Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Quantum Matching Theory (with new complexity-theoretic, combinatorial and topical insights on the nature of the quantum entanglement)

Gurvits, L. (Leonid)

Published Jan 1, 2002• 0 citations
OpenAlex

Abstract

Classical matching theory can be defined in terms of matrices with nonnegative entries. The notion of Positive operator, central in Quantum Theory, is a natural generalization of matrices with non-negative entries. Based on this point of view, we introduce a definition of perfect Quantum (operator) matching. We show that the new notion inherits many 'classical' properties, but not all of them. This new notion goes somewhere beyound matroids. For separable bipartite quantum states this new notion coinsides with the full rank property of the intersection of two corresponding geometric matroids. In the classical situation, permanents are naturally associated with perfects matchings. We introduce an analog of permanents for positive operators, called Quantum Permanent and show how this generalization of the permanent is related to the Quantum Entanglement. Besides many other things, Quantum Permanents provide new rational inequalities necessary for the separability of bipartite quantum states. Using Quantum Permanents, we give deterministic poly-time algorithm to solve Hidden Matroids Intersection Problem and indicate some 'classical' complexity difficulties associated with the Quantum Entanglement. Finally, we prove that the weak membership problem for the convex set of separable bipartite density matrices is NP-HARD.

Related papers

Quantum Matching Theory (with new complexity theoretic, combinatorial and topological insights on the nature of the Quantum Entanglement)

Gurvits, Leonid

Classical matching theory can be defined in terms of matrices with nonnegative entries. The notion of Positive operator, central in Quantum Theory, is a natural generalization of matrices with nonnegative entries. Based on this point of view, we introduce a definition of perfect Quantum (operator) matching . We show that the new notion inherits many "classical" properties, but not all of them . This new notion goes somewhere beyound matroids . For separable bipartite quantum states this new notion coinsides with the full rank property of the intersection of two corresponding geometric matroids . In the classical situation, permanents are naturally associated with perfects matchings. We introduce an analog of permanents for positive operators, called Quantum Permanent and show how this generalization of the permanent is related to the Quantum Entanglement. Besides many other things, Quantum Permanents provide new rational inequalities necessary for the separability of bipartite quantum states . Using Quantum Permanents, we give deterministic poly-time algorithm to solve Hidden Matroids Intersection Problem and indicate some "classical" complexity difficulties associated with the Quantum Entanglement. Finally, we prove that the weak membership problem for the convex set of separable bipartite density matrices is NP-HARD.

arXiv:quant-ph/0201022•Jan 7, 2002•14 citations

Classical deterministic complexity of Edmonds' Problem and quantum entanglement

Leonid Gurvits

This paper continues research initiated in quant-ph/0201022 . The main subject here is the so-called Edmonds' problem of deciding if a given linear subspace of square matrices contains a nonsingular matrix . We present a deterministic polynomial time algorithm to solve this problem for linear subspaces satisfying a special matroids motivated property, called in the paper the Edmonds-Rado property . This property is shown to be very closely related to the separability of bipartite mixed states . One of the main tools used in the paper is the Quantum Permanent introduced in quant-ph/0201022 .

ACM Symposium on Theory of Computing (STOC)•Jun 9, 2003•451 citations

Classical deterministic complexity of Edmonds' Problem and quantum entanglement

Leonid Gurvits

This paper continues research initiated in quant-ph/0201022 . The main subject here is the so-called Edmonds' problem of deciding if a given linear subspace of square matrices contains a nonsingular matrix . We present a deterministic polynomial time algorithm to solve this problem for linear subspaces satisfying a special matroids motivated property, called in the paper the Edmonds-Rado property . This property is shown to be very closely related to the separability of bipartite mixed states . One of the main tools used in the paper is the Quantum Permanent introduced in quant-ph/0201022 .

ACM Symposium on Theory of Computing (STOC)•Jan 1, 2003•30 citations

Encoding matroids into quantum states

Nathan Ferreira, Alison A. Silva, Giuliano G. La Guardia +1 more

Efficient representations of multipartite quantum states play a fundamental role in quantum information theory, providing both conceptual insight and practical tools for characterizing entanglement. Motivated by the axiomatic framework for graph states [Phys. Rev. A 85, 062313 (2012)] and its subsequent extension to hypergraph states [Phys. Rev. A 87, 022311 (2013)], we introduce an axiomatic construction of \emph{matroid states}, a new family of multipartite quantum states associated with matroids. Our constructions are based on a set of axioms analogous to those that define graph and hypergraph states, yielding a consistent quantum representation of arbitrary matroids. Two ways of constructing matroid states are proposed: the first is defined in terms of circuits, and the second in terms of independent sets. In both approaches, we establish the existence of universal global operators that satisfy desirable properties such as locality, symmetry, commutativity, and are associated with the combinatorial structure of matroids. Furthermore, we establish a hierarchy connecting graph, matroid, and hypergraph states within a unified framework. Additionally, we show how to obtain an arbitrary graph state by applying suitable families of matroid states, whose corresponding operators are the generators of the stabilizer subgroup of the graph state. These results identify matroid theory as a natural combinatorial language for the efficient description of multipartite quantum states and open new perspectives for the investigation of quantum entanglement and related combinatorial structures.

