Pure loss channel and binomial thinning

For $0\le \eta \le 1$, direct multiplication gives $C_{\eta }^{\ast }{C}_{\eta }=C_{\eta }{C}_{\eta }^{\ast }=I_2$, so the full unitary-matrix hypothesis of the beam-splitter theorem holds. The product occupation vectors $\mathrm{\mid }m⟩\otimes \mathrm{\mid }n⟩$ form a complete orthonormal family by the product-basis result of Hilbert tensor construction: inner product, completion and product bases, and the two-mode occupation vectors $\mathrm{\mid }m,n⟩$ form a complete orthonormal family by Finite-mode Fock to Schrödinger identification. Therefore the basis map defining $\mathcal{I}$ preserves inner products on a dense subspace and has dense range; it extends uniquely to a surjective isometry, hence a unitary. Also, the vacuum insertion $V_0\psi =\psi \otimes \mathrm{\mid }0⟩$ is an isometry by the elementary-tensor norm identity of Hilbert tensor construction: inner product, completion and product bases. For each finite-dimensional ancilla ${\mathbb{C}}^q$, $W_q:=I_q\otimes V_0$ is an isometry by [F6]. If $T\ge 0$ is trace class, then $W_{q}T{W}_q^{\ast }\ge 0$ because $⟨W_{q}T{W}_q^{\ast }x,x⟩=⟨T{W}_q^{\ast }x,W_q^{\ast }x⟩\ge 0$. For arbitrary trace-class $T$, write $T={\sum }_j{T}_{u_j,v_j}$ as a nuclear series by [F5]. Directly from the rank-one definition and the adjoint identity, $W_q{T}_{u_j,v_j}{W}_q^{\ast }x=⟨W_q^{\ast }x,v_j⟩W_q{u}_j=⟨x,W_q{v}_j⟩W_q{u}_j=T_{W_q{u}_j,W_q{v}_j}x$. The resulting rank-one series converges in trace norm since ${\sum }_j\mathrm{\parallel }{W}_q{u}_j\mathrm{\parallel }\mathrm{\parallel }{W}_q{v}_j\mathrm{\parallel }={\sum }_j\mathrm{\parallel }{u}_j\mathrm{\parallel }\mathrm{\parallel }{v}_j\mathrm{\parallel }<\mathrm{\infty }$, and its operator-norm sum is $W_{q}T{W}_q^{\ast }$. Since $W_q$ preserves inner products, [F5]'s rank-one trace formula gives $\mathrm{tr}(W_{q}T{W}_q^{\ast })={\sum }_j⟨W_q{u}_j,W_q{v}_j⟩={\sum }_j⟨u_j,v_j⟩=\mathrm{tr}T$. Thus adjoining vacuum is completely positive and trace-preserving.

For each finite ancilla ${\mathbb{C}}^q$, the amplification of the partial-trace map is, after the canonical unitary reassociation from [F6], the partial trace over the environment Fock space on ${\mathbb{C}}^q\widehat{\otimes }{\mathcal{F}}_s(\mathbb{C})\widehat{\otimes }{\mathcal{F}}_s(\mathbb{C})$. By [F2] it maps positive trace-class operators to positive trace-class operators and preserves their trace. Therefore the partial trace is completely positive and trace-preserving under [F3].

For $0<\eta <1$ and input $\mathrm{\mid }n⟩⟨n\mathrm{\mid }$, [F1] gives the joint pure state ${\sum }_{k=0}^n{c}_k\mathrm{\mid }k⟩\otimes \mathrm{\mid }n-k⟩$ with $c_k=\sqrt{\left(\genfrac{}{}{0px}{}{n}{k}\right)}{\eta }^{k\mathrm{/}2}(1-\eta {)}^{(n-k)\mathrm{/}2}$. Expanding its rank-one density and applying the rank-one partial-trace formula [F2], cross terms with $k\mathrm{\ne }\mathrm{\ell }$ vanish because $⟨n-\mathrm{\ell },n-k⟩={\delta }_{k\mathrm{\ell }}$. The reduced state is ${\sum }_k\mathrm{\mid }{c}_k{\mathrm{\mid }}^2\mathrm{\mid }k⟩⟨k\mathrm{\mid }$, which is the displayed mixture; its ideal number-PVM law is the binomial distribution by the declared outcome model [A1].

