Quantum Ruzsa Divergence to Quantify Magic

2024-01-25

AI Summary

This work addresses the fundamental challenge of quantifying quantum state “magic.” Methodologically, it introduces a novel theoretical framework grounded in quantum convolution and quantum entropy: (i) adapts the classical Ruzsa inequality to quantum information by defining the quantum Ruzsa divergence; (ii) establishes an entropy convergence theory under quantum convolution and proposes the convolutional strong subadditivity conjecture; and (iii) extends inverse sumset theory to the quantum regime, integrating stabilizer formalism and magic theory to develop new analytical tools. Key contributions include: (1) two axiomatically compliant magic measures—the quantum Ruzsa magic measure and quantum doubling constant; (2) proof that the quantum Ruzsa divergence satisfies the triangle inequality; (3) a quantum central limit theorem with explicit magic-gap control; and (4) a computable, robust quantification scheme applicable to arbitrary pure and mixed states.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Non-convex Optimization
• Knowledge Representation and Reasoning: Qualitative Reasoning

Application Category

• Security and Privacy: Large-scale security measurements
• Responsible Web: Measurement, analysis, and circumvention of Web censorship
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work

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.

Problem

• Study quantum entropy under quantum convolution.
• Introduce quantum Ruzsa divergence for stabilizers.
• Quantify magic in quantum states.

Innovation

• Quantum Ruzsa divergence introduced
• Quantum central limit theorem established
• New magic measures proposed