Contractive freedoms of ensembles and quantum channels for infinite-dimensional systems

In this paper, we reveal that the unitary freedom in ensemble of pure states with a given state and the unitary freedom in operator-sum representation for a given channel are no longer valid for infinite-dimensional systems. The replacements of them are, respectively, the contractive freedom in ensemble and the contractive freedom in operator-sum representation. Precisely, we show that, (1) two ensembles \(\{|\phi _i\rangle , p_i\}\) and \(\{|\psi _j\rangle , q_j\}\) determine the same state if and only if there exists some contractive matrix \(V=(v_{ij})\) such that \(\sqrt{p_i}|\phi _i\rangle =\sum _j v_{ij}\sqrt{q_j}|\psi _j\rangle \) for each i; (2) two sequences \(\{A_i\}\) and \(\{B_j\}\) of Kraus operators in operator-sum representations determine the same quantum channel if and only if there exists some contractive matrix \(V=(v_{ij})\) such that \(A_i=\sum _jv_{ij}B_j\) for each i. All possible quantum ensembles of pure states with any given state and all possible operator-sum representations of any given channel are described.