64c3814
==== TABLE OF CONTENTS ====
The-entropy-photon-number-inequality//main.tex
The-entropy-photon-number-inequality//references.tex
The-entropy-photon-number-inequality//sections/00-introduction.tex
The-entropy-photon-number-inequality//sections/01-preliminaries.tex
The-entropy-photon-number-inequality//sections/02-interpolation.tex
The-entropy-photon-number-inequality//sections/03-minimization.tex
The-entropy-photon-number-inequality//sections/04-metric.tex
The-entropy-photon-number-inequality//sections/05-stationarity.tex
The-entropy-photon-number-inequality//sections/06-broadcast.tex
===========================
==== BEGIN The-entropy-photon-number-inequality//main.tex ====
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\title{The entropy photon-number inequality}
\author{OpenAI}
\date{September 24, 2026}
\begin{document}
\maketitle
\begin{abstract}
We prove the entropy photon-number inequality for two independent bosonic inputs with finite mean energy, for every finite number of modes. This resolves the entropy photon-number conjecture in this finite-energy setting while allowing arbitrary entanglement among the modes within either input. As a consequence, we determine the exact minimum output entropy at fixed input entropy for every finite tensor power of an identical thermal attenuator, over finite-energy inputs that may be entangled across modes. We also obtain the exact classical capacity region of the degraded two-receiver pure-loss bosonic broadcast channel under a mean photon constraint.
\end{abstract}
{\small\tableofcontents}
\clearpage
\input{sections/00-introduction}
\input{sections/01-preliminaries}
\input{sections/02-interpolation}
\input{sections/03-minimization}
\input{sections/04-metric}
\input{sections/05-stationarity}
\input{sections/06-broadcast}
\input{references}
\end{document}
==== END The-entropy-photon-number-inequality//main.tex ====
==== BEGIN The-entropy-photon-number-inequality//references.tex ====
\begin{thebibliography}{99}
\bibitem{AndersonDuffin1969}
W.~N.~Anderson, Jr., and R.~J.~Duffin,
\emph{Series and parallel addition of matrices},
Journal of Mathematical Analysis and Applications \textbf{26} (1969), 576--594.
\href{https://doi.org/10.1016/0022-247X(69)90200-5}{\mbox{doi:10.1016/0022-247X(69)90200-5}}.
\bibitem{BeigiMehrabi2026}
S.~Beigi and H.~Mehrabi,
\emph{Monotonicity of the von Neumann entropy under quantum convolution},
Communications in Mathematical Physics \textbf{407} (2026), article 199.
\href{https://doi.org/10.1007/s00220-026-05703-4}{doi:10.1007/s00220-026-05703-4};
\href{https://arxiv.org/abs/2504.02206v3}{arXiv:2504.02206v3}.
\bibitem{BeigiRahimiKeshari2025}
S.~Beigi and S.~Rahimi-Keshari,
\emph{A meta logarithmic-Sobolev inequality for phase-covariant Gaussian channels},
Annales Henri Poincar\'e \textbf{26} (2025), 2737--2778.
\href{https://doi.org/10.1007/s00023-024-01487-2}{doi:10.1007/s00023-024-01487-2};
\href{https://arxiv.org/abs/2311.09572v2}{arXiv:2311.09572v2}.
\bibitem{Blachman1965}
N.~M.~Blachman,
\emph{The convolution inequality for entropy powers},
IEEE Transactions on Information Theory \textbf{11} (1965), 267--271.
\href{https://doi.org/10.1109/TIT.1965.1053768}{doi:10.1109/TIT.1965.1053768}.
\bibitem{CarlenMaas2017}
E.~A.~Carlen and J.~Maas,
\emph{Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance},
Journal of Functional Analysis \textbf{273} (2017), 1810--1869.
\href{https://doi.org/10.1016/j.jfa.2017.05.003}{doi:10.1016/j.jfa.2017.05.003};
\href{https://arxiv.org/abs/1609.01254}{arXiv:1609.01254}.
\bibitem{DSM2011}
S.~Das, N.~Sharma, and S.~Muthukrishnan,
\emph{On some special cases of the entropy photon-number inequality},
\href{https://arxiv.org/abs/1107.2365}{arXiv:1107.2365} (2011).
\bibitem{DePalmaThesis2017}
G.~De~Palma,
\emph{Gaussian optimizers and other topics in quantum information},
PhD thesis, defended 2016;
\href{https://arxiv.org/abs/1710.09395v1}{arXiv:1710.09395v1} (2017).
\bibitem{DP2019}
G.~De~Palma,
\emph{New lower bounds to the output entropy of multi-mode quantum Gaussian channels},
IEEE Transactions on Information Theory \textbf{65} (2019), 5959--5968.
\href{https://doi.org/10.1109/TIT.2019.2914434}{doi:10.1109/TIT.2019.2914434};
\href{https://arxiv.org/abs/1805.12469v3}{arXiv:1805.12469v3}.
\bibitem{DePalmaHuber2018}
G.~De~Palma and S.~Huber,
\emph{The conditional entropy power inequality for quantum additive noise channels},
Journal of Mathematical Physics \textbf{59} (2018), 122201.
\href{https://doi.org/10.1063/1.5027495}{doi:10.1063/1.5027495};
\href{https://arxiv.org/abs/1803.00470}{arXiv:1803.00470}.
\bibitem{DMG2014}
G.~De~Palma, A.~Mari, and V.~Giovannetti,
\emph{A generalization of the entropy power inequality to bosonic quantum systems},
Nature Photonics \textbf{8} (2014), 958--964.
\href{https://doi.org/10.1038/nphoton.2014.252}{doi:10.1038/nphoton.2014.252};
\href{https://arxiv.org/abs/1402.0404v2}{arXiv:1402.0404v2}.
\bibitem{DMLG2015}
G.~De~Palma, A.~Mari, S.~Lloyd, and V.~Giovannetti,
\emph{Multimode quantum entropy power inequality},
Physical Review A \textbf{91} (2015), 032320.
