ_ρ²(𝔤)的尖锐态无关界限：min_ρΔ_ρ²(𝔤)=min_{λ∈Λ(ℋ)}2⟨λ,δ⟩，其中ρ为ℋ上任意密度算子，Λ(ℋ)为标记各扇区的最高权重λ的集合，δ为Weyl向量。这表明终极不确定性完全由其本征对称结构控制。作为特例，该结果证实了我们此前的猜想：集体自旋-1/2系统的总不确定性下限仅取决于粒子数的奇偶性。我们还以多体自旋-1系统为例阐释了该框架，证明相同的对称扇区机制在自旋-1/2之外依然成立。
英文摘要
Uncertainty relations constrain the fluctuations of incompatible observables, but most familiar bounds depend on the quantum state. State-independent uncertainty relations instead ask how much fluctuation remains unavoidable for every quantum state. For observables generated by a continuous symmetry, sharp state-independent bounds are known when the symmetry representation is irreducible. Multipartite collective systems, however, generally appear as reducible tensor-product representations of a symmetry algebra, which raises the question of how to determine their total uncertainty. We resolve this problem for multipartite quantum systems with a compact semisimple symmetry algebra $\mathfrak g$. Exploiting the symmetry structure, we formulate a general framework for state-independent uncertainty based on representation theory. The total variance admits an exact decomposition into intrinsic fluctuations within irreducible sectors and a nonnegative dispersion between sectors. This yields the sharp state-independent bound for the total variance $Δ_ρ^2(\mathfrak g)$ on the multipartite Hilbert space $\mathcal H$ \[ \min_ρΔ_ρ^2(\mathfrak g) = \min_{λ\inΛ(\mathcal H)} 2\langleλ,δ\rangle, \] where $ρ$ is any density operator on $\mathcal H$, $Λ(\mathcal H)$ is the set of highest weights $λ$ labeling those sectors, and $δ$ is the Weyl vector. This demonstrates that the ultimate uncertainty is completely controlled by its intrinsic symmetry structure. As a special example, this result confirms our previous conjecture that the total uncertainty floor of collective spin-$1/2$ systems depends only on the parity of the particle number. We further illustrate the framework for multipartite spin-$1$ systems, demonstrating that the same symmetry-sector mechanism persists beyond spin-$1/2$.
Comments5+3pp
