Beyond Operator Systems (2312.13983v1)

Abstract: Operator systems connect operator algebra, free semialgebraic geometry and quantum information theory. In this work we generalize operator systems and many of their theorems. While positive semidefinite matrices form the underlying structure of operator systems, our work shows that these can be promoted to far more general structures. For instance, we prove a general extension theorem which unifies the well-known homomorphism theorem, Riesz' extension theorem, Farkas' lemma and Arveson's extension theorem. On the other hand, the same theorem gives rise to new vector-valued extension theorems, even for invariant maps, when applied to other underlying structures. We also prove generalized versions of the Choi-Kraus representation, Choi-Effros theorem, duality of operator systems, factorizations of completely positive maps, and more, leading to new results even for operator systems themselves. In addition, our proofs are shorter and simpler, revealing the interplay between cones and tensor products, captured elegantly in terms of star autonomous categories. This perspective gives rise to new connections between group representations, mapping cones and topological quantum field theory, as they correspond to different instances of our framework and are thus siblings of operator systems.

• The paper introduces a generalized framework using category theory that replaces traditional psd matrices with 'stems' to form general conic systems.
• It develops a unified extension theorem that generalizes classical results such as Arveson's extension theorem, Riesz' theorem, and Farkas' lemma.
• The framework simplifies duality and representation proofs, offering new insights for quantum information, operator algebra, and free semialgebraic geometry.

An Overview of the Paper "Beyond Operator Systems"

The paper "Beyond Operator Systems" presents a comprehensive extension of the traditional framework of abstract operator systems by introducing a generalized approach based on category theory. This work connects operator algebra, free semialgebraic geometry, and quantum information theory by generalizing the concept of operator systems beyond positive semidefinite (psd) matrices. The authors propose a new theoretical framework for understanding operator systems in terms of "general conic systems."

Generalization and Key Results

Operator systems typically rely on psd matrices as their structural backbone. However, this paper reveals that many properties attributed to operator systems are not inherent to the psd construction but hold true for broader categorical structures. The authors introduce the notion of "stems," which utilize star autonomous categories to construct operator systems without being confined to psd matrices alone. This generalization arises through specific functors called stems, providing a base structure representing a new category of systems termed general conic systems.

Extension Theorems

One of the paper's central results is the development of a general extension theorem that encompasses various classical theorems such as the classical homomorphism theorem, Riesz' extension theorem, Farkas' lemma, and Arveson's extension theorem. Additionally, this theorem yields new vector-valued extension theorems even in the presence of invariant maps, broadening the utility and applicability of these mathematical constructs within abstract operator systems.

New Representation and Duality Theorems

Alongside the extension results, the paper establishes generalized versions of several well-known theorems in operator theory: the Choi--Kraus representation and the Choi--Effros theorem among others. These lead to novel results within the context of operator systems, offering simpler and more concise proofs than previously available. A particular emphasis is placed on duality, where generalized dual systems are shown to unfold significantly simpler structural insights, offering dramatic simplifications of prior complex proofs.

Theoretical Implications

The theoretical implications extend beyond operator systems, providing a unified framework that captures various known results in operator algebra and free semialgebraic geometry. For instance, the results have implications on topics such as quantum channels and mapping cones, which are broadened by the newly identified structures.

Applications and Further Developments

The new framework opens up several avenues for application and further theoretical variance. The mapping of this structure to concrete mathematical and physical problems in quantum information theory, such as entanglement theory, could potentially pave the way for deeper insights into complex phenomena not previously analyzable through conventional methods.

Future Directions

This construction of general conic systems provides fertile ground for expanding areas such as generalized probabilistic theories, offering new methods to analyze quantum systems and structures defining quantum mechanics' mathematical base. The introduction of stems and their associated categories suggest that further investigation could yield additional powerful theorems in operator theory, potentially redefining the boundary between classical and quantum mathematical spaces and their operational definitions.

Overall, "Beyond Operator Systems" presents a significant stride in the mathematical abstraction of operator systems, enforcing versatile theories with wide theoretical and practical implications. These findings could reframe discussions in both mathematics and quantum mechanics, offering broader and deeper tools for researchers in the field.