\href{https://doi.org/10.1103/PhysRevA.91.032320}{doi:10.1103/PhysRevA.91.032320};
\href{https://arxiv.org/abs/1408.6410v2}{arXiv:1408.6410v2}.
\bibitem{DePalmaTrevisan2018}
G.~De~Palma and D.~Trevisan,
\emph{The conditional entropy power inequality for bosonic quantum systems},
Communications in Mathematical Physics \textbf{360} (2018), 639--662.
\href{https://doi.org/10.1007/s00220-017-3082-8}{doi:10.1007/s00220-017-3082-8};
\href{https://arxiv.org/abs/1706.00440v2}{arXiv:1706.00440v2}.
\bibitem{DTG2016passive}
G.~De~Palma, D.~Trevisan, and V.~Giovannetti,
\emph{Passive states optimize the output of bosonic Gaussian quantum channels},
IEEE Transactions on Information Theory \textbf{62} (2016), 2895--2906.
\href{https://doi.org/10.1109/TIT.2016.2547426}{doi:10.1109/TIT.2016.2547426}.
\bibitem{DTG2017attenuator}
G.~De~Palma, D.~Trevisan, and V.~Giovannetti,
\emph{Gaussian states minimize the output entropy of the one-mode quantum attenuator},
IEEE Transactions on Information Theory \textbf{63} (2017), 728--737.
\href{https://doi.org/10.1109/TIT.2016.2621748}{doi:10.1109/TIT.2016.2621748};
\href{https://arxiv.org/abs/1605.00441}{arXiv:1605.00441}.
\bibitem{DTG2017}
G.~De~Palma, D.~Trevisan, and V.~Giovannetti,
\emph{Gaussian states minimize the output entropy of one-mode quantum Gaussian channels},
Physical Review Letters \textbf{118} (2017), 160503.
\href{https://doi.org/10.1103/PhysRevLett.118.160503}{doi:10.1103/PhysRevLett.118.160503};
\href{https://arxiv.org/abs/1610.09970v2}{arXiv:1610.09970v2}.
\bibitem{DTG2017multimode}
G.~De~Palma, D.~Trevisan, and V.~Giovannetti,
\emph{Multimode Gaussian optimizers for the Wehrl entropy and quantum Gaussian channels},
\href{https://arxiv.org/abs/1705.00499}{arXiv:1705.00499} (2017).
\bibitem{GGLMS2004}
V.~Giovannetti, S.~Guha, S.~Lloyd, L.~Maccone, and J.~H.~Shapiro,
\emph{Minimum output entropy of bosonic channels: a conjecture},
Physical Review A \textbf{70} (2004), 032315.
\href{https://doi.org/10.1103/PhysRevA.70.032315}{doi:10.1103/PhysRevA.70.032315}.
\bibitem{GGLMSY2004}
V.~Giovannetti, S.~Guha, S.~Lloyd, L.~Maccone, J.~H.~Shapiro, and H.~P.~Yuen,
\emph{Classical capacity of the lossy bosonic channel: the exact solution},
Physical Review Letters \textbf{92} (2004), 027902.
\href{https://doi.org/10.1103/PhysRevLett.92.027902}{doi:10.1103/PhysRevLett.92.027902}.
\bibitem{GHG2015}
V.~Giovannetti, A.~S.~Holevo, and R.~Garc\'ia-Patr\'on,
\emph{A solution of Gaussian optimizer conjecture for quantum channels},
Communications in Mathematical Physics \textbf{334} (2015), 1553--1571.
\href{https://doi.org/10.1007/s00220-014-2150-6}{doi:10.1007/s00220-014-2150-6};
\href{https://arxiv.org/abs/1312.2251v2}{arXiv:1312.2251v2}.
\bibitem{GES2007}
S.~Guha, B.~I.~Erkmen, and J.~H.~Shapiro,
\emph{The entropy photon-number inequality and its consequences},
in 2008 Information Theory and Applications Workshop (2008), 128--130.
\href{https://doi.org/10.1109/ITA.2008.4601037}{doi:10.1109/ITA.2008.4601037};
\href{https://arxiv.org/abs/0710.5666v2}{arXiv:0710.5666v2} (2007).
\bibitem{GSE2007}
S.~Guha, J.~H.~Shapiro, and B.~I.~Erkmen,
\emph{Classical capacity of bosonic broadcast communication and a minimum output entropy conjecture},
Physical Review A \textbf{76} (2007), 032303.
\href{https://doi.org/10.1103/PhysRevA.76.032303}{doi:10.1103/PhysRevA.76.032303}.
\bibitem{Herglotz1911}
G.~Herglotz,
\emph{\"Uber Potenzreihen mit positivem, reellen Teil im Einheitskreis},
Berichte \"uber die Verhandlungen der K\"oniglich-S\"achsischen Gesellschaft
der Wissenschaften zu Leipzig, Mathematisch-Physische Klasse
\textbf{63} (1911), 501--511.
\bibitem{HuberKoenigVershynina2017}
S.~Huber, R.~K\"onig, and A.~Vershynina,
\emph{Geometric inequalities from phase space translations},
Journal of Mathematical Physics \textbf{58} (2017), 012206.
\href{https://doi.org/10.1063/1.4974224}{doi:10.1063/1.4974224};
\href{https://arxiv.org/abs/1606.08603}{arXiv:1606.08603}.
\bibitem{KS2014}
R.~K\"onig and G.~Smith,
\emph{The entropy power inequality for quantum systems},
IEEE Transactions on Information Theory \textbf{60} (2014), 1536--1548.
\href{https://doi.org/10.1109/TIT.2014.2298436}{doi:10.1109/TIT.2014.2298436};
\href{https://arxiv.org/abs/1205.3409v2}{arXiv:1205.3409v2}.
\bibitem{KuboAndo1980}
F.~Kubo and T.~Ando,
\emph{Means of positive linear operators},
Mathematische Annalen \textbf{246} (1980), 205--224.
