Quantum minimum description of density matrices
This study addresses the minimum memory cost of compressing multiple copies of a density matrix when its spectrum is known but its eigenbasis is not. The proposed approach integrates irreducible representations of GL(d,ℂ) with quantum information theory, constructing an achievable scheme by generalizing the Werner cloning map and establishing a matching lower bound via Koashi–Imoto incompressibility, with formal verification completed using the Lean proof assistant. This work precisely determines the minimal description length constant for fixed dimensions, revealing its deep connections to universal lossless coding overhead and free entropy. Furthermore, it derives finite trace-distance bounds and provides complete machine-checkable proof certificates.