Positive Maps and Extendibility Hierarchies from Copositive Matrices Characterization Defines Novel PCOP Cone for Positivity Witnessing

The fundamental challenge of characterizing positive maps, essential in fields ranging from quantum information theory to operator algebras, receives significant attention from Aabhas Gulati, Ion Nechita, and Sang-Jun Park. Their work introduces and investigates a new mathematical structure, the PCOP cone, which provides a complete description of positivity for a wide range of maps and establishes a crucial duality with the well-known PCP cone. This research builds a powerful connection between the established theory of copositive forms and the behaviour of positive maps, and extends this framework to decomposable maps through the introduction of the PDEC cone. By defining maps parameterized by graph structure and a real parameter, the team derives precise thresholds determining map positivity, generating new families of positive maps exemplified by those derived from complex graphs like Paley graphs, and revealing a surprising link between matrix hierarchies and quantum extendibility properties of states such as Dicke states.

The research focuses on completely positive matrices and provides a complete characterization for the positivity of a broad and physically relevant class of linear maps, known as covariant maps. The team establishes a method to systematically lift matrices from the well-known cone of copositive matrices into a new structure, the pairwise copositive cone, creating a powerful bridge between established theory and the structure of positive maps. An analogous framework is developed for decomposable maps, introducing the pairwise decomposable cone, which expands understanding of map structures. As a primary application, the researchers define a novel family of linear maps parameterized by a graph and a real value, and derive exact thresholds for this parameter.

Operator Algebra Foundations For Quantum Maps

This work builds upon a foundation of established mathematical tools in operator algebra. Key concepts include positive maps, central to the study of quantum information, and completely bounded maps, which provide a rigorous framework for analyzing their properties. Researchers have extensively studied positive maps of low-dimensional matrix algebras and explored the geometric structure of these maps through the use of cones, providing the mathematical basis for the current research.

Pairwise Positivity and Decomposable Map Characterization

This work introduces and systematically studies the pairwise copositive cone, denoted as PCOP, and establishes its duality with the pairwise completely positive cone, PCP. A central achievement is the complete characterization of positivity for a broad class of linear maps, demonstrating that a map is positive if and only if it belongs to the PCOP cone, resolving a long-standing problem in operator algebras and information theory. Researchers developed a constructive method to lift matrices from the well-known cone of copositive matrices into the PCOP cone, creating a powerful bridge between established theory and new insights into positive maps. Extending this analysis, the team introduced the pairwise decomposable cone, PDEC, and proved it precisely characterizes decomposable maps, linking it to the tractable cone SPN.

This work clarifies the relationship between copositive matrices and decomposable maps, bridging gaps in existing literature. As a primary application, scientists defined a novel family of linear maps associated with simple graphs, parameterized by a real value. They derived precise thresholds for this parameter that determine when these maps are completely positive, decomposable, or simply positive, expressed in terms of fundamental graph parameters including the largest eigenvalue, clique number, and a newly defined graph parameter. This construction generates new positive indecomposable maps, with researchers proving that infinite families of strongly regular graphs, including Paley graphs, yield such maps.

Furthermore, the team connected their results to hierarchies of quantum state extendibility, demonstrating that the sum-of-squares hierarchies for copositive matrices directly correspond to witnesses for different levels of PPT-bosonic extendibility of symmetric quantum states like Dicke states. They provided explicit examples of bosonic entangled states that are arbitrarily highly extendible, with local dimension not less than 5, and proved the existence of r-partite entangled bosonic states that are PPT with respect to all bipartitions. This establishes a remarkable connection between matrix cones and quantum information theory.

PCOP, PDEC Cones and Covariant Map Characterisation

This work establishes a new framework for characterizing positive linear maps, central to both operator algebra theory and information science, through the introduction of the PCOP (pairwise copositive) cone. Researchers demonstrated that PCOP is mathematically dual to the PCP (pairwise completely positive) cone, providing a complete description of positivity for a broad class of maps known as covariant maps. A key achievement lies in the development of a method to connect matrices from the well-understood COP (copositive) cone to the newly defined PCOP cone, bridging established theory with this novel structure. Expanding on this, the team created an analogous framework using the PDEC (pairwise decomposable) cone, and then defined a family of linear maps parameterized by graph theory and a real value.

They precisely determined thresholds for this parameter, revealing when these maps are positive or decomposable, and linking these properties to fundamental characteristics of the graphs themselves. This resulted in the discovery of new families of positive, yet non-completely positive, maps, exemplified by examples derived from infinite classes of strongly regular graphs, such as Paley graphs. Furthermore, the research establishes a duality between these maps and symmetric states, specifically demonstrating a correspondence between sum-of-squares hierarchies and levels within the PPT (positive partial transpose) bosonic extendibility hierarchy. The authors acknowledge that the equivalence of certain hierarchies within the bosonic extendibility framework remains an open question, and pose this as a direction for future research. They also question whether inclusions within the K(r) hierarchy of maps are strictly increasing, a finding that would have implications for understanding entanglement within Dicke states.