category of quantum automata
With the data from above definition we can now define also the category of quantum automata as follows.
Definition 0.2
The
category of quantum automata

is defined as an
algebraic category
whose objects are triples

(where

is either a Hilbert space or a rigged Hilbert space of quantum states and
operators
acting on

, and

is a measure related to the
quantum logic,

, and (quantum) transition probabilities of this quantum
system), and whose
morphisms
are defined between such triples by
homomorphisms
of Hilbert spaces,

, naturally compatible with the operators

, and by homomorphisms between the associated
Haar measure
systems.
An alternative definition is also possible based on Quantum Algebraic Topology.
Definition 0.3
A
quantum algebraic topology definition of the
category
of quantum algebraic
automata involves the objects specified above in
Definition 0.1 as quantum automaton triples

,
and
quantum automata
homomorphisms defined between such triples; these

morphisms are defined by
groupoid homomorphisms

and

, together
with unitarity preserving mappings
between unitary representations of

on rigged Hilbert spaces
(or
Hilbert space bundles).
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As of this snapshot date, this entry was owned by bci1.