A Noether theorem for stochastic operators on Schatten classes

Bartoszek, Krzysztof; Bartoszek, Wojciech

doi:10.1016/j.jmaa.2017.03.068

Cited by 4 publications

(25 citation statements)

References 9 publications

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“…In this article, we show that all the results in [7,21], and [8] can be unified by means of an abstract approach within dilation theory in C * -algebras for completely positive maps in the sense of Stinespring [34], more precisely, through the concepts of bimodule domains and multiplication domains of Choi [11]. For example, we show that the abstract results on propagation of fixed points for completely positive maps on C * -algebras that we get in Theorem 2.2 and Corollary 2.3 short cut completely the probabilistic tools in the proofs of the main results in [7] and [8]. Also, although the results in [8] apparently refer to a more general case of positive maps that may not be completely positive, our Corollary 2.3 shows that it is exactly the complete positivity that lies behind them.…”

Section: Introductionmentioning

confidence: 61%

“…For example, we show that the abstract results on propagation of fixed points for completely positive maps on C * -algebras that we get in Theorem 2.2 and Corollary 2.3 short cut completely the probabilistic tools in the proofs of the main results in [7] and [8]. Also, although the results in [8] apparently refer to a more general case of positive maps that may not be completely positive, our Corollary 2.3 shows that it is exactly the complete positivity that lies behind them. In addition, in the case studied in [21], we reveal what happens if the technical assumption of existence of a stationary strictly positive density operator is removed.…”

Section: Introductionmentioning

confidence: 84%

“…The equivalence of (ii)-(v) in the following theorem has been obtained in [8], compared with Theorem 3.2. Here, we show that these equivalences can be obtained as a direct consequence of Corollary 2.3.…”

Section: Dynamical Systems Of Stochastic/markov Maps: the Noncommutatmentioning

confidence: 97%

“…In this setting, they obtain several equivalent characterisations to the compatibility (commutation) of the dynamical stochastic system {S t } t≥0 with the quantum measurement M A 1/2 : for example, one of these equivalent characterisations refers to A and A 2 being fixed by the dual semigroup {S t } t≥0 and a second one refers to the commutation of the infinitesimal generator s of {S t } t≥0 with M A 1/2 . The approach used in [8] combines the probability theory methods as in [7] with operator theoretical methods. There are some important questions left unanswered in [8]: for example, how are these related to the results in [7] and [21] and to what extent is the additional condition that A 2 be fixed by the dual semigroup {S t } t≥0 really necessary?…”

Section: Introductionmentioning

confidence: 99%

“…The approach used in [8] combines the probability theory methods as in [7] with operator theoretical methods. There are some important questions left unanswered in [8]: for example, how are these related to the results in [7] and [21] and to what extent is the additional condition that A 2 be fixed by the dual semigroup {S t } t≥0 really necessary? It is another aim of our article to provide an answer to these questions.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Symmetries Versus Conservation Laws in Dynamical Quantum Systems: A Unifying Approach Through Propagation of Fixed Points

Gheondea

2018

Ann. Henri Poincaré

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We unify recent Noether-type theorems on the equivalence of symmetries with conservation laws for dynamical systems of Markov processes, of quantum operations, and of quantum stochastic maps, by means of some abstract results on propagation of fixed points for completely positive maps on C *-algebras. We extend most of the existing results with characterisations in terms of dual infinitesimal generators of the corresponding strongly continuous one-parameter semigroups. By means of an ergodic theorem for dynamical systems of completely positive maps on von Neumann algebras, we show the consistency of the condition on the standard deviation for dynamical systems of quantum operations, and hence of quantum stochastic maps as well, in case the underlying Hilbert space is infinite dimensional.

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Section: Introductionmentioning

confidence: 61%

Section: Introductionmentioning

confidence: 84%

Section: Dynamical Systems Of Stochastic/markov Maps: the Noncommutatmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Symmetries Versus Conservation Laws in Dynamical Quantum Systems: A Unifying Approach Through Propagation of Fixed Points

Gheondea

2018

Ann. Henri Poincaré

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Generalized Dobrushin Coefficients on Banach Spaces

Bartoszek

Beśka²,

Florek

2021

Bull. Iran. Math. Soc.

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The asymptotic behavior of iterates of bounded linear operators (not necessarily positive), acting on Banach spaces, is studied. Through the Dobrushin ergodicity coefficient, we generalize some ergodic theorems obtained earlier for classical Markov semigroups acting on $$L^1$$ L 1 (or positive operators on abstract state spaces).

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Study on the Core Groups of First Integrals and Folding Index for Mechanical Systems

Zhang

Chen

2019

Journal of Applied Mechanics

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In applying the Udwadia–Kalaba equation for constrained mechanical systems, a direct proof of the equivalence of first integrals and nonholonomic constraints is given, and it is demonstrated that the generalized force of the system is equivalent to the constraint force derived by all first integrals of the nonholonomic constraints. Furthermore, depending on whether complete information is included in the subsets of the first integrals or not, the concept of “multiple kernel” of the system is introduced, and then the core groups of the first integrals and the folding index, which reveals the “simplicity” of the system, are defined. Finally, the onefold system is discussed in detail, and the judgment method is given. To verify the feasibility of this method and illustrate the application of the multiple kernel theory, three examples are considered. The new concepts and results presented in this paper help reveal the inner structure of the general mechanical system, which forms the foundation of control theory of constraint motions, and the multiple kernel analysis of the complex systems can be a new research area of analytic mechanics in the future.

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A Noether theorem for stochastic operators on Schatten classes

Cited by 4 publications

References 9 publications

Symmetries Versus Conservation Laws in Dynamical Quantum Systems: A Unifying Approach Through Propagation of Fixed Points

Symmetries Versus Conservation Laws in Dynamical Quantum Systems: A Unifying Approach Through Propagation of Fixed Points

Generalized Dobrushin Coefficients on Banach Spaces

Study on the Core Groups of First Integrals and Folding Index for Mechanical Systems

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