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A Differential-Geometric Approach to Quantum Ignorance Consistent with Entropic Properties of Statistical Mechanics
Shannon Ray1,2, Paul M Alsing1, Carlo Cafaro3
1Air Force Research Laboratory, Rome, NY 13441, USA.
Entropy (Basel, Switzerland)
|May 27, 2023
Summary
This study introduces quantum coarse-graining (CG) to quantify missing information in quantum systems. The research demonstrates how systems evolve towards equilibrium, increasing entanglement and volume, crucial for understanding thermalization.
Area of Science:
- Quantum Information Theory
- Statistical Mechanics
- Quantum Many-Body Physics
Background:
- Understanding quantum system evolution and thermalization is key.
- Quantifying information loss in quantum systems is an ongoing challenge.
- Reduced density operators describe subsystems within larger quantum systems.
Purpose of the Study:
- To define and calculate the metric tensor and volume for the manifold of purifications of a reduced density operator.
- To introduce a quantum coarse-graining (CG) framework using "surfaces of ignorance" (SOI) as macrostates.
- To investigate the role of volume as a measure of missing information and its behavior during system evolution.
Main Methods:
- Construction of the metric tensor and volume for purification manifolds.
- Definition of quantum coarse-graining (CG) with macrostates as manifolds of purifications (SOI).
- Analysis of SOI generated from SU(2), SO(3), and SO(N) representations.
Main Results:
- Demonstrated that systems evolve from smaller to larger volume macrostates, increasing entanglement.
- Showed that equilibrium macrostates dominate the coarse-grained space, especially for large systems.
- Established that the volume function mirrors von Neumann entropy properties (zero for pure states, maximal for mixed states, concave).
Conclusions:
- The developed CG framework and volume measure are essential for typicality arguments in thermalization.
- The study provides insights into information dynamics and entanglement growth in quantum systems.
- The findings connect quantum information concepts to fundamental principles of statistical mechanics and coarse-graining.
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