Uniform vacuum detector loss scales the effective reflectances

Let $M,J$ be natural numbers with $M>0$, allowing $J=0$. Let two splitters, indexed by $b\in\{0,1\}$, have reflectances $R_b\in[0,1]$ and phases $\theta_b$, with $R_0\le R_1$, $\sqrt{R_0}e^{i\theta_0}\ne\sqrt{R_1}e^{i\theta_1}$, and $\theta_0=\theta_1=0$, so $R_0<R_1$. Let $0\le\eta\le1$. An occupation vector on $L$ modes is an element of $\mathbb N^L$, including the unique empty occupation vector when $L=0$. For a zero-phase splitter of reflectance $R$, define

$$A_R(n,k)= \begin{cases} \displaystyle\prod_{m=1}^{M}\sqrt{\binom{n_m}{k_m}}\, (\sqrt R)^{\,n_m-k_m}(\sqrt{1-R})^{\,k_m}, &k\le n\text{ componentwise},\\ 0,&\text{otherwise}. \end{cases}$$

For a square-summable complex probe $\psi$ on $I\times\mathbb N^M$, its returned kernel is

$$K_R^\psi((i,n),(j,v)) =\sum_{k\in\mathbb N^M} \psi(i,n+k)A_R(n+k,k) \overline{\psi(j,v+k)A_R(v+k,k)}.$$

For any kernel $K$ on the same index set, define its attenuation by the zero-phase splitter of reflectance $\eta$ as

$$(\mathcal A_\eta K)((i,n),(j,v)) =\sum_{k\in\mathbb N^M} A_\eta(n+k,k)\overline{A_\eta(v+k,k)} K((i,n+k),(j,v+k)).$$

Scaling splitter $b$ by $\eta$ means replacing $R_b$ by $\eta R_b$ and retaining its zero phase. For kernels $K,L$ on $X$, a resolution of $K$ means a representation $K(x,y)=\sum_j a_j u_j(x)\overline{u_j(y)}$ with nonnegative summable weights of sum one and vectors in $\ell^2(X;\mathbb C)$, normalized whenever their weight is positive and mutually orthogonal at distinct positive-weight indices; zero-weight vectors are unrestricted. Define $P(K,L)$ as the real infimum, over resolutions $(a_j,u_j)$, $(b_j,v_j)$ of $K,L$ and bounded complex-linear operators $E=E^*$ satisfying $0\le\operatorname{Re}\langle v,Ev\rangle\le\|v\|^2$ for every $v$, of

$$\frac12\sum_j a_j(1-\operatorname{Re}\langle u_j,Eu_j\rangle) +\frac12\sum_j b_j\operatorname{Re}\langle v_j,Ev_j\rangle;$$

an empty defining set has infimum zero. For $n\in\mathbb N^M$, let $f_n$ be the unit coordinate vector at $(0,n)$ on $\mathbb N^0\times\mathbb N^M$, where $0$ is the unique empty occupation vector, and put $|n|=\sum_m n_m$. Define

$$q_b(K,j)= \begin{cases} \displaystyle\binom Kj(\eta R_b)^j(1-\eta R_b)^{K-j},&j\le K,\\ 0,&j>K. \end{cases}$$

For arbitrary $\beta\in\mathbb C^J$ and $\alpha\in\mathbb C^M$, define

$$g(z,\ell)=e^{-|z|^2/2}\frac{z^\ell}{\sqrt{\ell!}}, \qquad \psi_{\beta,\alpha}(p,n) =\left(\prod_{j=1}^{J}g(\beta_j,p_j)\right) \left(\prod_{m=1}^{M}g(\alpha_m,n_m)\right),$$

regarded as an $\ell^2$ vector when its coefficients are square-summable and replaced by zero otherwise, and put $E_\alpha=\sum_m|\alpha_m|^2$. The declaration asserts, for every $n\in\mathbb N^M$ and $b\in\{0,1\}$, and for every $\beta,\alpha,b$ as above, respectively,

$$\mathcal A_\eta K_{R_b}^{f_n}=K_{\eta R_b}^{f_n}, \qquad \mathcal A_\eta K_{R_b}^{\psi_{\beta,\alpha}} =K_{\eta R_b}^{\psi_{\beta,\alpha}}.$$

It also asserts, for every $n$ and every $\beta,\alpha$, respectively,

$$P(\mathcal A_\eta K_{R_0}^{f_n},\mathcal A_\eta K_{R_1}^{f_n}) =\frac12\sum_{j=0}^{|n|}\min\{q_0(|n|,j),q_1(|n|,j)\},$$ $$P(\mathcal A_\eta K_{R_0}^{\psi_{\beta,\alpha}}, \mathcal A_\eta K_{R_1}^{\psi_{\beta,\alpha}}) =\frac{1-\sqrt{\,1-\exp\!\left(-\eta(\sqrt{R_1}-\sqrt{R_0})^2E_\alpha\right)}}{2}.$$

These assertions include zero total occupation, zero coherent signal energy, and both $\eta=0$ and $\eta=1$; at $\eta=0$, the scaled splitters coincide, and distinctness of the scaled splitters is not required.