We establish an entropic, quantum central limit theorem and quantum inverse sumset theorem in discrete-variable quantum systems describing qudits or qubits. Both results are enabled by using our recently-discovered quantum convolution. We show that the exponential rate of convergence of the entropic central limit theorem is bounded by the magic gap. We also establish an ``quantum, entropic inverse sumset theorem,'' by introducing a quantum doubling constant. Furthermore, we introduce a ``quantum Ruzsa divergence'', and we pose a conjecture called ``convolutional strong subaddivity,'' which leads to the triangle inequality for the quantum Ruzsa divergence. A byproduct of this work is a magic measure to quantify the nonstabilizer nature of a state, based on the quantum Ruzsa divergence.

翻译：我们在描述量子比特或量子四态的离散变量量子系统中建立了熵量子中心极限定理和量子逆和集定理。这两个结果均通过我们最近发现的量子卷积得以实现。我们证明了熵中心极限定理的指数收敛速度受魔法间隙的界限制。通过引入量子加倍常数，我们还建立了“量子熵逆和集定理”。此外，我们引入了“量子Ruzsa散度”，提出了称为“卷积强次可加性”的猜想，该猜想导出量子Ruzsa散度的三角不等式。本工作的一个副产品是基于量子Ruzsa散度的一种魔法度量，用于量化量子态的非稳定子特性。