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法态中的应用。我们首先建立了一个熵形式的量子中心极限定理（q-CLT），其收敛速率受魔法间隙的约束。同时，我们引入了一种基于量子卷积的新型量子散度——量子Ruzsa散度，用以研究量子态的稳定子结构。我们提出了“卷积强次可加性”不等式的猜想，该猜想可导出量子Ruzsa散度的三角不等式。此外，我们提出了两种新的魔法度量：魔法态的量子Ruzsa散度与量子倍增常数，用以量化量子态中的魔法含量。最后，通过运用量子卷积，我们将经典的逆和集理论推广至量子情形。这些结果为量子信息理论中稳定子态与魔法态的研究提供了新的视角。