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Beyond Operator Systems

Gemma De les Coves, Mirte van der Eyden, Tim Netzer

TL;DR

This work extends the theory of abstract operator systems beyond PSD cones to general conic structures by introducing stems: star autonomous functors from a base category to finite-dimensional vector spaces. It proves that central results—Choi–Kraus decompositions, Choi–Effros realizations, Arveson extension, duality, and tensor-product laws—still hold in this broader setting with simpler proofs. The framework unifies diverse areas (group representations, mapping cones, TFTs) under a common categorical-convex-analytic perspective, and yields new results such as vector-valued extensions and invariant-CP criteria. This generalization opens paths to general probabilistic theories, free mapping cones, and topological quantum field theories, providing a versatile toolkit for analyzing CP maps and cone structures in varied contexts.

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.

Beyond Operator Systems

TL;DR

This work extends the theory of abstract operator systems beyond PSD cones to general conic structures by introducing stems: star autonomous functors from a base category to finite-dimensional vector spaces. It proves that central results—Choi–Kraus decompositions, Choi–Effros realizations, Arveson extension, duality, and tensor-product laws—still hold in this broader setting with simpler proofs. The framework unifies diverse areas (group representations, mapping cones, TFTs) under a common categorical-convex-analytic perspective, and yields new results such as vector-valued extensions and invariant-CP criteria. This generalization opens paths to general probabilistic theories, free mapping cones, and topological quantum field theories, providing a versatile toolkit for analyzing CP maps and cone structures in varied contexts.

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.
Paper Structure (21 sections, 27 theorems, 146 equations, 17 figures, 1 table)

This paper contains 21 sections, 27 theorems, 146 equations, 17 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cones and tensor products
  4. Abstract operator systems
  5. Star autonomous categories
  6. Useful tools
  7. Theory of general conic systems
  8. Stems --- the underlying structures
  9. General conic systems
  10. Duality
  11. Generalizing theorems about abstract operator systems
  12. Tensor products
  13. More stems and applications
  14. Cone systems
  15. Two-sided systems and free mapping cones
  16. ...and 6 more sections

Key Result

Theorem 2.6

Let $X\subseteq {\rm Mat}_d$ be a (concrete) operator system, and let $\psi\colon X \to {\rm Mat}_t$ be a completely positive map. Then $\psi$ can be extended to a completely positive map $\phi\colon {\rm Mat}_d \to {\rm Mat}_t$.

Figures (17)

  • Figure 1: An abstract operator system on a real vector space $X$ consists of a convex cone inside $X \otimes \textrm{Her}_s$ for every $s$ ($\textrm{Her}_s$ are $s\times s$ complex Hermitian matrices), such that the cones are compatible under matrix contractions. These morphisms and underlying vector spaces, i.e. everything in green, will be formalized as a stem in general conic systems. The yellow cone will be referred to as 'the cone at the base level'.
  • Figure 2: A category consists of objects (black dots) and morphisms (black arrows in C, grey directional shapes in FVec). A stem $\mathcal{S}$ is a functor from a star autonomous category ${\tt C}$ to the category of finite-dimensional vector spaces ${\tt FVec}$ which picks out the vector spaces and the linear maps that will be used to build a general conic system, i.e., that play the role of everything green in \ref{['fig:aos_def']}. Here, the green dots and arrows in FVec are in the image of the stem, whereas the black/grey are not. For example, for aos the vector spaces of Hermitian matrices of all sizes and the linear maps given by matrix contractions would be green, i.e. in the image of $\mathcal{S}$.
  • Figure 3: a) A simplex cone in $\mathbb{R}^2$, b) a polyhedral cone in $\mathbb{R}^3$, c) the Lorentz cone in $\mathbb{R}^3$ and d) the psd cone $\textrm{Psd}_2 \subseteq \textrm{Her}_2 \cong \mathbb{R}^4$ are examples of convex cones (defined in \ref{['ex:cones']}).
  • Figure 4: There are many possible tensor products of convex cones. For any two convex cones, their tensor product $\otimes$ is in between their minimal tensor product $\stackunder[1.5pt]{$ ⊗ $}{}$ and maximal tensor product $\space\bar{\otimes}\space$.
  • Figure 5: A given cone in $X$, here the yellow cone (base cone), can be extended to an abstract operator system by choosing a cone in every other level in a compatible way (\ref{['fig:aos_def']}). Due to this compatibility constraint, there is a smallest and largest abstract operator system over the base cone, given by the minimal and maximal tensor product of the base cone and the psd cone (green) in any level. For general conic systems there are different vector spaces and different compatibility conditions (everything that is green in \ref{['fig:aos_def']}) and therefore different cones play the role of the psd cone.
  • ...and 12 more figures

Theorems & Definitions (84)

  • Example 2.1: Examples and facts about important cones (\ref{['fig:cones_examples']})
  • Example 2.2: Tensor products of some important cones
  • Definition 2.3: Abstract operator system (\ref{['fig:aos_def']})
  • Example 2.4: Examples of abstract operator systems
  • Definition 2.5: (Completely) Positive maps
  • Theorem 2.6: Arveson's extension theorem Paulsen03
  • Definition 2.7: Symmetric monoidal category nlab:symmetric_monoidal_category
  • Definition 2.8: Star autonomous category nlab:star_autonomous_category
  • Definition 2.9: Strong monoidal functor nlab:monoidal_functor
  • Example 2.10: Star autonomous functors
  • ...and 74 more