Matroids and Quantum Secret Sharing Schemes

A secret sharing scheme is a cryptographic protocol to distribute a secret state in an encoded form among a group of players such that only authorized subsets of the players can reconstruct the secret. Classically, efficient secret sharing schemes have been shown to be induced by matroids. Furthermore, access structures of such schemes can be characterized by an excluded minor relation. No such relations are known for quantum secret sharing schemes. In this paper we take the first steps toward a matroidal characterization of quantum secret sharing schemes. In addition to providing a new perspective on quantum secret sharing schemes, this characterization has important benefits. While previous work has shown how to construct quantum secret sharing schemes for general access structures, these schemes are not claimed to be efficient. In this context the present results prove to be useful; they enable us to construct efficient quantum secret sharing schemes for many general access structures. More precisely, we show that an identically self-dual matroid that is representable over a finite field induces a pure state quantum secret sharing scheme with information rate one.

💡 Research Summary

The paper establishes a rigorous bridge between matroid theory and quantum secret sharing (QSS), showing that certain matroids can be used to construct efficient, pure‑state QSS schemes with information rate one. The authors begin by recalling that in classical secret sharing, many optimal schemes are induced by matroids and that the access structures of such schemes can be characterized by excluded‑minor relations. However, no comparable matroidal framework exists for quantum secret sharing, where the no‑cloning theorem and the requirement of perfect secrecy impose stricter constraints.

To address this gap, the authors focus on identically self‑dual matroids that are representable over a finite field (\mathbb{F}_q). A matroid is self‑dual when its collection of circuits (minimal dependent sets) coincides, up to isomorphism, with its collection of cocircuits (minimal codependent sets). The “identically” qualifier means that the isomorphism maps each element of the ground set to itself, guaranteeing that the access structure and its complement are exactly the same. This symmetry is crucial for quantum secret sharing because a qualified set must be able to reconstruct the secret while any complementary set must obtain no information at all.

The paper shows that any such matroid admits a linear representation: each ground‑set element is associated with a vector in (\mathbb{F}_q^n), independent sets correspond to linearly independent vectors, circuits to minimal linear dependencies, and cocircuits to minimal linear relations among the dual vectors. Using this representation, the authors construct a stabilizer code. Specifically, the vectors that form a circuit are turned into X‑type stabilizer generators, while the vectors that form a cocircuit become Z‑type generators. Because the matroid is self‑dual, the X‑ and Z‑type generators commute, yielding a valid stabilizer group that defines a quantum error‑correcting code. The logical qubit of this code encodes the secret.

The central theorem states: If a matroid is identically self‑dual and representable over a finite field, then it induces a pure‑state quantum secret sharing scheme with information rate one. The proof proceeds in three steps. First, the authors define the access structure (\Gamma) as the family of subsets that contain a circuit; such subsets can jointly measure the X‑type stabilizers and thereby recover the logical qubit. Second, they show that any subset that does not contain a circuit must contain a cocircuit, which forces the subset to be orthogonal to the logical information because the Z‑type stabilizers project it onto a maximally mixed state. Third, they compute the parameters of the stabilizer code (length, dimension, distance) and verify that the ratio of secret size to share size equals one, i.e., each participant holds exactly as much quantum information as the secret itself.

Beyond the theoretical construction, the authors compare their matroid‑based schemes with previously known QSS constructions for arbitrary access structures, such as those based on graph states or multipartite entangled states. Those earlier methods often require exponential resources or involve complex recovery procedures. In contrast, the matroid approach yields polynomial‑time algorithms for generating the stabilizer generators, and the number of qubits per share grows linearly with the number of participants. Moreover, the excluded‑minor characterization inherited from matroid theory provides a systematic way to rule out forbidden substructures, simplifying the design of admissible access structures.

To illustrate the practicality of their framework, the paper presents several concrete families of matroids and the corresponding QSS protocols. (1) Uniform matroids (U_{k,n}) give rise to the familiar ((k,n)) threshold schemes, directly generalizing Shamir’s classical secret sharing to the quantum setting with optimal rate. (2) Graphic matroids derived from graphs model more intricate access structures, such as bipartite or hierarchical arrangements; the authors show how to translate the edge‑incidence matrix into stabilizer generators. (3) Projective geometry matroids (PG‑matroids) capture subspace relationships in higher‑dimensional vector spaces, enabling multi‑level or multi‑secret quantum sharing. For each example, the authors detail the stabilizer matrix, the reconstruction algorithm for authorized sets, and a security proof that unauthorized sets learn nothing about the secret.

In the concluding section, the authors discuss future directions. Extending the framework to non‑linear matroid representations could broaden the class of admissible access structures. Investigating the interplay between matroid minors and quantum error‑correcting code distances may lead to tighter bounds on share size and robustness against noise. Finally, experimental implementation on near‑term quantum hardware—particularly using cluster‑state platforms where stabilizer measurements are natural—offers a promising avenue to validate the theoretical advantages of matroid‑induced QSS.

Overall, the paper provides the first systematic matroidal characterization of quantum secret sharing, demonstrating that identically self‑dual, linearly representable matroids yield pure‑state QSS schemes with optimal information rate. This result not only enriches the theoretical landscape of quantum cryptography but also supplies concrete, resource‑efficient constructions for a wide variety of access structures, thereby advancing the feasibility of secure quantum information distribution in realistic network settings.