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Abstract
Noether’s theorem on constants of the motion of dynamical systems has recently been extended to classical dissipative systems (Markovian semigroups) by Baez and Fong [J. Math. Phys. 54, 013301 (2013)]. We show how to extend these results to the fully quantum setting of quantum Markov dynamics. For finitedimensional Hilbert spaces, we construct a mapping from observables to completely positive maps that leads to the natural analogue of their criterion of commutativity with the infinitesimal generator of the Markov dynamics. Using standard results on the relaxation of states to equilibrium under quantum dynamical semigroups, we are able to characterise the constants of the motion under quantum Markov evolutions in the infinitedimensional setting under the usual assumption of existence of a stationary strictly positive density matrix. In particular, the Noether constants are identified with the fixed point of the Heisenberg picture semigroup.
Original language  English 

Article number  022108 
Journal  Journal of Mathematical Physics 
Volume  56 
Early online date  18 Feb 2015 
DOIs  
Publication status  Published  2015 
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Projects
 1 Finished

Quantum Stochastic Analysis for Nanophotonic Circuit Design
Gough, J. (PI)
Engineering and Physical Sciences Research Council
01 Aug 2013 → 31 Mar 2015
Project: Externally funded research