A satisfactory theory of continuous measurements has to be developed according to the axioms of quantum mechanics, that is by introducing, more or less explicitly, the associated instruments (Sect. B.4). This approach requires the statistical formulation of quantum mechanics (see Sect. B.3). This chapter generalises to this framework the theory developed in Chap. 2 and it extends the results to the case of incomplete measurements. Now, the key notions are “statistical operator”, “stochastic master equation”, “master equation” and “quantum dynamical (or Markov) semigroup”.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
A. V. Skorohod, Linear stochastic differential equations and stochastic semigroups, Uspekhi Mat. Nauk. 37 (1982) 157–183.
R. Alicki, M. Fannes, On dilating quantum dynamical semigroups with classical Brownian motion, Lett. Math. Phys. 11 (1986) 259–262.
R. Alicki, M. Fannes, Dilations of quantum dynamical semigroups with classical Brownian motion, Commun. Math. Phys. 1108 (1987) 353–361.
A. S. Holevo, Stochastic representations of quantum dynamical semigroups, Proc. Steklov Inst. Math. 191 in Russian; English translation: Issue 2 (1992) 145–154.
A. S. Holevo, On dissipative stochastic equations in a Hilbert space, Probab. Theory Relat. Fields 104 (1996) 483–500.
H.-P. Breuer, F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, 2002).
P. Baldi, Equazioni differenziali stocastiche e applicazioni, Quaderno UMI 28 (Pitagora, Bologna, 2000).
A. Barchielli, V. P. Belavkin, Measurements continuous in time and a posteriori states in quantum mechanics, J. Phys. A: Math. Gen. 24 (1991) 1495–1514; arXiv:quant-ph/0512189.
A. Barchielli, On the quantum theory of measurements continuous in time, Rep. Math. Phys. 33 (1993) 21–34.
A. Barchielli, Stochastic differential equations and ‘a posteriori’ states in quantum mechanics. Int. J. Theor. Phys. 32 (1993) 2221–2232.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Barchielli, A., Gregoratti, M. (2009). The Stochastic Master Equation: Part I. In: Quantum Trajectories and Measurements in Continuous Time. Lecture Notes in Physics, vol 782. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01298-3_3
Download citation
DOI: https://doi.org/10.1007/978-3-642-01298-3_3
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-01297-6
Online ISBN: 978-3-642-01298-3
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)