A class of quantum channels based on a infinite-dimensional Hilbert space is constructed using quantum Bernoulli noises. Some basic properties of channels are discussed from the point of quantum probability theory. Then, a quantum-mechanical compound state of the input state and the output state generated by the quantum channel is constructed. Finally, the mutual information in quantum communication theory is defined by the compound state, and its fundamental properties are discussed, which proves that the compound state of quantum communication theory is reasonable.
[1] Davies E B.Quantum stochastic processes[J].Communications in Mathematical Physics,1970,19(2):83-105.
[2] Bruneau L,Joye A,Merkli M.Repeated interactions in open quantum systems[J].Journal of Mathematical Physics,2014,55(7):1-67.
[3] Carbone R,Pautrat Y.Irreducible decompositions and stationary states of quantum channels[J].Reports on Mathematical Physics,2016,77(3):293-313.
[4] Nielsen M A,Chuang I L.Quantum computation and quantum information[M].Cambridge:Cambridge University Press,2000:353-394.
[5] Ohya M.Entropy transmission in-dynamical systems[J].Journal of Mathematical Analysis and Applications,1984,100(1):222-235.
[6] Accardi L.Computations of mutual entropy in quantum Markov chains[J].Open Systems and Information Dynamics,1994,2(3):337-354.
[7] Ohmura K,Watanabe N.Quantum dynamical mutual entropy based on AOW entropy[J].Open Systems & Information Dynamics,2019,26(2):1950009.
[8] 张淑仪,席政军.量子假设检验互信息[J].计算机研究与发展,2021,58(9):1906-1914.
[9] Wang C S,Chai H F,Lu Y C.Discrete-time quantum Bernoulli noises[J].Journal of Mathematical Physics,2010,51(5):053528.
[10] Wang C S,Zhang J H.Localization of quantum Bernoulli noises[J].Journal of Mathematical Physics,2013,54(10):103502.
[11] Han Q,Chen Z H,Lu Z Q.Quantum entropy in terms of local quantum Bernoulli noises and related properties[J].Communications in Statistics Theory and Methods,2022,51(12):4210-4220.
[12] Han Q,Han Y Y,Kou Y X,et al.Quantum mutual entropy in terms of local quantum Bernoulli noises[J].Communications in Statistics Theory and Methods,2023,52(3):515-522.
[13] Han Q,Wang S,Gou L G,et al.Quantum conditional entropy based on local quantum Bernoulli noises[J].Reviews in Mathematical Physics,2023,36(2):2350037.
[14] Wang C S,Lu Y C,Chai H F.An alternative approach to Privault’s discrete-time chaotic calculus[J].Journal of Mathematical Analysis and Applications,2011,373(2):643-654.
[15] Kossakowski A,Ohya M,Watanabe N. Quantum dynamical entropy for completely positive map[J].Infinite Dimensional Analysis,Quantum Probability and Related Topics,1999,2(2):267-282.
[16] Ohya M.On compound state and mutual information in quantum information theory[J].IEEE Transactions on Information Theory,1983,29(5):770-774.
