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Categorical relations and bipartite entanglement in tensor cones for Toeplitz and Fejér-Riesz operator systems

Douglas Farenick

TL;DR

The paper studies separability and entanglement in tensor cones that arise from base cones of operator-system tensor products, focusing on Toeplitz and Fejér-Riesz systems and their duality. It develops a framework showing the base cones of operator-system tensor products are tensor cones, derives precise separability results (e.g., certain Toeplitz and Fejér-Riesz tensor products have separable bases), and identifies entanglement phenomena via universal matrices like $T_n$ that play a role analogous to the Choi matrix. The authors establish a robust matrix-CP criterion via Toeplitz constructions (e.g., $T_n$) and explore dualities, truncations, and categorical properties, including the absence of the weak expectation property and the behavior of C$^*$-envelopes. The results advance the understanding of entanglement structure in these structured operator systems and illuminate how dualities, quotients, and envelopes interact in this noncommutative setting, with implications for tensor-product theory in finite dimensions and for quantum-information-inspired frameworks.

Abstract

The present paper aims to understand separability and entanglement in tensor cones, in the sense of Namioka and Phelps, that arise from the base cones of operator system tensor products. Of particular interest here are the Toeplitz and Fejér-Riesz operator systems, which are, respectively, operator systems of Toeplitz matrices and Laurent polynomials of bounded degree (that is, trigonometric polynomials), and which are related in the operator system category through duality. Some additional categorical relationships established in this paper for Toeplitz and Fejér-Riesz operator systems. Of independent interest is a single matrix criterion, similar to the criterion involving the Choi matrix, for a linear map of the Fejér-Riesz operator system to be completely positive.

Categorical relations and bipartite entanglement in tensor cones for Toeplitz and Fejér-Riesz operator systems

TL;DR

The paper studies separability and entanglement in tensor cones that arise from base cones of operator-system tensor products, focusing on Toeplitz and Fejér-Riesz systems and their duality. It develops a framework showing the base cones of operator-system tensor products are tensor cones, derives precise separability results (e.g., certain Toeplitz and Fejér-Riesz tensor products have separable bases), and identifies entanglement phenomena via universal matrices like that play a role analogous to the Choi matrix. The authors establish a robust matrix-CP criterion via Toeplitz constructions (e.g., ) and explore dualities, truncations, and categorical properties, including the absence of the weak expectation property and the behavior of C-envelopes. The results advance the understanding of entanglement structure in these structured operator systems and illuminate how dualities, quotients, and envelopes interact in this noncommutative setting, with implications for tensor-product theory in finite dimensions and for quantum-information-inspired frameworks.

Abstract

The present paper aims to understand separability and entanglement in tensor cones, in the sense of Namioka and Phelps, that arise from the base cones of operator system tensor products. Of particular interest here are the Toeplitz and Fejér-Riesz operator systems, which are, respectively, operator systems of Toeplitz matrices and Laurent polynomials of bounded degree (that is, trigonometric polynomials), and which are related in the operator system category through duality. Some additional categorical relationships established in this paper for Toeplitz and Fejér-Riesz operator systems. Of independent interest is a single matrix criterion, similar to the criterion involving the Choi matrix, for a linear map of the Fejér-Riesz operator system to be completely positive.
Paper Structure (35 sections, 60 theorems, 140 equations)

This paper contains 35 sections, 60 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. Tensor cones and the category $\mathfrak S_1$
  3. Operator systems
  4. Real versus complex vector spaces
  5. Tensor cones with operator systems
  6. Toeplitz and Fejér-Riesz operator systems
  7. Statement of the main results
  8. Base cones of operator system tensor products are tensor cones
  9. Separability
  10. Entanglement
  11. Categorical relations
  12. Operator System Duality and Tensor Products
  13. Toeplitz and Fejér-Riesz Operator Systems
  14. Canonical linear bases
  15. Generalised circulants
  16. ...and 20 more sections

Key Result

Theorem 1.3

If ${\mathcal{R}}$ and ${\mathcal{T}}$ are finite-dimensional operator systems, then if and only if one of the cones ${\mathcal{R}}_+$ or ${\mathcal{T}}_+$ is simplicial.

Theorems & Definitions (118)

  • Definition 1.1
  • Theorem 1.3
  • proof
  • Example 1.4: Block-positive matrices
  • proof
  • Theorem 1.5: Duality
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • ...and 108 more