量子旋转性
Quantum Rotatability
摘要：
在arXiv：0807.0677中，Köstler和Speicher观察到，如果在量子置换下用更强的不变性条件取代交换序列，那么de Finetti关于可交换序列的定理有一个自由的类似物。本文研究了联合分布在量子正交变换下不变的非对易随机变量序列。我们证明了Freedman对条件独立高斯族的刻画的一个自由类比，即自伴随机变量的无穷序列是量子正交不变的当且仅当它们构成一个具有共同方差的算子值自由中心半圆族.同样，我们证明了一个非交换随机变量的无穷序列是量子酉不变的当且仅当它们形成一个具有共同方差的算子值自由中心圆族。我们提供了一个例子来表明，在经典的情况下，这些结果失败的有限序列。然后，我们给出了一个近似的距离有多远的一个有限的量子正交不变序列的分布是从一个运营商值的自由中心的半圆形家庭的共同方差。
英文摘要：
In arXiv:0807.0677, Köstler and Speicher observed that de Finetti's theorem on exchangeable sequences has a free analogue if one replaces exchangeability by the stronger condition of invariance under quantum permutations. In this paper we study sequences of noncommutative random variables whose joint distribution is invariant under quantum orthogonal transformations. We prove a free analogue of Freedman's characterization of conditionally independent Gaussian families, namely an infinite sequence of self-adjoint random variables is quantum orthogonally invariant if and only if they form an operator-valued free centered semicircular family with common variance. Similarly, we show that an infinite sequence of noncommutative random variables is quantum unitarily invariant if and only if they form an operator-valued free centered circular family with common variance. We provide an example to show that, as in the classical case, these results fail for finite sequences. We then give an approximation to how far the distribution of a finite quantum orthogonally invariant sequence is from that of an operator-valued free centered semicircular family with common variance.
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