Quantum minimum description of density matrices
Abstract
We study a variant of Schumacher compression. The task asks for the minimal memory cost for compressing many copies of a density matrix with known spectrum and unknown eigenbasis, without preserving its purification. For fixed dimension and distinct positive eigenvalues, we obtain the cost through its additive constant. Achievability uses a generalization of Werner's cloning map to GL(d,ℂ) irreducible representations, with finite trace-distance bounds controlled by highest-weight differences and row gaps. The matching converse follows from a quantitative form of Koashi--Imoto incompressibility for irreducible group orbits under a spectral-gap assumption. We also identify the relation between this quantum minimum description length and universal lossless coding overhead, and its connection to free entropy is explained in a companion letter. We provide a Lean certificate for our proofs.
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BibTeX
@misc{hayden2026quantum,
title = {{Quantum minimum description of density matrices}},
author = {Patrick Hayden and Alexander Maloney and Jinzhao Wang and Yuxiang Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06742},
eprint = {2610.06742},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06742},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Quantum minimum description of density matrices
AU - Patrick Hayden
AU - Alexander Maloney
AU - Jinzhao Wang
AU - Yuxiang Yang
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.06742
UR - https://arxiv.org/abs/2610.06742
AB - We study a variant of Schumacher compression. The task asks for the minimal memory cost for compressing many copies of a density matrix with known spectrum and unknown eigenbasis, without preserving its purification. For fixed dimension and distinct positive eigenvalues, we obtain the cost through its additive constant. Achievability uses a generalization of Werner's cloning map to GL(d,ℂ) irreducible representations, with finite trace-distance bounds controlled by highest-weight differences and row gaps. The matching converse follows from a quantitative form of Koashi--Imoto incompressibility for irreducible group orbits under a spectral-gap assumption. We also identify the relation between this quantum minimum description length and universal lossless coding overhead, and its connection to free entropy is explained in a companion letter. We provide a Lean certificate for our proofs.
ER -