Density of states of quantum systems from free probability theory: a brief overview

arXiv:2512.03850 · quant-ph, cond-mat.stat-mech · Submitted 2025-12-03 · Read on arXiv

quant-ph, cond-mat.stat-mech

Submitted: 2025-12-03

Updated: 2026-09-23

Comments: Minor corrections. References added

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Terminology

• Entanglement — 5× in this paper · explained in 3 episodes
• Adjacency Matrix — 2× in this paper · explained in 1 episode
• Generalization — 2× in this paper · explained in 13 episodes
• Gwangju Institute of Science and Technology — 2× in this paper
• Trace — 2× in this paper · explained in 2 episodes
• Accuracy — 1× in this paper · explained in 1 episode
• Additivity — 1× in this paper · explained in 1 episode
• Chebyshev Polynomials — 1× in this paper · explained in 2 episodes
• Consistency — 1× in this paper · explained in 3 episodes
• Cooperation — 1× in this paper · explained in 2 episodes
• National Research Foundation of Korea — 1× in this paper
• Quantum Circuits — 1× in this paper · explained in 2 episodes
• Quantum Physics — 1× in this paper · explained in 10 episodes
• SYK Model — 1× in this paper · explained in 2 episodes
• Self-consistency — 1× in this paper · explained in 2 episodes

Sources

• Lecture Notes on "Free Probability Theory"
• Mastering the Master Field
• Large N Gauge Theory -- Expansions and Transitions
• Various Shades of Blue's Functions
• Variational Master Field for Large-N Interacting Matrix Models - Free Random Variables on Trial
• Eigenstate Thermalization Hypothesis and Free Probability
• Full Eigenstate Thermalization via Free Cumulants in Quantum Lattice Systems
• Long-time Freeness in the Kicked Top
• Scaling of free cumulants in closed system-bath setups
• Free Independence and the Noncrossing Partition Lattice in Dual-Unitary Quantum Circuits
• Generalized Free Cumulants for Quantum Chaotic Systems
• Quantum Signatures of Chaos from Free Probability
• Free Probability in a Minimal Quantum Circuit Model
• Free Independence and Unitary Design from Random Matrix Product Unitaries
• Stability of Periodically Driven Topological Phases against Disorder
• Random matrix techniques in quantum information theory
• Beyond islands: A free probabilistic approach
• Free mutual information and higher-point OTOCs
• Non-commutative probability insights into the double-scaling limit SYK Model with constant perturbations: moments cumulants and $q$-independence
• Isotropic Entanglement

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