Exact operator identity for the quartic logarithm of a Gibbs minimizer

Let $F=\ell^2(\mathbb{N}^n,\mathbb{C})$ be the $n$-mode Fock space with number basis $(|k\rangle)_{k\in\mathbb{N}^n}$, total number $|k|=k_1+\dots+k_n$, annihilation operators $a_j$, and the diagonal weight $\hat N=1+\sum_ja_j^\dagger a_j$, i.e. $\hat N|k\rangle=(1+|k|)|k\rangle$. A state is a positive bounded operator on $F$ whose number-basis diagonal sums to $1$. For a state $\rho$ write $S(\rho)=-\operatorname{Tr}\rho\log\rho$, $E(\rho)=\sum_k|k|\,\rho_{kk}$ and $M_4(\rho)=\sum_k(1+|k|)^4\rho_{kk}$; $\rho$ has finite $m$-th moment if $\sum_k(1+|k|)^m\rho_{kk}<\infty$.

A Kraus flow consists of bounded operators $(V_i)_{i}$ on $F$ with $\sum_i\|V_ix\|^2=\|x\|^2$ for all $x$, a constant $F_0\ge0$ with $\sum_i\langle V_ix,\hat n V_ix\rangle\le\langle x,\hat nx\rangle+F_0\|x\|^2$ (where $\hat n=\sum_ja_j^\dagger a_j$, both sides in $[0,\infty]$), and a map $\mathcal F$ on states such that $\mathcal F(\rho)=\sum_{i,l}|V_iv_l\rangle\langle V_iv_l|$ whenever $\rho=\sum_l|v_l\rangle\langle v_l|$ with $\sum_l\|v_l\|^2=1$, and $\mathcal F$ preserves finite energy. Its Heisenberg dual is $\mathcal F^*(Z)=\sum_iV_i^*ZV_i$.

A Gibbs representation of $\rho$ with quartic coefficient $k$ consists of a positive bounded operator $B_\rho$, a Hilbert basis $(g_j)_{j}$ of $F$ and numbers $u_j>0$ with $\mathcal Z=\sum_je^{-u_j}<\infty$ such that $\hat N^{-2}\bigl(k+\hat N^{-3/2}B_\rho\hat N^{-3/2}\bigr)^{-1}\hat N^{-2}g_j=u_j^{-1}g_j$ for all $j$ and $\rho=\sum_j\frac{e^{-u_j}}{\mathcal Z}|g_j\rangle\langle g_j|$ (moreover $\rho$ is injective and has finite seventh moment). Formally $\rho=e^{-H}/\mathcal Z$ with $H=k\hat N^4+\hat N^{1/2}B_\rho\hat N^{1/2}$. Its quartic logarithm is the bounded operator

formally $\hat N^{-2}(-\log\rho)\hat N^{-2}$.

Theorem. Let $\rho$ have a Gibbs representation $(B_\rho,g,u)$ with quartic coefficient $k$, let $\mathcal F=(V_i)$ be a Kraus flow and $\tau$ a state. Let $r>0$ and $0<\delta\le1$ with $\delta\,\tau_r\le\tau$ in operator order, where $\tau_r$ is the thermal state with mean photon number $r$ per mode. Let $(b_j)_{j\in J}$ be a Hilbert basis and $p_j>0$, $\sum_jp_j=1$, with $\tau b_j=p_jb_j$. Let $\alpha,\beta\ge0$ and suppose that for all $x\in F$, in $[0,\infty]$,

Then there exist positive bounded operators $C,D$ on $F$ such that for all $x$

(in $[0,\infty]$), and

Formally $C=\hat N^{-2}\mathcal F^*(-\log\tau)\hat N^{-2}$ and $D=\hat N^{-2}\hat n\hat N^{-2}$, so the identity is the bounded, sandwiched version of the Euler–Lagrange equation $-\log\rho=k\hat N^4+\alpha\,\mathcal F^*(-\log\tau)+\beta\hat n+\log\mathcal Z$. Having it as an exact identity of bounded operators lets one pair $-\log\rho$ with unbounded generators rigorously.

Formalization Note The Gibbs representation is GibbsData n k ρ and $\mathcal L_\rho$ is G.quarticLogarithm; quadratic forms are compared in ℝ≥0∞ via ENNReal.ofReal; $\hat N^{-1/2}$ and $\hat N^{-2}$ are inversePower rootNumberWeight _ 1 and _ 4 (with rootNumberWeight $=\sqrt{1+|k|}$).