×

zbMATH — the first resource for mathematics

Noncommutative Poisson boundaries of unital quantum operations. (English) Zbl 1310.81019
Summary: In this paper, Poisson boundaries of unital quantum operations (also called Markov operators) are investigated. In the case of unital quantum channels, compact operators belonging to Poisson boundaries are characterized. Using the characterization of amenable groups by the injectivity of their von Neumann algebras, we will answer negatively some conjectures appearing in the work of A. Arias et al. [ibid. 43, No. 12, 5872–5881 (2002; Zbl 1060.81009)] about injectivity of the commuting algebra of the Kraus operators of unital quantum operations and their injective envelopes.
©2010 American Institute of Physics

MSC:
81P15 Quantum measurement theory, state operations, state preparations
81S25 Quantum stochastic calculus
46L07 Operator spaces and completely bounded maps
47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
47H10 Fixed-point theorems
46L10 General theory of von Neumann algebras
46L60 Applications of selfadjoint operator algebras to physics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Arias, A.; Gheonda, A.; Gudder, S., Fixed points of quantum operations, J. Math. Phys., 43, 5872, (2002) · Zbl 1060.81009
[2] Babillot, M., An introduction to Poisson boundaries of Lie groups, Probability Measures on Groups: Recent Directions and Trends, 1-90, (2006), Tata Inst. Fund. Res: Tata Inst. Fund. Res, Mumbai · Zbl 1181.60011
[3] Bény, C.; Kempf, A.; Kribs, D. W., Quantum error correction on infinite-dimensional Hilbert spaces, J. Math. Phys., 50, 062108, (2009) · Zbl 1216.81055
[4] Bratteli, O.; Jorgensen, P.; Kishimoto, A.; Werner, R. F., Pure states on \(O_d\), J. Oper. Theory, 43, 97, (2000) · Zbl 0992.46044
[5] Choi, M. D., Completely positive linear maps on complex matrix, Linear Algebra Appl., 10, 285, (1975) · Zbl 0327.15018
[6] Conway, J. B., A Course in Functional Analysis, (1990), Springer: Springer, Berlin · Zbl 0706.46003
[7] Evans, D. E.; Lewis, J. T., Dilations of Irreversible Evolutions in Algebraic Quantum Theory, (1977), Dublin Institute for Advanced Studies: Dublin Institute for Advanced Studies, Dublin
[8] Izumi, M., Non commutative Poisson boundaries, (2002) · Zbl 1037.46056
[9] Kaimanovich, V. A., Boundaries of Invariant Markov Operators: The identification Problem, 228, 127-176, (1996), Cambridge University Press: Cambridge University Press, Cambridge · Zbl 0848.60073
[10] Kaimanovich, V. A.; Edinburgh, S.; Picardello, M. A., Measure-theoretic boundaries of Markov chains, 0-2 laws and entropy, 145-180, (1992), Plenum Publishing Corp.: Plenum Publishing Corp., New York
[11] Kaimanovich, V. A.; Vershik, A. M., Random walks on discrete groups: Boundary and entropy, Ann. Probab., 11, 457, (1983) · Zbl 0641.60009
[12] Kribs, D. W., Quantum channels, wavelets, dilations and representations of \(O_n\), Proc. Edinb. Math. Soc., 46, 421, (2003) · Zbl 1051.46046
[13] Kümmerer, B., Quantum Markov Processes and Applications in Physics, 1866, 259-328, (2005), Springer: Springer, Berlin
[14] Nielsen, M. A.; Chuang, I. L., Quantum Computing and Quantum Information, (2000), Cambridge University Press: Cambridge University Press, Cambridge · Zbl 1049.81015
[15] Takesaki, M., Theory of Operator Algebras. II, 125, (2003), Springer-Verlag: Springer-Verlag, Berlin · Zbl 1059.46031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.