\href{https://doi.org/10.1007/BF01371042}{doi:10.1007/BF01371042}.
\bibitem{LesniewskiRuskai1999}
A.~Lesniewski and M.~B.~Ruskai,
\emph{Monotone Riemannian metrics and relative entropy on noncommutative probability spaces},
Journal of Mathematical Physics \textbf{40} (1999), 5702--5724.
\href{https://doi.org/10.1063/1.533053}{doi:10.1063/1.533053};
\href{https://arxiv.org/abs/math-ph/9808016}{arXiv:math-ph/9808016}.
\bibitem{PetzToth1993}
D.~Petz and G.~Toth,
\emph{The Bogoliubov inner product in quantum statistics},
Letters in Mathematical Physics \textbf{27} (1993), 205--216.
\href{https://doi.org/10.1007/BF00739578}{doi:10.1007/BF00739578}.
\bibitem{Shannon1948}
C.~E.~Shannon,
\emph{A mathematical theory of communication},
Bell System Technical Journal \textbf{27} (1948), 379--423, 623--656.
\href{https://doi.org/10.1002/j.1538-7305.1948.tb00917.x}{doi:10.1002/j.1538-7305.1948.tb00917.x}.
\bibitem{Shirokov2006}
M.~E.~Shirokov,
\emph{Entropy characteristics of subsets of states. I},
Izvestiya: Mathematics \textbf{70} (2006), 1265--1292.
\href{https://doi.org/10.1070/IM2006v070n06ABEH002346}{doi:10.1070/IM2006v070n06ABEH002346};
\href{https://arxiv.org/abs/quant-ph/0510073}{arXiv:quant-ph/0510073}.
\bibitem{Stam1959}
A.~J.~Stam,
\emph{Some inequalities satisfied by the quantities of information of Fisher and Shannon},
Information and Control \textbf{2} (1959), 101--112.
\href{https://doi.org/10.1016/S0019-9958(59)90348-1}{doi:10.1016/S0019-9958(59)90348-1}.
\bibitem{Umegaki1962}
H.~Umegaki,
\emph{Conditional expectation in an operator algebra, IV (Entropy and information)},
Kodai Mathematical Seminar Reports \textbf{14} (1962), 59--85.
\href{https://doi.org/10.2996/kmj/1138844604}{doi:10.2996/kmj/1138844604}.
\bibitem{Winter2016}
A.~Winter,
\emph{Tight uniform continuity bounds for quantum entropies: conditional entropy, relative entropy distance and energy constraints},
Communications in Mathematical Physics \textbf{347} (2016), 291--313.
\href{https://doi.org/10.1007/s00220-016-2609-8}{doi:10.1007/s00220-016-2609-8};
\href{https://arxiv.org/abs/1507.07775v6}{arXiv:1507.07775v6}.
\bibitem{YHD2011}
J.~Yard, P.~Hayden, and I.~Devetak,
\emph{Quantum broadcast channels},
IEEE Transactions on Information Theory \textbf{57} (2011), 7147--7162.
\href{https://doi.org/10.1109/TIT.2011.2165811}{doi:10.1109/TIT.2011.2165811};
\href{https://arxiv.org/abs/quant-ph/0603098v2}{arXiv:quant-ph/0603098v2}.
\end{thebibliography}
==== END The-entropy-photon-number-inequality//references.tex ====
==== BEGIN The-entropy-photon-number-inequality//sections/00-introduction.tex ====
\section{Introduction}\label{sec:introduction}
The entropy photon number of an \(n\)-mode bosonic state is the mean photon number per mode of the product of \(n\) identical one-mode thermal states with the same total entropy. Guha, Erkmen, and Shapiro conjectured that this quantity obeys a concavity inequality under beam-splitter mixing of independent inputs \cite{GES2007}. Their entropy photon-number inequality is a quantum counterpart of the classical entropy power inequality and has consequences for minimum output entropy and the information capacities of bosonic channels. We prove its two-input form for finite-energy states.
Write
\begin{equation}\label{eq:g-definition}
g(t)=(t+1)\log(t+1)-t\log t,\qquad t\ge0,
\end{equation}
with \(0\log0=0\). This continuous function increases from zero to infinity; its inverse is denoted by \(g^{-1}\). All entropies and logarithms in this paper use the natural base.
\begin{theorem}[Entropy photon-number inequality]\label{thm:epni}
Let \(n\ge1\) be finite, and let \(\rho_A,\rho_B\) be states on \(n\) bosonic modes, with annihilation operators \(a_1,\ldots,a_n\) and \(b_1,\ldots,b_n\), respectively, satisfying
\[
\Tr\rho_A\sum_{j=1}^n a_j^\dagger a_j<\infty,
\qquad
\Tr\rho_B\sum_{j=1}^n b_j^\dagger b_j<\infty.
\]
The joint input is \(\rho_A\otimes\rho_B\); no independence assumption is made among the modes within either state. For \(0\le\eta\le1\), let \(\rho_C\) be the reduced state of the modes
\[
c_j=\sqrt\eta\,a_j+\sqrt{1-\eta}\,b_j,\qquad 1\le j\le n,
\]
after passive beam-splitter mixing. Then
\begin{equation}\label{eq:epni}
g^{-1}\!\left(\frac{S(\rho_C)}n\right)
\ge \eta\,g^{-1}\!\left(\frac{S(\rho_A)}n\right)
+(1-\eta)\,g^{-1}\!\left(\frac{S(\rho_B)}n\right),
\end{equation}
where \(S(\rho)=-\Tr\rho\log\rho\).
\end{theorem}
Thus Theorem~\ref{thm:epni} resolves the entropy photon-number conjecture positively for finite-energy inputs, including arbitrary internal multimode entanglement. Thermal inputs with a common mean photon number within each port give equality, even when the means at the two ports differ.
