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Linear transformations between multipartite quantum systems that map the set of tensor product of idempotent matrices into idempotent matrix set. (English) Zbl 06203142
Summary: Let $H_n$ be the set of $n \times n$ complex Hermitian matrices and $\cal P_n$ (resp., $\cal T_n$) be the set of all idempotent (resp., tripotent) matrices in $H_n$. In $l$-partite quantum system $H_{m_{1}\cdots m_{l}} = \otimes^l_1 H_{m_i}$, $\otimes^l_1 \cal P_{m_i}$ (resp., $\otimes^l_1 \cal T_{m_i}$) denotes the set of all decomposable elements $\otimes^l_1 A_i$ such that $A_i \in \cal P_{m_i}$ (resp., $A_i \in \cal T_{m_i}$). In this paper, linear maps $\phi$ from $H_{m_{1}\cdots m_{l}}$ to $H_n$ with $n \leq m_1 \cdots m_l$ such that $\phi(\otimes^l_1 \cal P_{m_i}) \in \cal P_n$ are characterized. As its application, the structure of linear maps $\phi$ from $H_{m_{1}\cdots m_{l}}$ to $H_n$ with $n \leq m_1 \cdots m_l$ such that $\phi(\otimes^l_1 \cal T_{m_i}) \in \cal T_n$ is also obtained.
46Functional analysis
Full Text: DOI
[1] A. Fo\vsner, Z. Haung, C. K. Li, and N. S. Sze, “Linear preservers and quantum information science,” Linear and Multilinear Algebra, 2012. · doi:10.1080/03081087.2012.740029
[2] K. He, J. C. Hou, and C. K. Li, “A geometric characterization of invertible quantum measurement maps,” Journal of Functional Analysis, vol. 264, no. 2, pp. 464-478, 2013. · Zbl 1283.47024 · doi:10.1016/j.jfa.2012.11.005
[3] S. Friedland, C. K. Li, Y. T. Poon, and N. S. Sze, “The automorphism group of separable states in quantum information theory,” Journal of Mathematical Physics, vol. 52, no. 4, Article ID 042203, 8 pages, 2011. · Zbl 1316.81014 · doi:10.1063/1.3578015
[4] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, UK, 2000. · Zbl 1049.81015
[5] M. H. Lim, “Additive preservers of tensor product of rank one Hermitian matrices,” Electronic Journal of Linear Algebra, vol. 23, pp. 356-374, 2012. · Zbl 1253.15042 · http://www.math.technion.ac.il/iic/ela/ela-articles/articles/vol23_pp356-374.pdf
[6] Z. H. Zhou and M. Zhu, “Extended Cesáro operators between generalized Besov spaces and Bloch type spaces in the unit ball,” Journal of Function Spaces and Applications, vol. 7, no. 3, pp. 209-223, 2009. · Zbl 1221.47065 · doi:10.1155/2009/548956 · http://www.jfsa.net/
[7] H. Yao and B. Zheng, “Zero triple product determined matrix algebras,” Journal of Applied Mathematics, vol. 2012, Article ID 925092, 18 pages, 2012. · Zbl 1239.16031 · doi:10.1155/2012/925092
[8] A. El-Sayed Ahmed and M. A. Bakhit, “Properties of Toeplitz operators on some holomorphic Banach function spaces,” Journal of Function Spaces and Applications, vol. 2012, Article ID 517689, 18 pages, 2012. · Zbl 1263.47031 · doi:10.1155/2012/517689
[9] Z. Liao, S. Hu, D. Sun, and W. Chen, “Enclosed Laplacian operator of nonlinear anisotropic diffusion to preserve singularities and delete isolated points in image smoothing,” Mathematical Problems in Engineering, vol. 2011, Article ID 749456, 15 pages, 2011. · Zbl 1213.94019 · doi:10.1155/2011/749456 · eudml:223879
[10] O. Agratini, “Linear operators that preserve some test functions,” International Journal of Mathematics and Mathematical Sciences, vol. 2006, Article ID 94136, 11 pages, 2006. · Zbl 1121.41018 · doi:10.1155/IJMMS/2006/94136 · eudml:53454
[11] D. Huo, B. Zheng, and H. Liu, “Characterizations of nonlinear Lie derivations of B(X),” Abstract and Applied Analysis, vol. 2013, Article ID 245452, 7 pages, 2013. · Zbl 1271.47029
[12] G. H. Chan, M. H. Lim, and K. K. Tan, “Linear preservers on matrices,” Linear Algebra and Its Applications, vol. 93, pp. 67-80, 1987. · Zbl 0619.15003 · doi:10.1016/S0024-3795(87)90312-0
[13] C. H. Sheng, C. G. Cao, and J. Hu, “Linear preservers on complex Hermitian matrices,” Journal of Natural Science of Heilongjiang University, vol. 25, pp. 358-362, 2008 (Chinese). · Zbl 1199.15085
[14] L. Molnár, “Transformations on the set of all n-dimensional subspaces of a Hilbert space preserving principal angles,” Communications in Mathematical Physics, vol. 217, no. 2, pp. 409-421, 2001. · Zbl 1026.81006 · doi:10.1007/PL00005551
[15] L. Molnár, “Orthogonality preserving transformations on indefinite inner product spaces: generalization of Uhlhorn’s version of Wigner’s theorem,” Journal of Functional Analysis, vol. 194, no. 2, pp. 248-262, 2002. · Zbl 1010.46023 · doi:10.1006/jfan.2002.3970
[16] P. \vSemrl, “Applying projective geometry to transformations on rank one idempotents,” Journal of Functional Analysis, vol. 210, no. 1, pp. 248-257, 2004. · Zbl 1062.47039 · doi:10.1016/j.jfa.2003.07.009
[17] P. \vSemrl, “Maps on idempotent matrices over division rings,” Journal of Algebra, vol. 298, no. 1, pp. 142-187, 2006. · Zbl 1155.15302 · doi:10.1016/j.jalgebra.2005.08.010
[18] N. Jacobson, Lectures in Abstract Algebra, vol. 2, Van Nostrand, Toronto, Canada, 1953. · Zbl 0053.21204