quant-pharXiv:2607.02736•Jul 6, 2026•0 citations

Classical deterministic complexity of Edmonds' problem and Quantum Entanglement

Gurvits, Leonid

This paper continues research initiated in quant-ph/0201022 . The main subject here is the so-called Edmonds' problem of deciding if a given linear subspace of square matrices contains a nonsingular matrix . We present a deterministic polynomial time algorithm to solve this problem for linear subspaces satisfying a special matroids motivated property, called in the paper the Edmonds-Rado property . This property is shown to be very closely related to the separability of bipartite mixed states . One of the main tools used in the paper is the Quantum Permanent introduced in quant-ph/0201022 .

arXiv:quant-ph/0303055•Mar 11, 2003•1 citation

Quantum Perfect Matchings

David Cui, Laura Mančinska, Seyed Sajjad Nezhadi +1 more

Abstract We investigate quantum and nonsignaling generalizations of perfect matchings in graphs using nonlocal games. Specifically, we introduce nonlocal games that test for L-perfect matchings in bipartite graphs, perfect matchings in general graphs and hypergraphs, and fractional perfect matchings. Our definitions come from the fact that these games are classical property tests for the corresponding matching conditions. We use the existence of perfect quantum and nonsignaling strategies for these games to define quantum and nonsignaling versions of perfect matchings. Finally, we provide characterizations of when graphs exhibit these extended properties: For nonsignaling matchings, we give a complete combinatorial characterization. In particular, a graph has a nonsignaling perfect matching if and only if it admits a fractional perfect matching that has bounded value on triangles. In bipartite graphs, the nonsignaling L-perfect matching property is achieved exactly when the left component of the graph can be split into two disjoint subgraphs: one with a classical L-perfect matching and another with left-degree 2. In the quantum setting, we show that complete graphs $$K_n$$ K n with odd $$n \ge 7$$ n ≥ 7 have quantum perfect matchings. We prove that a graph has a quantum perfect matching if and only if the quantum independence number of its line graph is maximal, extending a classical relationship between perfect matchings and line graph independence numbers. For bipartite graphs, we establish that the L-perfect matching game does not exhibit quantum pseudotelepathy, but we characterize the quantum advantage for complete bipartite graphs $$K_{n,2}$$ K n , 2 . Additionally, we prove that deciding quantum perfect matchings in hypergraphs is undecidable and leaves open the question of its complexity in graphs.

