On reliability function of quantum communication channel

Let H be a Hilbert space. We consider quantum channel [4] with a finite input alphabet {1, ..., a} and with pure signal states Si = |ψi >< ψi|; i = 1, ..., a. Compound channel of length n is in the n–tensor product of the space H , i.e. Hn = H ⊗ . . . ⊗ H . An input block (codeword) u = (i1, . . . , in) , ik ∈ {1, . . . a}, for it means using the compound state ψu = ψi1 ⊗ . . . ⊗ ψin ∈ Hn and the corresponding density operator Su = |ψu >< ψu| in the space Hn . A code (C,X) of cardinality M in Hn is a collection of M pairs (u, X1), . . . , (u , XM), where X = {X1, . . . , XM , XM+1} is a quantum decision rule, i.e. some resolution of identity in Hn [4]. The conditional probability P (ui|uk) to make a decision in favor of message u provided that message u was transmitted is given by P (u|u) = Tr SukXi =< ψuk |Xiψuk > .