Research
Quantum minimum description of density matrices
Overview Research area: Quantum information theory — specifically quantum data compression (a variant of Schumacher compression), combined with representation theory of $\mathrm{GL}(d,\mathbb{C
- arXiv
- 2610.06742
- Published
- 2026-10-05
- Authors
- Patrick Hayden, Alexander Maloney, Jinzhao Wang, Yuxiang Yang
AI summary
## Overview **Research area:** Quantum information theory — specifically quantum data compression (a variant of Schumacher compression), combined with representation theory of $\mathrm{GL}(d,\mathbb{C})$ and free probability. **Technical level:** Advanced. The paper assumes fluency in quantum Shannon theory, Schur–Weyl duality, highest-weight representation theory, and trace-distance analysis. **Scope:** The paper determines the exact asymptotic memory cost — including the additive constant, not just the leading $\log n$ term — for compressing many copies of a density matrix whose spectrum is known but whose eigenbasis is unknown and whose purification need not be preserved. ## What This Paper Is About Standard Schumacher compression asks how much memory is needed to preserve a quantum state *together with its correlations to a reference*. This paper studies a different task: reconstructing many identical copies of a density matrix when you know its eigenvalues but not the basis they are written in, and you do not care about preserving any purification. The unknown basis still has to be stored somehow, so the state cannot simply be regenerated from a classical description. The authors pin down the minimum memory cost exactly through its constant correction term, and show the same quantity governs the overhead of universal lossless quantum data compression. ## Key Contributions 1. **Exact optimal memory cost through the additive constant.** For fixed dimension $d$ and a rank-$r$ state with distinct positive eigenvalues $x_1 > \cdots > x_r > 0$, the paper gives the quantity $L_{d,r}(n,x) = \frac{r(2d-r-1)}{2}\log n + \sum_{1\le iAuthors’ 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.