We already saw that there exist peculiar cases (Sects. 2.4.4, 2.5.2.1) in which no information on the quantum system is extracted by the continuous measurement. Obviously, in other cases we get some information on the system, but the question arises of how to quantify the gain in information. The answer coming out from the whole development of classical and quantum information theory is that this can be obtained by means of entropy-like quantities [1–3].
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
T. M. Cover, J. A. Thomas, Elements of Information Theory (Wiley, New York, 1991).
M. Ohya, D. Petz, Quantum Entropy and Its Use (Springer, Berlin, 1993).
M. A. Nielsen, I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000).
A. Barchielli, G. Lupieri, Instruments and channels in quantum information theory, Optika i Spektroskopiya 99 (2005) 443–450; Optics and Spectroscopy 99 (2005) 425–432; arXiv:quant-ph/0409019v1.
A. Barchielli, G. Lupieri, Instruments and mutual entropies in quantum information. In M. Bożejko, W. Młotkowski, J. Wysoczánski (eds.), Quantum Probability (Banach Center Publications, Warszawa, 2006), pp. 65–80; arXiv:quant-ph/0412116v1.
A. Barchielli, G. Lupieri, Quantum measurements and entropic bounds on information transmission, Quantum Inf. Comput. 6 (2006) 16–45; arXiv:quantph/0505090v1.
A. Barchielli, Entropy and information gain in quantum continual measurements. In P. Tombesi, O. Hirota (eds.), Quantum Communication, Computing, and Measurement 3 (Kluwer, New York, 2001) pp. 49–57; quant-ph/0012115.
A. Barchielli, G. Lupieri, Instrumental processes, entropies, information in quantum continual measurements. In O. Hirota (ed.), Quantum Information, Statistics, Probability (Rinton, Princeton, 2004) pp. 30–43; Quantum Inform. Compu. 4 (2004) 437–449; quant-ph/0401114.
A. Barchielli, G. Lupieri, Entropic bounds and continual measurements. In L. Accardi, W. Freudenberg, M. Schürmann (eds.), Quantum Probability and Infinite Dimensional Analysis, Quantum Probability Series QP-PQ Vol. 20 (World Scientific, Singapore, 2007) pp. 79–89; arXiv:quant-ph/0511090v1.
A. Barchielli, G. Lupieri, Information gain in quantum continual measurements In V. P. Belavkin and M. Guţă, Quantum Stochastic and Information (World Scientific, Singapore, 2008) pp. 325–345; arXiv:quant-ph/0612010v1.
A. Barchielli, Convolution semigroups in quantum probability, Semesterbericht Funktionalanalysis Tübingen, Band 12, Sommersemester (1987) pp. 29–42.
A. S. Holevo, A noncommutative generalization of conditionally positive definite functions. In L. Accardi, W. von Waldenfels (eds.), Quantum Probability and Applications III, Lecture Notes in Mathematics 1303 (Springer, Berlin, 1988) pp. 128–148.
A. Barchielli, Probability operators and convolution semigroups of instruments in quantum probability, Probab. Theory Rel. Fields 82 (1989) 1–8.
A. S. Holevo, Limit theorems for repeated measurements and continuous measurement processes. In L. Accardi, W. von Waldenfels (eds.), Quantum Probability and Applications IV, Lecture Notes in Mathematics 1396 (Springer, Berlin, 1989) pp. 229–257.
A. Barchielli, A. M. Paganoni, A note on a formula of Lévy-Khinchin type in quantum probability, Nagoya Math. J. 141 (1996) 29–43.
J. Gambetta, H. M. Wiseman, State and dynamical parameter estimation for open quantum systems, Phys. Rev. A 64 (2001) 042105.
A. Ferraro, S. Oliveres, M. G. A. Paris, Gaussian States in Quantum Information, (Bibliopolis, Napoli, 2005) arXiv:quant-ph/0503237v1.
H. P. Yuen and M. Ozawa, Ultimate information carrying limit of quantum systems, Phys. Rev. Lett. 70 (1993) 363–366.
A. S. Holevo, M. E. Shirokov, Continuous ensembles and the capacity of infinite dimensional quantum channels, Teor. Veroyatn. Primen. 50 (2005) 98–114; translation in Theory Probab. Appl. 50 (2006) 86–98; quant-ph/0408176.
A. S. Holevo, Some estimates for the amount of information transmittable by a quantum communication channel, Probl. Inform. Transm. 9 (1973) 177–183 (Engl. transl.: 1975).
B. Schumacher, M. Westmoreland, and W. K. Wootters, Limitation on the amount of accessible information in a quantum channel, Phys. Rev. Lett. 76 (1996) 3452–3455.
K. Jacobs, A bound on the mutual information for quantum channels with inefficient measurements, J. Math. Phys. 47 (2006) 012102; quant-ph/0412006.
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). Mutual Entropies and Information Gain in Quantum Continuous Measurements. 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_6
Download citation
DOI: https://doi.org/10.1007/978-3-642-01298-3_6
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)