In this work, we investigate the behavior of quantum entropy under quantum convolution and its application in quantifying magic. We first establish an entropic, quantum central limit theorem (q-CLT), where the rate of convergence is bounded by the magic gap. We also introduce a new quantum divergence based on quantum convolution, called the quantum Ruzsa divergence, to study the stabilizer structure of quantum states. We conjecture a ``convolutional strong subadditivity'' inequality, which leads to the triangle inequality for the quantum Ruzsa divergence. In addition, we propose two new magic measures, the quantum Ruzsa divergence of magic and quantum-doubling constant, to quantify the amount of magic in quantum states. Finally, by using the quantum convolution, we extend the classical, inverse sumset theory to the quantum case. These results shed new insight into the study of the stabilizer and magic states in quantum information theory.

翻译：本文研究了量子熵在量子卷积下的行为及其在量化魔力方面的应用。我们首先建立了一个熵的量子中心极限定理（q-CLT），其中收敛速率受魔力间隙的约束。我们还基于量子卷积引入了一种新的量子散度，称为量子Ruzsa散度，用于研究量子态的稳定子结构。我们猜想了一个“卷积强次可加性”不等式，该不等式引出了量子Ruzsa散度的三角不等式。此外，我们提出了两种新的魔力度量——量子Ruzsa魔力散度和量子加倍常数——用于量化量子态中魔力的多少。最后，通过利用量子卷积，我们将经典的反求和理论推广到量子情形。这些结果为量子信息理论中稳定子与魔力态的研究提供了新的洞见。