Convolution identity for thermal-balance defects through a two-stage beam splitter

Let $F=\ell^2(\mathbb{N}^n,\mathbb{C})$ be the $n$-mode Fock space with annihilation operators $a_j$, number operators $\hat n_j=a_j^\dagger a_j$ and the weight $\hat N=1+\sum_j\hat n_j$ (diagonal, $\hat N|k\rangle=(1+|k|)|k\rangle$); states are positive operators on $F$ with number-basis diagonal summing to $1$. A weighted column family of a state $\mu$ is a family $(x_l)$ in $F$ with $\sum_l\|x_l\|^2<\infty$ such that $y_l=\hat N^{-3}x_l$ satisfy $\sum_l\|y_l\|^2=1$ and $\sum_l|y_l\rangle\langle y_l|=\mu$.

Put $h(r)=\log(r+1)-\log r$ for $r>0$, let $L$ be the logarithmic mean ($L(x,y)=\frac{y-x}{\log y-\log x}$, $L(x,x)=x$), $f(x)=x/\sinh x$ ($f(0)=1$), and let

be the thermal weight. A thermal spectral datum $\mathcal S=(e,p,r)$ on $F$ is a Hilbert basis $(e_k)_{k\in I}$, weights $p_k>0$ with $\sum_kp_k=1$, and $r>0$; put $\nu_{kl}=(\log p_k-\log p_l)/h(r)$ and $m_{kl}=h(r)\,L(p_k,p_l)\,w(r,\nu_{kl})>0$. Its observable map sends a bounded operator $Z$ to

so that $\|\mathrm{obs}_{\mathcal S}(Z)\|^2=Q^{\mathcal S}_w(Z):=\sum_{k,l}m_{kl}|\langle e_k,Ze_l\rangle|^2$. The centered space $\mathcal C(\mathcal S)\subseteq\ell^2(I\times I)$ is the orthogonal complement of $\mathrm{obs}_{\mathcal S}(\mathrm{Id})$ (the vector with diagonal entries $\sqrt{p_k}$ and zero off-diagonal entries), and $\mathcal Z(\mathcal S)$ is the span of the operators $|e_k\rangle\langle e_l|-\delta_{kl}p_k\mathrm{Id}$. Finally let $w_*(r)=\inf_{\nu\in\mathbb{R}}w(r,\nu)$ (which is $\ge r>0$) and $\chi(r)=\bigl(h(r)\,w_*(r)\bigr)^{-1/2}$.

A spectral logarithm datum $G$ for a state $\rho$ consists (among other data) of an eigenbasis $(b_l)$ of $\rho$ with eigenvalues $p_l>0$, $\sum_lp_l=1$, and spectral columns $x_l$ with $\hat N^{-3}x_l=\sqrt{p_l}\,b_l$ (a weighted column family of $\rho$). For mode $j$ put $\mu_j=\sum_lp_l\langle b_l,a_jb_l\rangle$ (the mean of $a_j$) and $c_{lm}=\langle b_l,(a_j-\mu_j)b_m\rangle$.

For $t>0$ let $\mathcal S_G(t)$ be the thermal spectral datum $(b,p,t)$ and $d_{lm}=(t+1)p_m-tp_l$. The defect functional of $G$ at $t$ in mode $j$ is the bounded functional on $\ell^2$

so that $\Delta^{G,t}_j(\mathrm{obs}(Z))=\sum_{l,m}d_{lm}\overline{c_{lm}}\langle b_l,Zb_m\rangle$ is a weak form of the thermal balance $(t+1)a_j$ versus $t\,a_j^\dagger$.

Setting. Let $\eta,\theta\in(0,1)$, $\kappa\in[0,1]$, $d>0$, $a_0,a_1>0$. Let $\rho_0,\rho_1,\tau_D,m,\gamma,\omega,\rho^\circ$ be states with $m=\mathcal B_\eta(\rho_0,\rho_1)$ and $\gamma=\mathcal B_\theta(m,\tau_D)$, both in the number-basis sense of the beam-splitter output relation. For $\eta\in[0,1]$ and states $\rho_0,\rho_1$, write $\mathcal B_\eta(\rho_0,\rho_1)$ for the beam-splitter output (the reduced state of the retained port after the passive rotation of transmissivity $\eta$ in every mode applied to $\rho_0\otimes\rho_1$). Put $\lambda_0=\theta\eta$, $\lambda_1=\theta(1-\eta)$ and $r_0=\theta(\eta a_0+(1-\eta)a_1)+(1-\theta)d$. For a bounded $Z$ let $A_0(Z)$ be the partial expectation, over the second input in state $\rho_1$, of the Heisenberg dual of the two-stage map (mix the two inputs at $\eta$, then mix the result with $\tau_D$ at $\theta$, keep the retained port) applied to $Z$, and $A_1(Z)$ the analogous partial expectation over the first input in state $\rho_0$. With $\mathrm c_i(T)=T-(\sum_lp^{(i)}_l\langle b^{(i)}_l,Tb^{(i)}_l\rangle)\mathrm{Id}$ the centering in the eigenbasis of $\rho_i$, the normalized slices are $N_i(Z)=\lambda_i^{-1/2}\,\mathrm c_i(A_iZ)$. Assume $r_0>0$, that $\tau_D$ and $\omega$ are the number-diagonal thermal states with mean photon numbers $d$ and $r_0$ per mode, and that $\rho^\circ=(1-\kappa)\gamma+\kappa\omega$. Let $G_0,G_1$ be spectral logarithm data for $\rho_0,\rho_1$, $G^\circ$ one for $\rho^\circ$, and let $y^{V}=(\hat N^{-3}v_l)$ and $y^{O}=(\hat N^{-3}o_l)$ come from weighted column families $v$ of $\gamma$ and $o$ of $\omega$ (weighted column families of $\tau_D$ and $m$ are also given).

Theorem. For every mode $j$ and every bounded operator $Z$ on $F$,

where $\overline W=(r_0+1)\sum_l\langle a_jy^V_l,y^V_l\rangle-r_0\sum_l\langle y^V_l,a_j^\dagger y^V_l\rangle$ is the weak mean of $\gamma$ in mode $j$ and $\mathcal M(Z)=\sqrt{(1-\kappa)\kappa}\,\bigl(\sum_l\langle y^V_l,Zy^V_l\rangle-\sum_l\langle y^O_l,Zy^O_l\rangle\bigr)$.

The output defect functional is thus a weighted sum ("convolution") of the two input defect functionals composed with the normalized conditional slices of the beam-splitter dual, plus an explicit error from replacing a fraction $\kappa$ of the output by a thermal state.

Formalization Note $\Delta^{G,t}_j$ is G.ambientDefect t ht j, $\mathrm{obs}$ is (G.thermalSpace t ht).observable, $N_i$ is normalizedSlices … (twoPortLambda η θ) i, $r_0$ is outputReference η θ a d, $\overline W$ is V.weakMean r₀ j, and $\mathcal M$ is EnsembleL2.mixtureFunctional … (1-κ). The beam-splitter relations are IsBeamSplitterOutput.