[Paper Review] Polymatroids and polyquantoids

František Matúš|arXiv (Cornell University)|Oct 30, 2012

Quantum Information and Cryptography16 references6 citations

TL;DR

This paper introduces polyquantoids as the quantum analogues of polymatroids, establishing a one-to-one correspondence between tight selfdual polymatroids and polyquantoids via linear mappings. The key contribution is that ideal quantum secret sharing schemes correspond precisely to selfdual matroids through this duality, extending classical results on matroids and secret sharing to the quantum domain.

ABSTRACT

When studying entropy functions of multivariate probability distributions, polymatroids and matroids emerge. Entropy functions of pure multiparty quantum states give rise to analogous notions, called here polyquantoids and quantoids. Polymatroids and polyquantoids are related via linear mappings and duality. Quantum secret sharing schemes that are ideal are described by selfdual matroids. Expansions of integer polyquantoids to quantoids are studied and linked to that of polymatroids.

Motivation & Objective

• To formalize quantum entropy functions using polyquantoids, the quantum analogue of polymatroids.
• To establish a duality between tight selfdual polymatroids and polyquantoids via linear mappings.
• To characterize ideal quantum secret sharing schemes through the structure of quantoids and selfdual matroids.
• To extend the concept of free expansions from integer polymatroids to integer polyquantoids, mirroring matroid theory in the quantum setting.

Proposed method

• Define polyquantoids as pairs (N, e) with normalized, complementary, and submodular rank functions e.
• Introduce linear mappings e ↦ e^∧ and h ↦ h^∨ that form a mutually inverse bijection between polyquantoids and tight selfdual polymatroids.
• Use duality of set functions h ↦ h′ to analyze selfdual and tight properties, with h′ defined via complementarity and singleton adjustments.
• Apply the mappings to show that entropic polyquantoids arise from von Neumann entropy of pure quantum states on tensor product Hilbert spaces.
• Define free expansions of integer polymatroids and extend the construction to polyquantoids, preserving tightness and selfduality.
• Prove that expansions of integer polyquantoids to quantoids mirror the role of matroids in classical polymatroid theory.

Experimental results

Research questions

• RQ1 How do polyquantoids generalize polymatroids in the context of quantum entropy functions?
• RQ2 What is the precise correspondence between tight selfdual polymatroids and polyquantoids?
• RQ3 Which quantum secret sharing schemes are ideal, and how are they characterized via quantoids?
• RQ4 Can the classical notion of free expansion of integer polymatroids be extended to polyquantoids, and what properties are preserved?
• RQ5 What is the quantum analogue of the classical result that ideal secret sharing is governed by matroids?

Key findings

• The mappings e ↦ e^∧ and h ↦ h^∨ establish a one-to-one correspondence between polyquantoids and tight selfdual polymatroids.
• Ideal quantum secret sharing schemes are characterized by quantoids that correspond to tight selfdual matroids.
• The expansion of an integer polyquantoid to a quantoid preserves tightness and selfduality, analogous to the classical case.
• The 2-factor of a free expansion of a tight, selfdual, even-valued polymatroid yields another tight, selfdual polymatroid.
• The duality mapping h ↦ h′ preserves submodularity and is an involution, with selfdual functions being fixed points.
• Polyquantoids arise naturally from the von Neumann entropy of pure quantum states, just as polymatroids arise from Shannon entropy.