For the channel consequences, write \(\tau_r\) for the \(n\)-mode product thermal state with mean photon number \(r\) in each mode, including the vacuum at \(r=0\); thus \(S(\tau_r)=ng(r)\). Let \(\mathcal E_{\eta,N_B}\) be the one-mode thermal attenuator obtained by retaining \(c=\sqrt\eta\,a+\sqrt{1-\eta}\,b\) when the environment mode \(b\) is thermal with mean photon number \(N_B\) and is independent of the input. Its tensor power \(\mathcal E_{\eta,N_B}^{\otimes n}\) uses the same parameters in every mode and the joint input \(\rho\otimes\tau_{N_B}\).
\begin{corollary}[Constrained entropy minimum for identical thermal attenuators]\label{cor:thermal-attenuator}
Let \(n\ge1\) be finite, let \(s,N_B\ge0\) be finite, and let \(0\le\eta\le1\). Every state \(\rho\) on \(n\) modes with
\(\Tr\rho\sum_{j=1}^n a_j^\dagger a_j<\infty\) satisfies
\begin{equation}\label{eq:thermal-attenuator-bound}
\frac1n S\!\left(\mathcal E_{\eta,N_B}^{\otimes n}(\rho)\right)
\ge g\!\left(\eta g^{-1}\!\left(\frac{S(\rho)}n\right)
+(1-\eta)N_B\right).
\end{equation}
In particular, the exact constrained minimum is
\begin{equation}\label{eq:thermal-attenuator-minimum}
\min_{\substack{\rho\text{ a state on }n\text{ modes}\\
\Tr\rho\sum_{j=1}^n a_j^\dagger a_j<\infty\\
S(\rho)=ns}}
\frac1n S\!\left(\mathcal E_{\eta,N_B}^{\otimes n}(\rho)\right)
=g\!\left(\eta g^{-1}(s)+(1-\eta)N_B\right).
\end{equation}
It is attained by the product thermal input \(\tau_{g^{-1}(s)}\). No independence assumption is imposed among the modes of \(\rho\); only the environment is required to be independent of that input.
\end{corollary}
\begin{proof}
The environment \(\tau_{N_B}\) has finite total mean photon number \(nN_B\) and entropy \(ng(N_B)\). The retained output has finite energy because passive mixing preserves combined total number, so its entropy is finite by Lemma~\ref{lem:energy-compactness}. Apply Theorem~\ref{thm:epni} to the independent inputs \(\rho\) and \(\tau_{N_B}\), and then apply the increasing function \(g\), to obtain \eqref{eq:thermal-attenuator-bound}.
For \(r=g^{-1}(s)<\infty\), the state \(\tau_r\) has finite total mean photon number \(nr\) and entropy \(ns\). Proposition~\ref{prop:thermal-mixing} gives
\[
\mathcal E_{\eta,N_B}^{\otimes n}(\tau_r)
=\tau_{\eta r+(1-\eta)N_B},
\]
whose entropy is \(n g(\eta r+(1-\eta)N_B)\). This attains the lower bound and proves \eqref{eq:thermal-attenuator-minimum}. The endpoints are included: at \(\eta=0\) the retained state is the environment, and at \(\eta=1\) it is the input. When \(s=0\) or \(N_B=0\), the corresponding thermal state is the vacuum, covered by the zero-mean case of Proposition~\ref{prop:thermal-mixing}.
\end{proof}
Corollary~\ref{cor:thermal-attenuator} concerns identical passive thermal attenuators with an independent product thermal environment. It asserts attainment, not uniqueness, and makes no claim for mode-dependent attenuations, amplifiers, or additive-noise channels.
The vacuum case also determines the classical capacity region of a
degraded pure-loss broadcast channel. Here one input mode per use is
passively split with auxiliary vacuum modes independent of the input
between receivers \(B,C\) and an inaccessible loss output. The power
transmissivities satisfy
\(0\le\eta_C<\eta_B\) and \(\eta_B+\eta_C\le1\), so \(C\)'s marginal
is obtained from \(B\)'s by a further pure-loss attenuation.
Section~\ref{sec:broadcast} gives the exact region for independent
classical messages and separate collective decoding, without feedback
or receiver cooperation. Its finite mean photon constraint is averaged
over the codebook and uses, and its finite-energy codewords may be
entangled across uses. The vacuum-port specialization of
Theorem~\ref{thm:epni} supplies the entropy input assumed in the
converse of Guha, Shapiro, and Erkmen~\cite{GSE2007};
coherent-state superposition coding supplies achievability.
\subsection{Context and earlier results}
Shannon introduced entropy power in his theory of communication
\cite{Shannon1948}; the Fisher-information proofs of Stam and Blachman
\cite{Stam1959,Blachman1965} became central to its later development.
For independent classical random vectors, the entropy power inequality
compares the entropy of a sum with the entropies of its summands. Its
bosonic counterpart replaces addition by passive mixing of quantum modes.
The analogy is especially natural for thermal states: their mean photon
numbers add with the beam-splitter weights, and their entropies are given
by the same function \(g\) appearing in Theorem~\ref{thm:epni}.
The formulation of Guha, Erkmen, and Shapiro
\cite[Section~II.B]{GES2007} makes this thermal comparison the central
conjecture and explains its consequences for bosonic communication.
Earlier work had established the capacity of the pure-loss bosonic channel
\cite{GGLMSY2004} and connected noisy-channel capacity questions with
minimum output entropy \cite{GGLMS2004}. The constrained minimum-output
problem also appears in the bosonic broadcast-channel analysis of Guha,
Shapiro, and Erkmen \cite{GSE2007}. Early EPnI special cases included
one-mode number-diagonal inputs with two nonzero probabilities and a
vacuum second port \cite{DSM2011}.
