# Deriving Dagger Compactness

@article{Tull2020DerivingDC, title={Deriving Dagger Compactness}, author={Sean Tull}, journal={Electronic Proceedings in Theoretical Computer Science}, year={2020} }

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.

## References

SHOWING 1-10 OF 26 REFERENCES

### Dagger Compact Closed Categories and Completely Positive Maps: (Extended Abstract)

- MathematicsElectron. Notes Theor. Comput. Sci.
- 2007

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

- MathematicsJ. Philos. Log.
- 2013

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)

- MathematicsElectron. 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

- MathematicsQPL
- 2017

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

- Computer ScienceInt. J. Softw. Informatics
- 2014

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

- PhysicsElectron. Notes Theor. Comput. Sci.
- 2008

### Process-theoretic characterisation of the Hermitian adjoint

- Mathematics
- 2016

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

- MathematicsLog. 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

- Mathematics
- 2010

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…