Past exam of the introduction to quantum information course of the University of Oxford / 2017 / 3 / b / Solution

For any column vector $v$ and positive semidefinite input $\varrho$, the Kraus representation givesThus the output is positive semidefinite. The cyclic property of the trace and completeness of the Kraus operators giveMore strongly, tensoring with an arbitrary ancilla replaces $A_k$ by $A_k\otimes I$, so the same argument proves complete positivity rather than only positivity on single-system density matrices.

If $\mathcal{F}$ has Kraus operators $B_j$, the composition of quantum channels hasThis is again a trace-preserving Kraus representation. The composition of completely positive maps is completely positive even without the trace-preserving assumption; in the general case, the product Kraus operators still establish complete positivity.