# Fantastic Quantum Theories and Where to Find Them

@article{Gogioso2017FantasticQT, title={Fantastic Quantum Theories and Where to Find Them}, author={Stefano Gogioso}, journal={arXiv: Quantum Physics}, year={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 semirings, and which give rise to some semiring-based notion of classical non-determinism via the Born rule. The models obtained with our construction possess many of the familiar structures used in Categorical Quantum Mechanics. We also provide a bestiary of increasingly exotic examples: some well…

## 18 Citations

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## References

SHOWING 1-10 OF 76 REFERENCES

### Operational theories and Categorical quantum mechanics

- Physics
- 2012

A central theme in current work in quantum information and quantum foundations is to see quantum mechanics as occupying one point in a space of possible theories, and to use this perspective to…

### Quantum and Classical Structures in Nondeterminstic Computation

- PhysicsQI
- 2009

It is shown that classical structures in the category of relations correspond to direct sums of abelian groups in the standard Hilbert space model of quantum theories.

### Strong Complementarity and Non-locality in Categorical Quantum Mechanics

- Physics2012 27th Annual IEEE Symposium on Logic in Computer Science
- 2012

The diagrammatic calculus substantially simplifies (and sometimes even trivialises) many of the derivations, and provides new insights, and the diagrammatic computation of correlations clearly shows how local measurements interact to yield a global overall effect.

### Categorical Quantum Mechanics I: Causal Quantum Processes

- Philosophy
- 2018

We derive the category-theoretic backbone of quantum theory from a process ontology. More specifically, we treat quantum theory as a theory of systems, processes and their interactions.
In this…

### Fully graphical treatment of the quantum algorithm for the Hidden Subgroup Problem

- Mathematics, Computer Science
- 2017

This work provides the first fully diagrammatic proof of correctness for the abelian HSP protocol, showing that strongly complementary observables are the key ingredient to its success.

### A Complete Graphical Calculus for Spekkens’ Toy Bit Theory

- Physics
- 2014

In this paper, a graphical language for Spekkens’ toy theory is developed and it is shown that it is complete: meaning any equality that can be derived using other formalisms can also be derived entirely graphically.

### Division Algebras and Quantum Theory

- Mathematics
- 2012

Quantum theory may be formulated using Hilbert spaces over any of the three associative normed division algebras: the real numbers, the complex numbers and the quaternions. Indeed, these three…

### A Royal Road to Quantum Theory (or Thereabouts)

- PhysicsQPL
- 2016

A representation of finite-dimensional probabilistic systems in terms of Euclidean Jordan algebras is obtained, in a strikingly easy way, from simple assumptions, which provides a framework within which real, complex and quaternionic QM can play happily together and allows some (but not too much) room for more exotic alternatives.

### Almost Quantum Theory

- Mathematics
- 2016

Modal quantum theory (MQT) is a “toy model” of quantum theory in which amplitudes are elements of a general field. The theory predicts, not the probabilities of a measurement result, but only whether…

### Phase Groups and the Origin of Non-locality for Qubits

- Physics, MathematicsElectron. Notes Theor. Comput. Sci.
- 2011