# PlanetPhysics/Category of Quantum Automata

Let us recall that as a Quantum Algebraic Topology object, a quantum automaton is defined by the quantum triple ${\displaystyle Q_{A}=({\mathcal {G}},\mathbb {H} -\Re _{G},Aut({\mathcal {G}})}$), where ${\displaystyle {\mathcal {G}}}$ is a (locally compact) quantum groupoid, ${\displaystyle \mathbb {H} -\Re _{G}}$ are the unitary representations of ${\displaystyle {\mathcal {G}}}$ on rigged Hilbert spaces ${\displaystyle \Re _{G}}$ of quantum states and quantum operators on the Hilbert space ${\displaystyle \mathbb {H} }$, and ${\displaystyle Aut({\mathcal {G}})}$ is the transformation, or automorphism 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 ${\displaystyle {\mathcal {\mathbb {Q} }}_{A}}$ is defined as an algebraic category category of logic algebras whose objects are triples ${\displaystyle (\mathbb {H} ,\Delta :\mathbb {H} \rightarrow \mathbb {H} ,\mu )}$ (where ${\displaystyle \mathbb {H} }$ is either a Hilbert space or a rigged Hilbert space of quantum states and operators acting on ${\displaystyle \mathbb {H} }$, and ${\displaystyle \mu }$ is a measure related to the quantum logic, ${\displaystyle LM}$, and (quantum) transition probabilities of this quantum system), and whose morphisms are defined between such triples by homomorphisms of Hilbert spaces, ${\displaystyle \emptyset :\mathbb {H} \rightarrow \mathbb {H} }$, naturally compatible with the operators ${\displaystyle \Delta }$, and by homomorphisms between the associated Haar measure systems.

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

A quantum algebraic topology definition of the Cod of quantum algebraic automata involves the objects specified above in Definition 0.1 as quantum automaton triples ${\displaystyle (Q_{A})}$, and quantum automata homomorphisms defined between such triples; these ${\displaystyle Q_{A}}$ morphisms are defined by groupoid homomorphisms ${\displaystyle h:{\mathcal {G}}\rightarrow {\mathcal {G}}^{*}}$ and ${\displaystyle \alpha :Aut({\mathcal {G}})\rightarrow Aut({\mathcal {G}}^{*})}$, together with unitarity preserving mappings ${\displaystyle u}$ between unitary representations of ${\displaystyle {\mathcal {G}}}$ on rigged Hilbert spaces (or Hilbert space bundles).