Quantum entropy growth follows new submodular inequality patterns
Submodularity of entropy under quantum convolution
Information Theory
Summary
Entropy measures how much uncertainty or disorder is in a system. The paper explores how entropy behaves when combining multiple quantum inputs using a process called quantum convolution. The authors found a mathematical structure called submodularity that governs the growth of entropy in these combinations, extending familiar classical ideas into the quantum world. This leads to several new inequalities that help compare and bound entropy changes when inputs are combined in quantum states.
What this means in practice
- •For quantum information engineers: Design better quantum data processing methods by applying new entropy inequality frameworks to optimize quantum state combinations.
- •For quantum cryptography developers: Use new bounds on quantum entropy growth to improve security proofs in quantum cryptographic protocols.
A theory result. No direct application yet.
Authors
Milad M. Goodarzi
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.