离散量子高斯与中心极限定理
Discrete Quantum Gaussians and Central Limit Theorem
AI总结：
该研究将稳定子态命名为离散量子高斯，建立了量子卷积相关的原理与量子中心极限定理，证明其收敛性并界定收敛速率，还将结果推广至量子信道。
摘要：
我们引入量子卷积和一个概念框架，用于研究离散变量（DV）量子系统中的量子态。所有结果表明，稳定子态在DV量子系统中所起的作用，类似于高斯态在连续变量系统中的作用，因此我们将稳定子态命名为“离散量子高斯”。例如，我们证明两个稳定子态的卷积仍是稳定子态，且稳定子态使量子熵和费舍尔信息均取极值。我们建立了“最大熵原理”“量子卷积的热力学第二定律”以及“量子中心极限定理”。后者基于对零均值量子态的卷积进行迭代，我们证明其收敛于稳定子态，并通过定义在该态特征函数支撑集上的“魔术间隙”来界定量子中心极限定理的指数收敛速率。我们通过若干实例阐释了这些一般性结果，并将其中许多结果推广至量子信道。
英文摘要：
We introduce a quantum convolution and a conceptual framework to study states in discrete-variable (DV) quantum systems. All our results suggest that stabilizer states play a role in DV quantum systems similar to the role that Gaussian states play in continuous-variable systems. Hence we suggest the name ``discrete quantum Gaussians'' for stabilizer states. For example, we prove that the convolution of two stabilizer states is another stabilizer state, and that stabilizer states extremize both quantum entropy and Fisher information. We establish a ``maximal entropy principle,'' a ``second law of thermodynamics for quantum convolution,'' and a ``quantum central limit theorem.'' The latter is based on iterating the convolution of a zero-mean quantum state, which we prove converges to a stabilizer state. We bound the exponential rate of convergence of the quantum central limit theorem by the ``magic gap,'' defined in terms of the support of the characteristic function of the state. We elaborate on our general results with a discussion of some examples, as well as extending many of them to quantum channels.
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