K\"onig and Smith \cite{KS2014} established a quantum entropy power
inequality using quantum Fisher information and a heat evolution. Their
results include the linear bound
\[
S(\rho_C)\ge\eta S(\rho_A)+(1-\eta)S(\rho_B)
\]
for every transmissivity, and the exponential entropy power bound for a
balanced beam splitter. De Palma, Mari, and Giovannetti \cite{DMG2014}
proved the exponential bound for every transmissivity,
\begin{equation}\label{eq:exponential-qepi}
e^{S(\rho_C)/n}
\ge\eta e^{S(\rho_A)/n}+(1-\eta)e^{S(\rho_B)/n}.
\end{equation}
These results are formulated for an arbitrary finite number of modes and independent
inputs, with no product assumption within a group. De Palma, Mari, Lloyd, and Giovannetti
\cite{DMLG2015} subsequently treated more general linear mixing of
several independent groups of modes.
De Palma and Trevisan later gave a rigorous finite-energy treatment of the
Fisher-information and heat-flow steps using an integral formulation;
their conditional inequality with a trivial memory system yields
\eqref{eq:exponential-qepi} for arbitrary independent finite-energy inputs
\cite[Theorem~6]{DePalmaTrevisan2018}.
The entropy photon-number inequality requires a sharper comparison than
\eqref{eq:exponential-qepi} when the input entropy levels differ. When
$S(\rho_A)=S(\rho_B)$, the linear entropy bound already gives the EPnI
value by monotonicity of $g^{-1}$. For unequal entropy levels, thermal
inputs attain EPnI, whereas the exponential bound does not recover their
exact output entropy. This distinction matters for the
minimum output entropy problem with an input-entropy constraint: the
bound must reproduce the exact thermal comparison at unequal entropy
levels.
The EPnI itself was also known for two independent Gaussian inputs in any
finite number of modes, including Gaussian correlations within either input;
De Palma proves this case using symplectic eigenvalue comparisons
\cite[Appendix~A.8]{DePalmaThesis2017}.
The Gaussian optimizer theorem of Giovannetti, Holevo, and
Garc\'ia-Patr\'on \cite{GHG2015} settled the unconstrained minimum-output
problem for broad classes of Gaussian channels. Fixing a positive input
entropy is a different optimization problem. For one-mode gauge-covariant
channels, De Palma, Trevisan, and Giovannetti established optimality of
passive rearrangement at fixed spectrum \cite{DTG2016passive}, then the
sharp entropy bound for the quantum-limited attenuator
\cite{DTG2017attenuator}, and finally the sharp bound for one-mode
phase-covariant Gaussian channels with thermal noise
\cite[Theorem~4]{DTG2017}. Their multimode extension
\cite[Theorem~11]{DTG2017multimode} covers inputs diagonal in some
product basis, including classical correlations in that basis.
Later advances address further parts of this landscape. De Palma
\cite{DP2019} proved the sharp constrained bound for multimode
entanglement-breaking Gaussian attenuators and amplifiers with thermal
environments, and
improved lower bounds for other parameter regimes. Beigi and
Rahimi-Keshari \cite{BeigiRahimiKeshari2025} proved a one-mode
meta-logarithmic-Sobolev inequality that also gives a variational proof of
the one-mode constrained minimum-output result. Beigi and Mehrabi
\cite{BeigiMehrabi2026} established subset-convolution inequalities for
exponential entropy power and entropy monotonicity under repeated
symmetric quantum convolution of an identical input.
Theorem~\ref{thm:epni} supplies the thermal comparison for two freely
varying inputs, including arbitrary entanglement within either port.
Corollary~\ref{cor:thermal-attenuator} specializes it to the exact
finite-mode constrained minimum for identical thermal attenuators,
without a product or separability restriction on the finite-energy input.
Thermal states enter the proof of Theorem~\ref{thm:epni} as reference states and as limits of
regularized minimizers. The original inputs need not be Gaussian,
number-diagonal, or independent across modes.
\subsection{Proof strategy}
The difficulty is to compare the two freely varying inputs with thermal
states at their respective entropy levels. We fix those thermal reference
parameters near a hypothetical strictly violating pair. This turns the
contradiction into a negative minimum of a linear combination of three
entropies, after adding penalties that ensure compactness and regularity.
The reference parameters remain fixed throughout the variational argument.
An independent analytic ingredient is an interpolation theorem for positive diagonal quadratic forms, proved in Section~\ref{sec:interpolation}. For positive scalars $m$ and real parameters $x,y$, it transforms four comparisons with weights
\[
mf(x)e^{x},\quad mf(x)e^{-x},\quad
mf(y)e^{y},\quad mf(y)e^{-y},
\qquad f(x)=\frac{x}{\sinh x},\quad f(0)=1,
\]
into the comparison with weight \(mf(x)f(y)/f(x+y)\). No compatibility is required between the linear maps and the diagonal multipliers. A Stieltjes representation for the derivative of an implicitly defined function is the essential analytic step. The theorem is formulated for finitely many spectral parameter values on source tests and permits unrestricted nonnegative target sums, so it also applies to the infinite matrix spaces used below.
Section~\ref{sec:minimization} develops the variational argument. Assuming a strict violation, we add a thermal port, a small thermal replacement of the output, a relative-entropy penalty, and a fourth-moment penalty. The resulting functional has faithful Gibbs minimizers with enough moment control for all subsequent traces. Varying one input at a time yields a component contraction in the logarithmic-mean metric \emph{at those minimizers}. It is not asserted as a universal channel inequality.
Section~\ref{sec:metric} combines two product-port variance estimates with the component contractions. These supply precisely the four hypotheses of the interpolation theorem. Its conclusion controls the norm of a centered thermal-balance defect: a linear functional measuring the creation--annihilation balance after subtracting the state's mean. In Section~\ref{sec:stationarity}, pairing the Gibbs identities with the relaxation generators gives an exact cancellation of the centered energy terms. As the fourth-moment penalty vanishes, the remaining defects and means tend to zero; their number-basis identities force both limiting inputs to be thermal. Entropy continuity then contradicts the original strictly negative objective value.
