Editing Quantum operation (section)

==Dynamics==
For a non-relativistic quantum mechanical system, its [[time evolution]] is described by a [[one-parameter group]] of automorphisms {α<sub>''t''</sub>}<sub>''t''</sub> of ''Q''.  This can be narrowed to unitary transformations: under certain weak technical conditions (see the article on [[quantum logic]] and the Varadarajan reference), there is a strongly continuous one-parameter group {''U''<sub>''t''</sub>}<sub>''t''</sub> of unitary transformations of the underlying Hilbert space such that the elements ''E'' of ''Q'' evolve according to the formula
:<math> \alpha_t(E) = U^*_t E U_t. </math>
The system time evolution can also be regarded dually as time evolution of the statistical state space.  The evolution of the statistical state is given by a family of operators {β<sub>''t''</sub>}<sub>''t''</sub> 
such that
<math display="block"> \operatorname{Tr}(\beta_t(S) E) = \operatorname{Tr}(S \alpha_{-t}(E)) =  \operatorname{Tr}(S U _t E U^*_t ) = \operatorname{Tr}( U^*_t S U _t E ).</math>

Clearly, for each value of ''t'', ''S'' → ''U''*<sub>''t''</sub> ''S'' ''U''<sub>''t''</sub> is a quantum operation.  Moreover, this operation is ''reversible''.

This can be easily generalized: If ''G'' is a connected [[Lie group]] of symmetries of ''Q''  satisfying the same weak continuity conditions, then the [[Group action (mathematics)|action]] of any element ''g'' of ''G'' is given by a unitary operator ''U'':
<math display="block"> g \cdot E = U_g E U_g^*. </math>
This mapping ''g'' → ''U''<sub>''g''</sub> is known as a [[projective representation]] of ''G''.  The mappings ''S'' → ''U''*<sub>''g''</sub> ''S'' ''U''<sub>''g''</sub> are reversible quantum operations.