Defect estimate for the regularized output in terms of the input defects and means

Let $F=\ell^2(\mathbb{N}^n,\mathbb{C})$ be the $n$-mode Fock space with number basis $(|k\rangle)$, $|k|=k_1+\dots+k_n$, annihilation operators $a_j$, $\hat n_j=a_j^\dagger a_j$, and the diagonal weight $\hat N=1+\sum_j\hat n_j$ (so $\hat N|k\rangle=(1+|k|)|k\rangle$). States are positive operators on $F$ with number-basis diagonal summing to $1$; $E(\rho)=\sum_k|k|\rho_{kk}$, and $\rho$ has finite $m$-th moment if $\sum_k(1+|k|)^m\rho_{kk}<\infty$. For $r>0$, $\tau_r$ denotes the thermal state with mean photon number $r$ per mode, $\langle k|\tau_r|l\rangle=\delta_{kl}\prod_i\frac1{r+1}(\frac r{r+1})^{k_i}$, and $h(r)=\log(r+1)-\log r$.

Thermal setup. Fix $\eta,\theta,\kappa\in(0,1)$, $\varepsilon>0$, $d>0$, $r_1,r_2>0$, and put

Let $L$ be the logarithmic mean, $f(x)=x/\sinh x$, $w(r,\nu)=\frac{1}{h(r)}\frac{f(h(r)\nu/2)f(-h(r)/2)}{f(h(r)\nu/2-h(r)/2)}$ the thermal weight, $w_*(r)=\inf_\nu w(r,\nu)$ and $\chi(r)=(h(r)w_*(r))^{-1/2}$. The setup assumes the slack conditions

and the smallness $\kappa\le\varepsilon^2/\bigl(4(\varepsilon+2)\chi(r_0)^2\bigr)$. For $\zeta>0$ the associated regularization uses transmissivities $\eta$ and $\theta$, environment $\tau_d$, replacement $\tau_{r_0}$ with weight $\kappa$, and constants $c=1/h(r_0)$, $a_i=(1+\varepsilon)w_i/h(r_i)$, $b_i=\varepsilon w_i$ and $\zeta$; its output is $\omega(\rho,\sigma)=(1-\kappa)\,\mathcal B_\theta(\mathcal B_\eta(\rho,\sigma),\tau_d)+\kappa\tau_{r_0}$ and its objective is $\Phi_\zeta(\rho,\sigma)=cS(\omega)-a_1S(\rho)-a_2S(\sigma)+b_1E(\rho)+\zeta M_4(\rho)+b_2E(\sigma)+\zeta M_4(\sigma)$.

For a spectral logarithm datum $G$ (eigenbasis $(b_l)$, eigenvalues $p_l$) of a state and a mode $j$, let $\mu_j=\sum_lp_l\langle b_l,a_jb_l\rangle$, $c_{lm}=\langle b_l,(a_j-\mu_j)b_m\rangle$, and for $t>0$ let $\Delta^{G,t}_j$ be the defect functional $x\mapsto\sum_{l,m}\frac{(t+1)p_m-tp_l}{\sqrt{m_{lm}}}\overline{c_{lm}}\,x_{lm}$, where $m_{lm}=h(t)L(p_l,p_m)w(t,\frac{\log p_l-\log p_m}{h(t)})$, restricted to the centered space (the orthogonal complement in $\ell^2$ of the vector with diagonal entries $\sqrt{p_l}$). Put

For a basis $e$ and weights $v$ let $Q_{e,v}(T)=\sum_{i,l}L(v_i,v_l)|\langle e_i,Te_l\rangle|^2$ and $\mathrm c_{e,v}(T)=T-(\sum_iv_i\langle e_i,Te_i\rangle)\mathrm{Id}$, and let $\mathcal Z(e,v)$ be the span of $|e_i\rangle\langle e_l|-\delta_{il}v_i\mathrm{Id}$. Let $A_1(Z)$ be the partial expectation, over the second input in state $\sigma$, of the Heisenberg dual of the two-stage map (mix the inputs at $\eta$, then mix with $\tau_d$ at $\theta$, keep the retained port) applied to $Z$, and $A_2(Z)$ the partial expectation over the first input in state $\rho$.

Theorem. Let $\zeta>0$, let $\rho,\sigma$ be states with finite sixth moments, and let $G_1,G_2,G_0$ be spectral logarithm data of $\rho$, $\sigma$ and $\omega(\rho,\sigma)$ (bases $e^{(i)}$, probabilities $v^{(i)}$). Assume for all $Z\in\mathcal Z(e^{(0)},v^{(0)})$

Then

So the total thermal-balance defect of the output is controlled by the weighted input defects, with an $\varepsilon/2$ relative loss that also involves the input means; this is one half of the cancellation argument for the minimizers.

Formalization Note The data are the structure ThermalSetup and its regularization P.reg n ζ hζ; $\mathsf D_t$ and $\mathsf M$ are defectSquared and meanSquared; $A_1,A_2$ are rawFirstSlice P.η P.θ … σ (thermalState n P.d P.hd) and rawSecondSlice … ρ ….