The auxiliary regularizations have linked roles: the relative-entropy penalty gives energy coercivity and supplies the strict margin in the final defect and mean estimate, while the thermal port supplies the slack needed for the associated Hessian comparisons. Output replacement controls the output logarithm, and the moment penalty justifies the unbounded-generator calculations. Only the last penalty tends to zero in the concluding compactness argument. This avoids assuming differentiability of entropy along an unbounded evolution or convergence of the input energies.
Section~\ref{sec:preliminaries} supplies the entropy, compactness, and thermal-mixing facts used in this argument. Each of the specialized interpolation, minimization, and limiting statements is proved within the paper.
==== END The-entropy-photon-number-inequality//sections/00-introduction.tex ====
==== BEGIN The-entropy-photon-number-inequality//sections/01-preliminaries.tex ====
\section{Bosonic states, energy, and entropy}\label{sec:preliminaries}
We work on the \(n\)-mode Fock space
\(\mathcal H_n=\ell^2(\N_0^n)\), with number basis
\(\{|\mathbf k\rangle:\mathbf k\in\N_0^n\}\). For the annihilation operators \(d_j\),
\[
d_j|\mathbf k\rangle=\sqrt{k_j}\,|\mathbf k-\mathbf e_j\rangle,
\qquad N=\sum_{j=1}^n d_j^\dagger d_j,
\qquad W=\id+N.
\]
A state is a positive trace-class operator of trace one. Expectations of positive unbounded operators are understood as increasing limits of bounded spectral truncations. Form inequalities have their usual quadratic-form meaning on the indicated domains.
For \(r>0\), put
\begin{equation}\label{eq:thermal-definition}
h(r)=\log(1+1/r),\qquad
\tau_r=(r+1)^{-n}e^{-h(r)N}.
\end{equation}
The product geometric law gives
\begin{equation}\label{eq:thermal-moments}
\Tr\tau_rN=nr,\qquad S(\tau_r)=ng(r).
\end{equation}
The state \(\tau_r\) is faithful and has all number moments finite. At \(r=0\), its limiting state is the multimode vacuum.
\subsection{Entropy estimates and compactness}
We first record the entropy facts needed in the infinite-dimensional argument.
The relative entropy introduced by Umegaki \cite{Umegaki1962} gives the
Gibbs variational principle. For oscillator energy bounds, the compactness
and entropy-continuity statements below are special cases of the general
energy-constrained theory \cite{Shirokov2006,Winter2016}; we include the
elementary cutoff proof used here.
\begin{lemma}[Relative entropy and the Gibbs variational inequality]\label{lem:gibbs-variational}
Let \(\sigma\) be a faithful state, and suppose both \(S(\rho)\) and \(\Tr\rho(-\log\sigma)\) are finite. Then
\[
D(\rho\Vert\sigma):=-S(\rho)+\Tr\rho(-\log\sigma)\ge0,
\]
with equality if and only if \(\rho=\sigma\).
Consequently, if \(K\) is bounded below and \(0<Z=\Tr e^{-K}<\infty\), its Gibbs state \(Z^{-1}e^{-K}\) uniquely minimizes \(\Tr\rho K-S(\rho)\) among states for which these expressions are finite, provided that Gibbs state belongs to the admitted class.
\end{lemma}
\begin{proof}
Choose eigenvectors \(\psi_j\) of \(\rho\) with positive eigenvalues \(p_j\), and set \(q_j=\langle\psi_j,\sigma\psi_j\rangle\). Scalar Jensen applied to the spectral measure of \(\sigma\) gives
\[
-\langle\psi_j,\log\sigma\,\psi_j\rangle\ge-\log q_j.
\]
The elementary inequality \(p\log(p/q)\ge p-q\) and \(\sum_jq_j\le1\) now imply nonnegativity. Equality requires \(p_j=q_j\), no mass of \(\sigma\) outside the support of \(\rho\), and equality in each Jensen inequality. The last condition makes each \(\psi_j\) an eigenvector of \(\sigma\) with eigenvalue \(q_j\), so \(\rho=\sigma\). The Gibbs assertion follows by substituting \(-\log\sigma=K+\log Z\).
\end{proof}
\begin{lemma}[Energy bound and compactness]\label{lem:energy-compactness}
For each finite \(E\ge0\), the set of states with \(\Tr\rho N\le E\) is compact in trace norm. On this set entropy is continuous and obeys
\begin{equation}\label{eq:entropy-energy}
0\le S(\rho)\le n g(E/n).
\end{equation}
Every finite-energy state is a trace-norm limit of normalized finite-number truncations with uniformly bounded energies and convergent entropies.
\end{lemma}
\begin{proof}
Diagonal entropy in any orthonormal basis bounds von Neumann entropy above: apply concavity of \(-t\log t\) to each diagonal entry expressed in a spectral decomposition, then sum the nonnegative terms. Compare the number-basis probabilities with the geometric probabilities of \(\tau_{E/n}\). Summing
\(-p\log p\le-p\log q+q-p\) gives, when \(E>0\),
\[
S(\rho)\le n\log(1+E/n)+h(E/n)E=ng(E/n).
\]
For \(E=0\), the state is the vacuum and the assertion is immediate.
Let \(P_K=\mathbf1_{\{N\le K\}}\), \(Q_K=\id-P_K\), and \(q=\Tr\rho Q_K\). The rank of \(P_K\) is finite, and \(q\le E/(K+1)\). Factoring off \(\rho^{1/2}\) and applying the Hilbert--Schmidt Cauchy--Schwarz inequality bounds each off-diagonal block by \(\sqrt q\) in trace norm. Hence \(\rho\) is uniformly approximated by its finite-dimensional compression; normalizing the latter changes its trace norm by at most \(q\). This proves precompactness. The energy bound is closed by lower semicontinuity of positive spectral expectations, so the set is compact.