arXiv:2502.05136•Annales Henri Poincaré•Oct 14, 2025•0 citations
Classical matching theory can be defined in terms of matrices with nonnegative entries. The notion of Positive operator, central in Quantum Theory, is a natural generalization of matrices with non-negative entries. Based on this point of view, we introduce a definition of perfect Quantum (operator) matching. We show that the new notion inherits many 'classical' properties, but not all of them. This new notion goes somewhere beyound matroids. For separable bipartite quantum states this new notion coinsides with the full rank property of the intersection of two corresponding geometric matroids. In the classical situation, permanents are naturally associated with perfects matchings. We introduce an analog of permanents for positive operators, called Quantum Permanent and show how this generalization of the permanent is related to the Quantum Entanglement. Besides many other things, Quantum Permanents provide new rational inequalities necessary for the separability of bipartite quantum states. Using Quantum Permanents, we give deterministic poly-time algorithm to solve Hidden Matroids Intersection Problem and indicate some 'classical' complexity difficulties associated with the Quantum Entanglement. Finally, we prove that the weak membership problem for the convex set of separable bipartite density matrices is NP-HARD.
Classical matching theory can be defined in terms of matrices with nonnegative entries. The notion of Positive operator, central in Quantum Theory, is a natural generalization of matrices with nonnegative entries. Based on this point of view, we introduce a definition of perfect Quantum (operator) matching . We show that the new notion inherits many "classical" properties, but not all of them . This new notion goes somewhere beyound matroids . For separable bipartite quantum states this new notion coinsides with the full rank property of the intersection of two corresponding geometric matroids . In the classical situation, permanents are naturally associated with perfects matchings. We introduce an analog of permanents for positive operators, called Quantum Permanent and show how this generalization of the permanent is related to the Quantum Entanglement. Besides many other things, Quantum Permanents provide new rational inequalities necessary for the separability of bipartite quantum states . Using Quantum Permanents, we give deterministic poly-time algorithm to solve Hidden Matroids Intersection Problem and indicate some "classical" complexity difficulties associated with the Quantum Entanglement. Finally, we prove that the weak membership problem for the convex set of separable bipartite density matrices is NP-HARD.
This paper continues research initiated in quant-ph/0201022 . The main subject here is the so-called Edmonds' problem of deciding if a given linear subspace of square matrices contains a nonsingular matrix . We present a deterministic polynomial time algorithm to solve this problem for linear subspaces satisfying a special matroids motivated property, called in the paper the Edmonds-Rado property . This property is shown to be very closely related to the separability of bipartite mixed states . One of the main tools used in the paper is the Quantum Permanent introduced in quant-ph/0201022 .
This paper continues research initiated in quant-ph/0201022 . The main subject here is the so-called Edmonds' problem of deciding if a given linear subspace of square matrices contains a nonsingular matrix . We present a deterministic polynomial time algorithm to solve this problem for linear subspaces satisfying a special matroids motivated property, called in the paper the Edmonds-Rado property . This property is shown to be very closely related to the separability of bipartite mixed states . One of the main tools used in the paper is the Quantum Permanent introduced in quant-ph/0201022 .
Efficient representations of multipartite quantum states play a fundamental role in quantum information theory, providing both conceptual insight and practical tools for characterizing entanglement. Motivated by the axiomatic framework for graph states [Phys. Rev. A 85, 062313 (2012)] and its subsequent extension to hypergraph states [Phys. Rev. A 87, 022311 (2013)], we introduce an axiomatic construction of \emph{matroid states}, a new family of multipartite quantum states associated with matroids. Our constructions are based on a set of axioms analogous to those that define graph and hypergraph states, yielding a consistent quantum representation of arbitrary matroids. Two ways of constructing matroid states are proposed: the first is defined in terms of circuits, and the second in terms of independent sets. In both approaches, we establish the existence of universal global operators that satisfy desirable properties such as locality, symmetry, commutativity, and are associated with the combinatorial structure of matroids. Furthermore, we establish a hierarchy connecting graph, matroid, and hypergraph states within a unified framework. Additionally, we show how to obtain an arbitrary graph state by applying suitable families of matroid states, whose corresponding operators are the generators of the stabilizer subgroup of the graph state. These results identify matroid theory as a natural combinatorial language for the efficient description of multipartite quantum states and open new perspectives for the investigation of quantum entanglement and related combinatorial structures.
This paper continues research initiated in quant-ph/0201022 . The main subject here is the so-called Edmonds' problem of deciding if a given linear subspace of square matrices contains a nonsingular matrix . We present a deterministic polynomial time algorithm to solve this problem for linear subspaces satisfying a special matroids motivated property, called in the paper the Edmonds-Rado property . This property is shown to be very closely related to the separability of bipartite mixed states . One of the main tools used in the paper is the Quantum Permanent introduced in quant-ph/0201022 .
Abstract We investigate quantum and nonsignaling generalizations of perfect matchings in graphs using nonlocal games. Specifically, we introduce nonlocal games that test for L-perfect matchings in bipartite graphs, perfect matchings in general graphs and hypergraphs, and fractional perfect matchings. Our definitions come from the fact that these games are classical property tests for the corresponding matching conditions. We use the existence of perfect quantum and nonsignaling strategies for these games to define quantum and nonsignaling versions of perfect matchings. Finally, we provide characterizations of when graphs exhibit these extended properties: For nonsignaling matchings, we give a complete combinatorial characterization. In particular, a graph has a nonsignaling perfect matching if and only if it admits a fractional perfect matching that has bounded value on triangles. In bipartite graphs, the nonsignaling L-perfect matching property is achieved exactly when the left component of the graph can be split into two disjoint subgraphs: one with a classical L-perfect matching and another with left-degree 2. In the quantum setting, we show that complete graphs KnK_nKn​ K n with odd n≥7n \ge 7n≥7 n ≥ 7 have quantum perfect matchings. We prove that a graph has a quantum perfect matching if and only if the quantum independence number of its line graph is maximal, extending a classical relationship between perfect matchings and line graph independence numbers. For bipartite graphs, we establish that the L-perfect matching game does not exhibit quantum pseudotelepathy, but we characterize the quantum advantage for complete bipartite graphs Kn,2K_{n,2}Kn,2​ K n , 2 . Additionally, we prove that deciding quantum perfect matchings in hypergraphs is undecidable and leaves open the question of its complexity in graphs.