Quantum Ruzsa Divergence to Quantify Magic
Kaifeng Bu, Weichen Gu, Arthur Jaffe
TL;DR
This work develops a quantum analog of Ruzsa divergence by leveraging a DV quantum convolution to quantify magic and the stabilizer structure of quantum states. The authors prove an entropic q-CLT showing convergence to stabilizer states at a rate governed by the magic gap MG(ρ), and define the quantum Ruzsa divergence D_Rz with a symmetrized form to capture non-stabilizer structure. They introduce two magic measures, the quantum Ruzsa divergence of magic M_Rz and the quantum-doubling constant δ_q, and establish an inverse-sumset perspective for quantifying proximity to stabilizer states, along with a conjectured convolutional strong subadditivity that would yield a triangle inequality for D_Rz. Special cases confirm several properties, and the framework offers a computationally tractable route to assess quantum magic, with potential extensions to bosonic systems and broader resource-theory applications. Overall, the paper provides a unified mathematical toolkit connecting quantum convolution, stabilizer theory, and additive-combinatorics to advance understanding of quantum advantage.
Abstract
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.