For completeness, write \(H_2(q)=-q\log q-(1-q)\log(1-q)\). Pinching into the two number blocks changes the entropy by a quantity between zero and \(H_2(q)\). To see the upper bound directly, split each vector in a pure eigenensemble of \(\rho\) into its two projected vectors. Concavity of \(H_2\) bounds the increase of the ensemble-weight entropy by \(H_2(q)\); the entropy of a pure-state mixture is at most its weight entropy. The latter fact follows by applying diagonal entropy to the Gram operator of the weighted vectors, which has the mixture's nonzero eigenvalues. The lower bound follows from the same concavity argument for pinching.
The pinched entropy equals
\[
H_2(q)+(1-q)S\!\left(\frac{P_K\rho P_K}{1-q}\right)
+qS\!\left(\frac{Q_K\rho Q_K}{q}\right),
\]
where zero-mass terms are omitted. The last term is bounded by
\[
qn\,g\!\left(\frac{E}{qn}\right),
\]
which tends to zero uniformly as \(q\downarrow0\). The low-number entropy term is continuous in the finite-dimensional compression. These bounds uniformly approximate \(S(\rho)\) by continuous finite-dimensional expressions and prove its continuity under the common energy bound.
Finally, normalized number compressions converge in trace norm. Their energies are at most \(E/(1-q)\), and hence are uniformly bounded for all sufficiently large \(K\). The established continuity proves entropy convergence.
\end{proof}
In particular, entropy grows only logarithmically with energy for fixed \(n\), whereas a positive linear energy penalty is coercive. For \(r>0\), we shall repeatedly use the identity
\begin{equation}\label{eq:thermal-relative-entropy}
D(\rho\Vert\tau_r)=-S(\rho)+n\log(r+1)+h(r)\Tr\rho N.
\end{equation}
\subsection{Passive mixing}
For any finite collection of ports, a real orthogonal rotation of their mode operators, performed identically for every mode index, is implemented by a unitary on their joint Fock space. One way to construct it is to rotate the creation operators acting on the joint vacuum; the commutation relations show that the resulting number vectors again form an orthonormal basis. This unitary preserves combined total number. A retained subsystem therefore has energy no greater than the combined input energy. On each finite total-number sector, the unitary depends continuously on the rotation; consequently the induced channels are trace-norm continuous on every fixed state as the rotation varies.
\begin{proposition}[Thermal mixing]\label{prop:thermal-mixing}
Suppose independent ports carry the \(n\)-mode states \(\tau_{r_s}\), and the retained modes are \(\sum_s\sqrt{\lambda_s}\,d_{s,j}\), where \(\lambda_s\ge0\) and \(\sum_s\lambda_s=1\). Their joint retained state is
\[
\tau_{\sum_s\lambda_s r_s}.
\]
\end{proposition}
\begin{proof}
The coherent-state representation of a thermal state is
\[
\tau_r=\int_{\C^n}|z\rangle\langle z|\,
\frac{e^{-\lVert z\rVert^2/r}}{(\pi r)^n}\,d^{2n}z.
\]
Indeed, inserting the coefficients
\(\langle\mathbf k|z\rangle=e^{-\lVert z\rVert^2/2}z^{\mathbf k}/\sqrt{\mathbf k!}\)
and integrating gives the diagonal geometric probabilities in \eqref{eq:thermal-definition}. Products of coherent states transform by the same linear rotation of their amplitudes, as follows from their annihilation eigenvector equations or creation-operator expansion. Independent centered circular complex Gaussians with covariances \(r_s\id\) consequently produce retained covariance \((\sum_s\lambda_s r_s)\id\). The displayed representation proves the claim. Zero reference means follow by continuity.
\end{proof}
This thermal representation and its relation to entropy photon number are also the starting point of the original formulation \cite[Section~II.B]{GES2007}. In the argument below it is only the auxiliary reference states that have this special form.
==== END The-entropy-photon-number-inequality//sections/01-preliminaries.tex ====
==== BEGIN The-entropy-photon-number-inequality//sections/02-interpolation.tex ====
\section{Interpolation of diagonal form comparisons}\label{sec:interpolation}
The proof requires an interpolation principle for quadratic forms. Its
hypotheses involve four simple weights. In the later application at the
regularized minima, the target weight makes the squared norm of a centered
thermal-balance defect equal to the normalized entropy-production pairing
plus a centered-energy correction. This is the identity used in the
stationarity cancellation. We prove the principle here, including the
analytic property on which it depends.
Set
\begin{equation}\label{int:fb}
f(x)=\frac{x}{\sinh x},\qquad b(x)=x\coth x,\qquad f(0)=b(0)=1.
\end{equation}
Both functions are positive and even on $\R$. On Hilbert spaces
$\mathcal H_j$, $0\le j\le k$, fix coordinate representations in which all
weights below act by scalar multiplication. A positive weight $X_j$
defines the possibly unbounded form
\[
Q_{X,j}(Z)=\sum_{\alpha}X_j(\alpha)|Z_\alpha|^2;
\]
fixed positive coordinate measures can be absorbed into the coordinates.
Let $\mathcal T\subset\mathcal H_0$ be a linear test space, let
$L_i:\mathcal T\to\mathcal H_i$ be linear, and let $w_i>0$ for $1\le i\le k$.
We say that the family $X=(X_j)_{j=0}^k$ \emph{satisfies comparison} if
\begin{equation}\label{int:comparison}
Q_{X,0}(Z)\ge\sum_{i=1}^k w_i Q_{X,i}(L_iZ)
\qquad(Z\in\mathcal T).
\end{equation}
All sums on the right are interpreted as nonnegative coordinate sums.
\begin{theorem}[Four-weight interpolation]\label{thm:interpolation}
At every coordinate of every space, let $m>0$ and $x,y\in\R$ be given.
Suppose each $Z\in\mathcal T$ is supported on finitely many values of the
parameter tuple $(m,x,y)$, and $\mathcal T$ is invariant under scalar
multiplication by functions of this tuple. If the four families
\begin{equation}\label{int:four-weights}
mf(x)e^x,\quad mf(x)e^{-x},\quad mf(y)e^y,\quad mf(y)e^{-y}
\end{equation}
satisfy comparison, then so does
\begin{equation}\label{int:target-weight}
F=m\frac{f(x)f(y)}{f(x+y)}.
