Channel Diamond Norm One

By [L1], it suffices to use the ancilla $R={\mathbb{C}}^d$, where $d=\dim H$. Let $\mathrm{\Lambda }=\mathrm{\Phi }\otimes {\mathrm{id}}_R$. By [L4], its Kraus operators are $K_j\otimes I_R$, and [L6] ensures this factorwise Kraus formula agrees with the amplification on every operator. Its trace-preserving normalization follows from [L2]. Define ${\mathrm{\Lambda }}^{\ast }(B)={\sum }_j(K_j^{\ast }\otimes I_R)B(K_j\otimes I_R)$. It is completely positive by [L3] and unital because ${\sum }_j(K_j^{\ast }\otimes I_R)(K_j\otimes I_R)=I$. If $\mathrm{\parallel }B{\mathrm{\parallel }}_{\mathrm{\infty }}\le 1$, then $\left(\begin{array}{cc}I& B\\ B^{\ast }& I\end{array}\right)$ is nonnegative: for any $(x,y)$ its quadratic form is at least $\mathrm{\parallel }x{\mathrm{\parallel }}^2+\mathrm{\parallel }y{\mathrm{\parallel }}^2-2\mathrm{\parallel }x\mathrm{\parallel }\mathrm{\parallel }y\mathrm{\parallel }\ge 0$ by [F2]–[F3]. Complete positivity of ${\mathrm{\Lambda }}^{\ast }$ applied with the two-dimensional auxiliary space preserves this block positivity. For $T={\mathrm{\Lambda }}^{\ast }(B)$, test the resulting block operator on $(-Ty,y)$; its quadratic form is $\mathrm{\parallel }y{\mathrm{\parallel }}^2-\mathrm{\parallel }Ty{\mathrm{\parallel }}^2$, so $\mathrm{\parallel }{\mathrm{\Lambda }}^{\ast }(B){\mathrm{\parallel }}_{\mathrm{\infty }}\le 1$.

For any input operator $X$, the Kraus formula for ${\mathrm{\Lambda }}^{\ast }$ and cyclicity [F2] give $\mathrm{tr}(B^{\ast }\mathrm{\Lambda }(X))=\mathrm{tr}({\mathrm{\Lambda }}^{\ast }(B{)}^{\ast }X)$. Trace-norm duality [L3] then gives $\mathrm{\parallel }\mathrm{\Lambda }(X){\mathrm{\parallel }}_1={\max }_{\mathrm{\parallel }B{\mathrm{\parallel }}_{\mathrm{\infty }}\le 1}\mathrm{\mid }\mathrm{tr}(B^{\ast }\mathrm{\Lambda }(X))\mathrm{\mid }={\max }_{\mathrm{\parallel }B{\mathrm{\parallel }}_{\mathrm{\infty }}\le 1}\mathrm{\mid }\mathrm{tr}({\mathrm{\Lambda }}^{\ast }(B{)}^{\ast }X)\mathrm{\mid }\le \mathrm{\parallel }X{\mathrm{\parallel }}_1,$ using step 1.1. Thus the induced norm at ancilla dimension $d$ is at most one, and [L1] gives $\mathrm{\parallel }\mathrm{\Phi }{\mathrm{\parallel }}_{\diamond }\le 1$. Conversely choose unit vectors $x\in H$ and $e\in {\mathbb{C}}^d$. The rank-one projector $P=\mathrm{\mid }x\otimes e⟩⟨x\otimes e\mathrm{\mid }$ is a density operator by [L5]. The amplified map $\mathrm{\Lambda }$ is completely positive by [L3] and trace-preserving by [L2] and [L4], so $\mathrm{\Lambda }(P)$ is a density operator and has trace norm one by [F1]. Therefore the induced norm at $d$ is at least one, and [L1] gives $\mathrm{\parallel }\mathrm{\Phi }{\mathrm{\parallel }}_{\diamond }\ge 1$.