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 $\mathrm{GL}(d,\mathbb C)$ 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.

Quantum Physics (quant-ph); Information Theory (cs.IT); Mathematical Physics (math-ph)

arXiv:2610.06742 [quant-ph]

(or arXiv:2610.06742v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2610.06742 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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