Submodularity of entropy under quantum convolution
Abstract
We develop a submodular framework for the von Neumann entropy of discrete quantum convolutions, providing a noncommutative counterpart to the direct side of entropic additive combinatorics. We first introduce globally weighted quantum convolutions, which form compatible families indexed by admissible subsets of a fixed collection of inputs. Our main theorem reveals a polymatroidal geometry underlying their entropy growth: relative to any fixed admissible input block, the entropy gains admit a normalized, monotone, submodular extension to all subsets of the remaining inputs. The theorem yields convolutional strong subadditivity, quantum Ruzsa triangle inequality, and quantum entropic Plünnecke--Ruzsa inequalities for arbitrary input states. For repeated inputs, it gives sharp comparisons of entropy growth across admissible scales; in particular, the quantum doubling constant controls all higher admissible convolution entropies with optimal exponents. Together, these results bring submodular methods from additive combinatorics into the quantum setting and provide a systematic route to broad families of convolutional entropy inequalities.
Related
- The twisted convolution identity and ghost r-SICs from finite quantum dilogarithms
- Quantum gates from the middle convolution of twisted Burau representations
- Discounted Hitting Domination on Graphs with Submodularity, Complexity and Exact Algorithms
- On the Log-submodularity for zonoids: from Mixed Volume inequalities to the Hypercube
- Poisson Exchange Beyond Submodularity: Effective Approximation Algorithms for Offline and Online Subset Selection over Matroids
BibTeX
@misc{goodarzi2026submodularity,
title = {{Submodularity of entropy under quantum convolution}},
author = {Milad M. Goodarzi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.40211},
eprint = {2609.40211},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.40211},
}
RIS
TY - UNPB
TI - Submodularity of entropy under quantum convolution
AU - Milad M. Goodarzi
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.40211
UR - https://arxiv.org/abs/2609.40211
AB - We develop a submodular framework for the von Neumann entropy of discrete quantum convolutions, providing a noncommutative counterpart to the direct side of entropic additive combinatorics. We first introduce globally weighted quantum convolutions, which form compatible families indexed by admissible subsets of a fixed collection of inputs. Our main theorem reveals a polymatroidal geometry underlying their entropy growth: relative to any fixed admissible input block, the entropy gains admit a normalized, monotone, submodular extension to all subsets of the remaining inputs. The theorem yields convolutional strong subadditivity, quantum Ruzsa triangle inequality, and quantum entropic Plünnecke--Ruzsa inequalities for arbitrary input states. For repeated inputs, it gives sharp comparisons of entropy growth across admissible scales; in particular, the quantum doubling constant controls all higher admissible convolution entropies with optimal exponents. Together, these results bring submodular methods from additive combinatorics into the quantum setting and provide a systematic route to broad families of convolutional entropy inequalities.
ER -