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This research establishes a rigorous structural correspondence between quantum states and matroid theory, allowing key properties like entanglement to be characterized through purely combinatorial terms. The findings reveal that genuine entanglement in these specific states is directly linked to the connectivity of their underlying matroids, while local measurements induce transformations corresponding to matroid minors. This work bridges abstract algebraic structures with quantum mechanics, offering new mathematical tools for analyzing state duality and symmetry without relying on traditional Hilbert space formalisms alone.
Quantum states supported by matroids
arXiv:2607.15548v1 Announce Type: new Abstract: In this work, we establish a structural correspondence between quantum states and matroid theory. This connection demonstrates that key properties of quantum states, including entanglement and measurement, can be characterized in purely combinatorial terms via matroids, despite the apparent conceptual distance between these two fields. Using this framework, we show that a matroid-supported state is genuinely entangled when its underlying matroid is connected. Moreover, a uniform superposition over all bases of a matroid is genuinely entangled if and only if the matroid is connected. We also demonstrate that a local measurement in the $Z$-basis on such a state yields another matroid-supported state, whose underlying matroid is a minor of the original one. Inspired by matroid duality, we further propose a notion of quantum state duality, uncovering a deep structural symmetry in state transformations.
Abstract:In this work, we establish a structural correspondence between quantum states and matroid theory. This connection demonstrates that key properties of quantum states, including entanglement and measurement, can be characterized in purely combinatorial terms via matroids, despite the apparent conceptual distance between these two fields. Using this framework, we show that a matroid-supported state is genuinely entangled when its underlying matroid is connected. Moreover, a uniform superposition over all bases of a matroid is genuinely entangled if and only if the matroid is connected. We also demonstrate that a local measurement in the $Z$-basis on such a state yields another matroid-supported state, whose underlying matroid is a minor of the original one. Inspired by matroid duality, we further propose a notion of quantum state duality, uncovering a deep structural symmetry in state transformations.
This paper has been published in Physical Review A. Commements are welcome
arXiv:2607.15548 [quant-ph]
(or arXiv:2607.15548v1 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2607.15548 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
Phys. Rev. A 113, 062452,2026
https://doi.org/10.1103/mx1w-k8fg Focus to learn more DOI(s) linking to related resources
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