For input $\mathrm{\mid }\alpha ⟩⟨\alpha \mathrm{\mid }$, [F1] gives the product output $\mathrm{\mid }t\alpha ⟩⟨t\alpha \mathrm{\mid }\otimes \mathrm{\mid }r\alpha ⟩⟨r\alpha \mathrm{\mid }$. The rank-one partial-trace formula [F2] leaves the first factor unchanged; with $t=\sqrt{\eta }$, this is $\mathrm{\mid }\sqrt{\eta }\alpha ⟩⟨\sqrt{\eta }\alpha \mathrm{\mid }$.

At $\eta =0$, the transported beam-splitter expansion [F1] gives ${\tilde{U}}_0(\mathrm{\mid }m⟩\otimes \mathrm{\mid }0⟩)=\mathrm{\mid }0⟩\otimes \mathrm{\mid }m⟩$ for every $m\in {\mathbb{N}}_0$; at $\eta =1$ it gives ${\tilde{U}}_1(\mathrm{\mid }m⟩\otimes \mathrm{\mid }0⟩)=\mathrm{\mid }m⟩\otimes \mathrm{\mid }0⟩$. The finite linear span of the one-mode occupation basis is dense by [F4], so boundedness of the unitaries extends these identities to ${\tilde{U}}_0(\psi \otimes \mathrm{\mid }0⟩)=\mathrm{\mid }0⟩\otimes \psi$ and ${\tilde{U}}_1(\psi \otimes \mathrm{\mid }0⟩)=\psi \otimes \mathrm{\mid }0⟩$ for all one-mode vectors $\psi$.

Conjugation by ${\tilde{U}}_{\eta }$ is completely positive because unitary conjugation preserves positivity on every finite ancilla amplification. It preserves trace for every trace-class input by [F5], since $\mathrm{tr}({\tilde{U}}_{\eta }T{\tilde{U}}_{\eta }^{\ast })=\mathrm{tr}(T{\tilde{U}}_{\eta }^{\ast }{\tilde{U}}_{\eta })=\mathrm{tr}T$; by [F6], $I_q\otimes {\tilde{U}}_{\eta }$ is unitary and the same identity holds on every finite ancilla amplification. Composition with step 1.1 preserves both properties.

By steps 1.1, 1.2 and 2.1, the composition defining ${\mathrm{\Phi }}_{\eta }$ maps trace-class operators to trace-class operators, is completely positive under every finite ancilla, and preserves trace. It is CPTP on the full infinite-dimensional one-mode trace-class space; no finite-dimensional Choi theorem is used.

For rank-one $T_{u,v}$, the partial-trace formula [F2] applied to the two isometries in step 1.5 yields ${\mathrm{\Phi }}_0(T_{u,v})=⟨u,v⟩\mathrm{\mid }0⟩⟨0\mathrm{\mid }=\mathrm{tr}(T_{u,v})\mathrm{\mid }0⟩⟨0\mathrm{\mid }$ and ${\mathrm{\Phi }}_1(T_{u,v})=T_{u,v}$. By [F5], every trace-class input has a nuclear series of rank-one terms converging in trace norm. The partial trace is trace-norm contractive by [F2], and bounded isometric and unitary conjugations are trace-norm continuous by [F5]; the trace is trace-norm continuous by [F5]. Passing to the limit gives ${\mathrm{\Phi }}_0(\rho )=\mathrm{tr}(\rho )\mathrm{\mid }0⟩⟨0\mathrm{\mid }$ and ${\mathrm{\Phi }}_1(\rho )=\rho$. In particular these are vacuum and identity on density operators.