PlanetPhysics/Category of Quantum Automata

Let us recall that as a [[../Triangulation Methods For Quantized Spacetimes 2|Quantum Algebraic Topology]] [[../Trivial Groupoid|object]], a [[../LQG2|quantum automaton]] is defined by the quantum triple $Q_A=(𝒢,ℍ-{\mathrm{\Re }}_G,Aut(𝒢)$), where $𝒢$ is a (locally compact) [[../Weak Hopf Algebra|quantum groupoid]], $ℍ-{\mathrm{\Re }}_G$ are the unitary [[../Categorical Group Representation|representations]] of $𝒢$ on [[../I3|rigged Hilbert spaces]] ${\mathrm{\Re }}_G$ of quantum states and [[../Quantum Operator Algebra 5|quantum operators]] on the [[../Norm Induced By Inner Product|Hilbert space]] $ℍ$, and $Aut(𝒢)$ is the transformation, or automorphism [[../Equivalence Relation|equivalence relation]] of quantum transitions that represents all flip-flop quantum transitions of one cubit each between the permitted quantum states of the quantum automaton.

With the data from above definition we can now define also the category of quantum automata as follows.

The category of quantum automata ${\mathbb{Q}}_A$ is defined as an algebraic category [[../Category Of Logic Algebras|category of logic algebras]] whose objects are triples $(ℍ,\mathrm{\Delta }:ℍ\to ℍ,\mu )$ (where $ℍ$ is either a Hilbert space or a rigged Hilbert space of quantum states and [[../Quantum Operator Algebra 4|operators]] acting on $ℍ$, and $\mu$ is a measure related to the [[../LQG2|quantum logic]], $LM$, and (quantum) transition probabilities of this quantum [[../Similarity And Analogous Systems Dynamic Adjointness And Topological Equivalence|system]]), and whose [[../Trivial Groupoid|morphisms]] are defined between such triples by [[../Trivial Groupoid|homomorphisms]] of Hilbert spaces, $\mathrm{\varnothing }:ℍ\to ℍ$, naturally compatible with the operators $\mathrm{\Delta }$, and by homomorphisms between the associated [[../Higher Dimensional Quantum Algebroid|Haar measure]] systems.

An alternative definition is also possible based on Quantum Algebraic Topology .

A quantum algebraic topology definition of the [[../Cod|Cod]] of quantum [[../Co Intersections|algebraic]] automata involves the objects specified above in Definition 0.1 as quantum automaton triples $(Q_A)$, and [[../Quantum Computers|quantum automata]] homomorphisms defined between such triples; these $Q_A$ morphisms are defined by [[../Equivalence Relation|groupoid homomorphisms]] $h:𝒢\to {𝒢}^{*}$ and $\alpha :Aut(𝒢)\to Aut({𝒢}^{*})$, together with unitarity preserving mappings $u$ between unitary representations of $𝒢$ on rigged Hilbert spaces (or [[../Hilbert Bundle|Hilbert space bundles]]).