Quantum Ruzsa Divergence to Quantify Magic
This work addresses the challenge of characterizing non-stabilizer resources in quantum computing, offering foundational tools for quantum information theory, though it appears incremental in building on classical concepts.
The authors tackled the problem of quantifying magic in quantum states by establishing a quantum central limit theorem with convergence rate bounded by the magic gap and introducing new measures like quantum Ruzsa divergence. They extended classical inverse sumset theory to quantum cases, providing new insights into stabilizer and magic states in quantum information theory.
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.