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%%% Primary Title: category of quantum automata
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%%% Owner: bci1
%%% Author(s): bci1
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\begin{document}
\begin{definition}
Let us recall that as a \htmladdnormallink{Quantum Algebraic Topology}{http://planetphysics.us/encyclopedia/TriangulationMethodsForQuantizedSpacetimes2.html} \htmladdnormallink{object}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}, a \emph{\htmladdnormallink{quantum automaton}{http://planetphysics.us/encyclopedia/LQG2.html}} is defined by the \emph{quantum
triple} $Q_A =(\grp,\H -\Re_G, Aut(\grp)$), where $\grp$ is a \emph{(locally compact) \htmladdnormallink{quantum groupoid}{http://planetphysics.us/encyclopedia/WeakHopfAlgebra.html}},
$\H -\Re_G$ are the unitary \htmladdnormallink{representations}{http://planetphysics.us/encyclopedia/CategoricalGroupRepresentation.html} of $\grp$ on \htmladdnormallink{rigged Hilbert spaces}{http://planetphysics.us/encyclopedia/I3.html} $\Re_G$ of quantum states and \htmladdnormallink{quantum operators}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra5.html} on the \htmladdnormallink{Hilbert space}{http://planetphysics.us/encyclopedia/NormInducedByInnerProduct.html} $\H$, and $Aut(\grp)$ is the transformation, or
\emph{automorphism \htmladdnormallink{groupoid}{http://planetphysics.us/encyclopedia/EquivalenceRelation.html} of quantum transitions} that represents all flip-flop quantum transitions of one cubit each between the permitted quantum states of the quantum automaton.
\end{definition}
With the data from above definition we can now define also the category of quantum automata as follows.
\begin{definition} The \emph{category of quantum automata} $\mathcal{\Q}_A$
is defined as an {\em \htmladdnormallink{algebraic category}{http://planetphysics.us/encyclopedia/CategoryOfLogicAlgebras.html} whose objects are triples} $(\H, \Delta: \H \rightarrow \H, \mu)$ (where $\H$ is either a Hilbert space or a rigged Hilbert space of quantum states and \htmladdnormallink{operators}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra4.html} acting on
$\H$, and $\mu$ is a measure related to the \htmladdnormallink{quantum logic}{http://planetphysics.us/encyclopedia/LQG2.html}, $LM$, and (quantum) transition probabilities of this quantum \htmladdnormallink{system}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html}), and whose \htmladdnormallink{morphisms}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} are defined between such triples by \htmladdnormallink{homomorphisms}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} of Hilbert spaces,
$\O: \H \rightarrow \H$, naturally compatible with the operators $\Delta$, and by homomorphisms between the associated \htmladdnormallink{Haar measure}{http://planetphysics.us/encyclopedia/HigherDimensionalQuantumAlgebroid.html} systems.
\end{definition}
An alternative definition is also possible based on {\em Quantum Algebraic Topology}.
\begin{definition} A \emph{quantum algebraic topology} definition of the {\em \htmladdnormallink{category}{http://planetphysics.us/encyclopedia/Cod.html} of quantum \htmladdnormallink{algebraic}{http://planetphysics.us/encyclopedia/CoIntersections.html} automata} involves the objects specified above in \textbf{Definition 0.1} as quantum automaton triples $(Q_A)$,
and \htmladdnormallink{quantum automata}{http://planetphysics.us/encyclopedia/QuantumComputers.html} homomorphisms defined between such triples; these $Q_A$ morphisms are defined by
\htmladdnormallink{groupoid homomorphisms}{http://planetphysics.us/encyclopedia/EquivalenceRelation.html} $h: \grp \rightarrow \grp ^*$ and $\alpha: Aut(\grp) \rightarrow Aut(\grp ^*)$, together
with unitarity preserving mappings \emph{$u$} between unitary representations of $\grp$ on rigged Hilbert spaces
(or \htmladdnormallink{Hilbert space bundles}{http://planetphysics.us/encyclopedia/HilbertBundle.html}).
\end{definition}
\end{document}
