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Quantum relations in the general setting: composition and adjacency operators

Matthew Daws

Abstract

Quantum relations in the sense of Weaver are $M'$-bimodules, for a von Neumann algebra $M$, these generalising actual relations on a set $X$ when $M=\ell^\infty(X)$. Similarly, relations between two sets can be generalised as bimodules over the commutants of two algebras. We make an explicit study of this idea, developing some tools to check that constructions are well-defined. Motivation comes from Kornell's concept of a Quantum Set (for algebras which are sums of matrix algebras), and we find that $*$-homomorphisms correspond to certain quantum relations, extending unpublished work of Kornell. We find a functor from completely positive maps to quantum relations, related to the idea of taking a noisy communication channel and reducing it to its underlying ``relation''. As with Quantum Graphs, at least in finite-dimensions, quantum relations correspond to ``adjacency operators'', certain CP maps depending on a choices of faithful functional on the algebras. We develop some tools to deal with the non-Schur-idempotent case, and show links with our functor from CP maps, and work of Verdon. We explicitly compute the adjacency operator of a $*$-homomorphism.

Quantum relations in the general setting: composition and adjacency operators

Abstract

Quantum relations in the sense of Weaver are -bimodules, for a von Neumann algebra , these generalising actual relations on a set when . Similarly, relations between two sets can be generalised as bimodules over the commutants of two algebras. We make an explicit study of this idea, developing some tools to check that constructions are well-defined. Motivation comes from Kornell's concept of a Quantum Set (for algebras which are sums of matrix algebras), and we find that -homomorphisms correspond to certain quantum relations, extending unpublished work of Kornell. We find a functor from completely positive maps to quantum relations, related to the idea of taking a noisy communication channel and reducing it to its underlying ``relation''. As with Quantum Graphs, at least in finite-dimensions, quantum relations correspond to ``adjacency operators'', certain CP maps depending on a choices of faithful functional on the algebras. We develop some tools to deal with the non-Schur-idempotent case, and show links with our functor from CP maps, and work of Verdon. We explicitly compute the adjacency operator of a -homomorphism.
Paper Structure (11 sections, 46 theorems, 86 equations)

This paper contains 11 sections, 46 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Acknowledgements
  4. Quantum sets and general quantum relations
  5. Functions between quantum sets
  6. Channels to relations
  7. Pullbacks
  8. Adjacency operators
  9. CP maps to quantum relations revisited
  10. Ordering
  11. Quantum functions

Key Result

Lemma 2.2

Let $\mathcal{X},\mathcal{Y}$ be quantum sets and regard $L^\infty(\mathcal{X}) \subseteq \mathcal{B}(\ell^2(\mathcal{X}))$ and the same for $\mathcal{Y}$. A quantum relation from $L^\infty(\mathcal{X})$ to $L^\infty(\mathcal{Y})$ is of the form where for each $X\in\mathcal{X}, Y\in\mathcal{Y}$ we have that $V_{X,Y} \subseteq \mathcal{B}(X,Y)$ is a subspace, and we regard $V_{X,Y} \subseteq \math

Theorems & Definitions (114)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • Corollary 2.7
  • ...and 104 more