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Some properties of generalized quantum operations. (English) Zbl 1270.81045
Summary: Let \(\mathcal{B}(\mathcal{H})\) be the set of all bounded linear operators on the separable Hilbert space \(\mathcal{H}\). A (generalized) quantum operation is a bounded linear operator defined on \(\mathcal{B}(\mathcal{H})\), which has the form \(\varPhi_{\mathcal{A}}(X)=\sum_{i=1}^{\infty}A_{i}XA_{i}^{*}\), where \(A_{i}\in\mathcal{B}(\mathcal{H})\) (\(i=1,2,\dots\)) satisfy \(\sum_{i=1}^{\infty}A_{i}A_{i}^{*}\leq I\) in the strong operator topology. In this paper, we establish the relationship between the (generalized) quantum operation \(\varPhi_{\mathcal{A}}\) and its dual \(\varPhi_{\mathcal {A}}^{†}\) with respect to the set of fixed points and the noiseless subspace. In particular, we also partially characterize the extreme points of the set of all (generalized) quantum operations and give some equivalent conditions for the correctable quantum channel.

81P45 Quantum information, communication, networks (quantum-theoretic aspects)
81R15 Operator algebra methods applied to problems in quantum theory
Full Text: DOI
[1] Arias, A.; Gheondea, A.; Gudder, S., Fixed points of quantum operations, J. Math. Phys., 43, 5872-5881, (2002) · Zbl 1060.81009
[2] Busch, P.; Singh, J., Lüders theorem for unsharp quantum measurements, Phys. Lett. A, 249, 10-12, (1998)
[3] Choi, M.D.; Kribs, D.W., Method to find quantum noiseless subsystems, Phys. Rev. Lett., 96, (2006)
[4] Bény, C.; Kempf, A.; Kribs, D.W., Quantum error correction on infinite-dimensional Hilbert spaces, J. Math. Phys., 50, (2009) · Zbl 1216.81055
[5] Du, H.K.; Wang, Y.Q.; Xu, J.L., Applications of the generalized Lüders theorem, J. Math. Phys., 49, (2008) · Zbl 1153.81352
[6] Nagy, G., On spectra of Lüders operations, J. Math. Phys., 49, (2008) · Zbl 1153.81410
[7] Kribs, D.W.; Laflamme, R.; Poulin, D., Unified and generalized approach to quantum error correction, Phys. Rev. Lett., 94, (2005) · Zbl 1152.81760
[8] Kribs, D.W., Quantum channels, wavelets, dilations and representations of \({\mathcal{O}}_{n}\), Proc. Edinb. Math. Soc., 46, 421-433, (2003) · Zbl 1051.46046
[9] Knill, E.; Laflamme, R., Theory of quantum error-correcting codes, Phys. Rev. A, 55, 900, (1997) · Zbl 1032.68080
[10] Knill, E.; Laflamme, R.; Viola, L., Theory of quantum error correction for general noise, Phys. Rev. Lett., 84, 2525, (2000) · Zbl 0956.81008
[11] Lim, B.J., Noncommutative Poisson boundaries of unital quantum operations, J. Math. Phys., 51, (2010) · Zbl 1310.81019
[12] Liu, W.H.; Wu, J.D., Fixed points of commutative Lüders operations, J. Phys. A, Math. Theor., 43, (2010) · Zbl 1201.81008
[13] Li, Y., Characterizations fixed points of quantum operations, J. Math. Phys., 52, (2011) · Zbl 1317.81014
[14] Li, Y., Fixed points of dual quantum operations, J. Math. Anal. Appl., 382, 172-179, (2011) · Zbl 1230.81006
[15] Liu, W.; Wu, J., On fixed points of Lüders operation, J. Math. Phys., 50, (2009) · Zbl 1283.47056
[16] Long, L.; Zhang, S., Fixed points of commutative super-operators, J. Phys. A, Math. Theor., 44, (2011) · Zbl 1211.81082
[17] Magajna, B., Fixed points of normal completely positive maps on \(B\)(\(H\)), J. Math. Anal. Appl., 389, 1291-1302, (2012) · Zbl 1266.47059
[18] Prunaru, B., Fixed points for Lüders operations and commutators, J. Phys. A, Math. Theor., 44, (2011) · Zbl 1215.81010
[19] Shen, J.; Wu, J.D., Generalized quantum operations and almost sharp quantum effects, Rep. Math. Phys., 66, 367-374, (2010) · Zbl 1381.81017
[20] Wang, Y.Q.; Du, H.K.; Dou, Y.N., Note on generalized quantum gates and quantum operations, Int. J. Theor. Phys., 47, 2268, (2008) · Zbl 1160.81350
[21] Zhang, H.Y.; Ji, G.X., Normality and fixed points associated to commutative row contractions, J. Math. Anal. Appl., 400, 247-253, (2013) · Zbl 1296.47020
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