On Complexity for Open System Dynamics View Full Text


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Chapter Info

DATE

2018-06-14

AUTHORS

Noboru Watanabe

ABSTRACT

It is important to investigate the dynamics of state change and the complexity of states of systems. Ohya introduced a new concept (so-called information dynamics) synthesizing the research schemes of several complicated systems. There are two types of complexities in ID, (1) a complexity of state describing system itself and (2) a transmitted complexity between two systems. The entropy and the mutual entropy of classical and quantum systems are the examples of those complexities. Based on the relative entropy of Umegaki, the quantum mutual entropy was introduced by Ohya in 1983, and it was extended to general quantum systems by using the relative entropy of Araki and Uhlmann. One can discuss the classical coding theorems based on the mean entropy and the mean mutual entropy defined by the classical dynamical entropy. The mean entropy and the mean mutual entropy in quantum system were introduced by Ohya (Rep Math Phys 27:19–47, 1989) and Muraki and Ohya (Lett Math Phys 36:327–335, 1996). The dynamical entropy in quantum systems was studied by several authors (Accardi et al., Open Syst Inf Dyn 4:71–87, 1997; Accardi et al., Rep Math Phys 38:457–469, 1996; Alicki and Fannes, Lett Math Phys 32:75–82, 1994; Benatti, Deterministic Chaos in Infinite Quantum Systems. Springer, Berlin,1993; Choda, Ergod Theory Dyn Syst 16(6):1197–1206, 1996; Connes et al., Commun Math Phys 112:691–719, 1987; Connes and Størmer, Acta Math 134:289–306, 1975; Emch, Z Wahrscheinlichkeitstheory verw Gebiete 29:241, 1974; Hudetz, J Math Phys 35(8):4303–4333, 1994; Kossakowski et al., Infinite Dimens Anal Quantum Probab Relat Top 2(2):267–282, 1999; Muraki and Ohya, Lett Math Phys 36:327–335, 1996; Ohya, Rep Math Phys 27:19–47, 1989; Park, Lett Math Phys 32:63–74, 1994; Voiculescu, Commun Math Phys 170:249, 1995; Watanabe, Found Phys 41:549–563, 2011; Watanabe, Philos Trans R Soc A 374:20150240-1–20150240-13, 2016; Watanabe, Int J Quantum Inf 14(04):640005-1–1640005-11, 2016). The dynamical entropy (KOW entropy) by means of completely positive maps was defined in Kossakowski et al. (Infinite Dimens Anal Quantum Probab Relat Top 2(2):267–282, 1999), and the dynamical entropy (AOW entropy) through the quantum Markov process was introduced in Accardi et al. (Open Syst Inf Dyn 4:71–87, 1997). Based on the KOW entropy, the generalized AOW entropy was defined. These entropies represent the complexity of dynamical systems.In this chapter, we discuss about complexity of open system dynamics according to Watanabe (Philos Trans R Soc A 374:20150240-1–20150240-13, 2016; Int J Quantum Inf 14(04):640005-1–1640005-11, 2016). More... »

PAGES

229-254

Book

TITLE

Quantum Foundations, Probability and Information

ISBN

978-3-319-74970-9
978-3-319-74971-6

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/978-3-319-74971-6_17

DOI

http://dx.doi.org/10.1007/978-3-319-74971-6_17

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1104552080


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