 Solution

A unital quantum channel preserves the identity operator: $\Phi (I)=I$. For equal input and output dimensions, this is equivalent to preserving the maximally mixed density operator.

For the amplitude damping channel,Thus the channel is not unital for $p>0$; it is unital only at $p=0$, when it is the identity channel. In particular, it biases the maximally mixed state toward the atomic ground state. The trace-preservation condition involves ${\sum }_j{A}_j^{†}{A}_j$, whereas unitality involves the opposite products ${\sum }_j{A}_j{A}_j^{†}$.