Deriving Dagger Compactness

  title={Deriving Dagger Compactness},
  author={Sean Tull},
  journal={Electronic Proceedings in Theoretical Computer Science},
  • S. Tull
  • Published 11 July 2019
  • Mathematics
  • Electronic Proceedings in Theoretical Computer Science
Dagger compact structure is a common assumption in the study of physical process theories, but lacks a clear interpretation. Here we derive dagger compactness from more operational axioms on a category. We first characterise the structure in terms of a simple mapping of states to effects which we call a 'state dagger', before deriving this in any category with 'completely mixed' states and a form of purification, as in quantum theory. 



Symmetry, Compact Closure and Dagger Compactness for Categories of Convex Operational Models

This paper characterize compact closure of symmetric monoidal categories of convex operational models in two ways: as a statement about the existence of teleportation protocols, and as the principle that every process allowed by that theory can be realized as an instance of a remote evaluation protocol—hence, as a form of classical probabilistic conditioning.

Finite Dimensional Hilbert Spaces are Complete for Dagger Compact Closed Categories (Extended Abstract)

  • P. Selinger
  • Mathematics
    Electron. Notes Theor. Comput. Sci.
  • 2011
We show that an equation follows from the axioms of dagger compact closed categories if and only if it holds in finite dimensional Hilbert spaces.

Purity through Factorisation

A construction is given that identifies the collection of pure processes within a theory containing both pure and mixed processes, and defines a pure subcategory in the framework of symmetric monoidal categories.

Distinguishability and Copiability of Programs in General Process Theories

We propose a notion of state distinguishability that does not refer to probabilities, but rather to the ability of a set of states to serve as programs for a desired set of gates. Using this notion,

Axiomatic Description of Mixed States From Selinger's CPM-construction

  • B. Coecke
  • Physics
    Electron. Notes Theor. Comput. Sci.
  • 2008

Process-theoretic characterisation of the Hermitian adjoint

We show that the physical principle "the adjoint associates to each state a `test' for that state" fully characterises the Hermitian adjoint for pure quantum theory, therefore providing the adjoint

A Categorical Reconstruction of Quantum Theory

  • S. Tull
  • Mathematics
    Log. Methods Comput. Sci.
  • 2020
We reconstruct finite-dimensional quantum theory from categorical principles. That is, we provide properties ensuring that a given physical theory described by a dagger compact category in which one

A Survey of Graphical Languages for Monoidal Categories

This article is intended as a reference guide to various notions of monoidal categories and their associated string diagrams. It is hoped that this will be useful not just to mathematicians, but also