# Higher-order CPM Constructions

@article{Gogioso2019HigherorderCC, title={Higher-order CPM Constructions}, author={Stefano Gogioso}, journal={Electronic Proceedings in Theoretical Computer Science}, year={2019} }

We define a higher-order generalisation of the CPM construction based on arbitrary finite abelian group symmetries of symmetric monoidal categories. We show that our new construction is functorial, and that its closure under iteration can be characterised by seeing the construction as an algebra for an appropriate monad. We provide several examples of the construction, connecting to previous work on the CPM construction and on categorical probabilistic theories, as well as upcoming work on…

## 4 Citations

### CPM Categories for Galois Extensions

- PhilosophyElectronic Proceedings in Theoretical Computer Science
- 2021

By considering a generalisation of the CPM construction, we develop an infinite hierarchy of probabilistic theories, exhibiting compositional decoherence structures which generalise the traditional…

### Fantastic Quantum Theories and Where to Find Them

- Mathematics
- 2017

We present a uniform framework for the treatment of a large class of toy models of quantum theory. Specifically, we will be interested in theories of wavefunctions valued in commutative involutive…

### Hyper-decoherence in Density Hypercubes

- Computer ScienceElectronic Proceedings in Theoretical Computer Science
- 2021

This work demonstrates the existence of a probabilistic hyper-decoherence of density hypercubes to quantum systems and calculate the associated hyper-phase group.

### Density Hypercubes, Higher Order Interference and Hyper-decoherence: A Categorical Approach

- Computer ScienceQI
- 2018

This work uses the recently introduced double-dilation construction by Zwart and Coecke to construct a new categorical probabilistic theory of density hypercubes and shows that the theory displays higher-order interference of order up to fourth.

## References

SHOWING 1-10 OF 22 REFERENCES

### A Bestiary of Sets and Relations

- MathematicsArXiv
- 2015

This work provides a treatment of pure state and mixed state quantum mechanics in the category fRel of finite sets and relations and concludes by proving that fRel is local, but not without some unexpected twists.

### Completely positive projections and biproducts

- Physics
- 2013

The recently introduced CP*-construction unites quantum channels and classical systems, subsuming the earlier CPM-construction in categorical quantum mechanics. We compare this construction to two…

### The sheaf-theoretic structure of non-locality and contextuality

- Mathematics
- 2011

It is shown that contextuality, and non-locality as a special case, correspond exactly to obstructions to the existence of global sections, and a linear algebraic approach to computing these obstructions is described, which allows a systematic treatment of arguments for non- Locality and contextuality.

### A Graph Theoretic Perspective on CPM(Rel)

- MathematicsQPL
- 2015

This work investigates the category CPM(Rel) of completely positive maps in Rel, which is isomorphic to a category of sets and graphs between them and demonstrates the possibly surprising phenomenon of a pure state that can be given as a mixture of two mixed states.

### Environment and Classical Channels in Categorical Quantum Mechanics

- Computer ScienceCSL
- 2010

We present a both simple and comprehensive graphical calculus for quantum computing. We axiomatize the notion of an environment, which together with the axiomatic notion of classical structure…

### Fantastic Quantum Theories and Where to Find Them

- Mathematics
- 2017

We present a uniform framework for the treatment of a large class of toy models of quantum theory. Specifically, we will be interested in theories of wavefunctions valued in commutative involutive…

### Categorical Probabilistic Theories

- Physics
- 2017

A minimal number of operationally motivated axioms are adopted which provide clean categorical foundations, in the style of CQM, for the treatment of the problems that OPTs are concerned with.

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

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

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

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