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Title: Noether’s theorem for dissipative quantum dynamical semi-groups

Noether’s theorem on constants of the motion of dynamical systems has recently been extended to classical dissipative systems (Markovian semi-groups) 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 finite-dimensional 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 semi-groups, we are able to characterise the constants of the motion under quantum Markov evolutions in the infinite-dimensional 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 semi-group.
Authors:
 [1] ;  [2] ;  [3]
  1. Aberystwyth University, Aberystwyth SY23 3BZ, Wales (United Kingdom)
  2. Section de Mathématiques and Bernoulli Center, École Polytechnique Fédérale de Lausanne, Lausanne CH 1015 (Switzerland)
  3. Mechanics and Mathematics Faculty, Moscow State University, Moscow 119991 (Russian Federation)
Publication Date:
OSTI Identifier:
22403099
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Mathematical Physics; Journal Volume: 56; Journal Issue: 2; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; DENSITY MATRIX; HEISENBERG PICTURE; HILBERT SPACE; MAPPING; MAPS; MARKOV PROCESS; NATURAL ANALOGUE; RELAXATION