Free entropy and quantum minimum description length
Abstract
In free probability, Voiculescu's free entropy counts matrix microstates that approximate an operator. It is a non-commutative analog of the Shannon entropy, distinct from any quantum entropy of a density operator, but it has had no physical meaning in quantum theory. We define a finite-dimensional physical free entropy at finite resolution and show that it directly characterizes, up to an explicit constant, the minimal quantum memory needed to program many copies of a state with known eigenvalues and unknown eigenbasis. This task is a variant of Schumacher compression, and we call its optimal cost the quantum minimum description length. We briefly discuss how free entropy differs from quantum entropies and its potential applications in quantum physics.
Related
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BibTeX
@misc{hayden2026free,
title = {{Free entropy and quantum minimum description length}},
author = {Patrick Hayden and Alexander Maloney and Jinzhao Wang and Yuxiang Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06741},
eprint = {2610.06741},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06741},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Free entropy and quantum minimum description length
AU - Patrick Hayden
AU - Alexander Maloney
AU - Jinzhao Wang
AU - Yuxiang Yang
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.06741
UR - https://arxiv.org/abs/2610.06741
AB - In free probability, Voiculescu's free entropy counts matrix microstates that approximate an operator. It is a non-commutative analog of the Shannon entropy, distinct from any quantum entropy of a density operator, but it has had no physical meaning in quantum theory. We define a finite-dimensional physical free entropy at finite resolution and show that it directly characterizes, up to an explicit constant, the minimal quantum memory needed to program many copies of a state with known eigenvalues and unknown eigenbasis. This task is a variant of Schumacher compression, and we call its optimal cost the quantum minimum description length. We briefly discuss how free entropy differs from quantum entropies and its potential applications in quantum physics.
ER -