Two Roads to Classicality

@inproceedings{Coecke2017TwoRT,
  title={Two Roads to Classicality},
  author={Bob Coecke and John H. Selby and Sean Tull},
  year={2017}
}
Mixing and decoherence are both manifestations of classicality within quantum theory, each of which admit a very general category-theoretic construction. We show under which conditions these two 'roads to classicality' coincide. This is indeed the case for (finite-dimensional) quantum theory, where each construction yields the category of C*-algebras and completely positive maps. We present counterexamples where the property fails which includes relational and modal theories. Finally, we… 
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19 Citations
Background Citations
13
Methods Citations
7

19 Citations

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A no-go theorem for theories that decohere to quantum mechanics

  • Ciarán M. LeeJohn H. Selby
  • Philosophy, Physics
    Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
  • 2018
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This work demonstrates the existence of a probabilistic hyper-decoherence of density hypercubes to quantum systems and calculate the associated hyper-phase group.

A Diagrammatic Derivation of the Hermitian Adjoint

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References

SHOWING 1-10 OF 38 REFERENCES

Categories of quantum and classical channels

A construction that turns a category of pure state spaces and operators into a categories of observable algebras and superoperators is introduced, providing elegant abstract notions of preparation and measurement.

Categorical Formulation of Finite-Dimensional Quantum Algebras

We describe how †-Frobenius monoids give the correct categorical description of certain kinds of finite-dimensional ‘quantum algebras’. We develop the concept of an involution monoid, and use it to…

Almost Quantum Theory

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…

Leaks: Quantum, Classical, Intermediate and More

It is shown that defining a notion of purity for processes in general process theories has to make reference to the leaks of that theory, a feature missing in standard definitions; hence, a refined definition is proposed and the resulting notion ofurity for quantum, classical and intermediate theories is studied.

Classical and quantum structuralism

In recent work, symmetric dagger-monoidal (SDM) categories have emerged as a convenient categorical formalization of quantum mechanics. The objects represent physical systems, the morphisms physical…

Strong Complementarity and Non-locality in Categorical Quantum Mechanics

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.

Axiomatizing complete positivity

There are two ways to turn a categorical model for pure quantum theory into one for mixed quantum theory, both resulting in a category of completely positive maps, and this work extends this axiomatization to the latter by introducing decoherence structures.

Categorical Quantum Mechanics I: Causal Quantum Processes

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…

Environment and Classical Channels in Categorical Quantum Mechanics

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…

Limited Holism and Real-Vector-Space Quantum Theory

This paper considers in this paper a class of theories more holistic than quantum theory in that they are constrained only by “bilocal tomography”: the state of any composite system is determined by the statistics of measurements on pairs of components.