\end{equation}
\end{theorem}
Finitely many parameter values do not mean finitely many nonzero
coordinates. In particular, a centered finite matrix can have an infinite
identity tail, all at the same parameter value. The hypothesis permits
this situation, which will occur in our application.
\subsection{Operations that preserve comparison}\label{int:operations}
The operations below belong to the parallel-sum and operator-mean
calculus of Anderson--Duffin and Kubo--Ando
\cite{AndersonDuffin1969,KuboAndo1980}. We give direct form proofs,
including the limits needed for unbounded weights.
Throughout this subsection, the auxiliary weights are positive functions
of the parameter tuple $(m,x,y)$.
Positive linear combinations preserve \eqref{int:comparison}. So do
pointwise limits of such weights: on a source test there are
only finitely many parameter values, so its form converges; on the target
spaces Fatou's lemma gives the required inequality.
Comparisons also persist under the \emph{parallel sum}
\begin{equation}\label{int:parallel}
X:Y=(X^{-1}+Y^{-1})^{-1}.
\end{equation}
Indeed, coordinatewise,
\[
(X:Y)|z|^2=\min_{u+v=z}\bigl(X|u|^2+Y|v|^2\bigr),
\qquad
u=\frac{Y}{X+Y}z,\quad v=\frac{X}{X+Y}z.
\]
The source minimizers $U,V$ lie in $\mathcal T$. Apply the comparisons for
$X$ and $Y$ to them, then use $L_iU+L_iV=L_iZ$ and the coordinatewise
minimum on each target space. Target minimizers need not belong to any
specified test space.
Consequently, if $X,Y$ satisfy comparison, so do their logarithmic mean
$\ell(X,Y)=(X-Y)/(\log X-\log Y)$ and $XY/\ell(X,Y)$,
with the continuous value $\ell(X,X)=X$.
The identities that implement these operations are
\begin{equation}\label{int:log-means}
\frac1{\ell(X,Y)}=\int_0^1\frac{dt}{(1-t)X+tY},
\qquad
\frac{XY}{\ell(X,Y)}=\int_0^1\frac{dt}{t/X+(1-t)/Y}.
\end{equation}
For the first identity, a positive quadrature followed by inversion
is a finite parallel sum: if $q_j>0$ and $V_j>0$, then
\[
\left(\sum_{j=1}^M\frac{q_j}{V_j}\right)^{-1}
= (V_1/q_1):\cdots:(V_M/q_M).
\]
Here $V_j=(1-t_j)X+t_jY$ already satisfies comparison. The second
identity is a positive average of weighted parallel sums. Pointwise limits give the integrals; zero coefficients
are omitted or obtained by a limit. Thus no rule asserting preservation
under multiplication of weights is being used.
Apply these two operations to each opposite-exponent pair in
\eqref{int:four-weights}. Since their logarithmic mean is $m$, we obtain
comparison for the three families
\begin{equation}\label{int:basic-weights}
m,\qquad ms,\qquad mt,
\qquad s=f(x)^2,\quad t=f(y)^2.
\end{equation}
The next step constructs one further weight from these three.
\subsection{A Stieltjes representation}
The function $x\mapsto f(x)^2$ decreases from $1$ to $0$ on $[0,\infty)$.
Define $B$ on $(0,1]$ by
\begin{equation}\label{int:B-definition}
B\bigl(f(x)^2\bigr)=b(x).
\end{equation}
The scalar quadratic \eqref{int:quadratic}, used at the end of the proof,
determines the candidate for the additional weight. It pairs the
four original weights on $\xi\mp\phi$ and $\xi\pm\phi$ and adds
$E|\phi|^2$. For a candidate $E=me>0$, write $a=b(x)+b(y)$. Its minimum is
\[
\frac m2\left(a-\frac{(x-y)^2}{a+2e}\right)|\xi|^2.
\]
For $x^2\ne y^2$, the addition formula gives
$F/m=(yb(x)+xb(y))/(x+y)$. Equating the minimum with $F|\xi|^2$ therefore
requires, within this quadratic construction,
\[
(a+2e)(b(x)-b(y))=x^2-y^2,
\qquad
E=\frac{m(t-s)}{2(B(s)-B(t))},
\quad s=f(x)^2,\quad t=f(y)^2,
\]
where the last equality uses $b(z)^2=f(z)^2+z^2$. The final minimization
will verify the value in all cases. It remains to obtain comparison for
this candidate from the known weights. Its reciprocal has the integral
form, initially for $0<s,t<1$,
\begin{equation}\label{int:E-integral}
\frac1E=\frac2m\int_0^1
-B'\bigl(\theta s+(1-\theta)t\bigr)\,d\theta,
\qquad s=f(x)^2,\quad t=f(y)^2.
\end{equation}
The lemma below will show that this defines a positive continuous weight
also when $s=1$ or $t=1$. A Stieltjes representation of $-B'$ turns
the reciprocal into a limit of positive combinations of $1/m$ and
reciprocals of
$\theta(ms)+(1-\theta)(mt)+rm$, with $r\ge0$.
The parallel-sum argument above will then construct $E$ from the three
known weights. We now prove the required representation.
\begin{lemma}\label{int:stieltjes}
The function $B$ extends holomorphically to the upper half-plane and
across the positive real axis, is real and strictly decreasing there,
and satisfies $B(1)=1$, $B'(1)=-1$. There are a constant $c\ge0$ and a
positive measure $\nu$ on $[0,\infty)$ such that
\begin{equation}\label{int:stieltjes-form}
-B'(s)=c+\int_{[0,\infty)}\frac{d\nu(r)}{s+r}
\quad(s>0),\qquad
\int_{[0,\infty)}\frac{d\nu(r)}{1+r}<\infty.
\end{equation}
In particular $-B'(s)>0$ for every $s>0$.
\end{lemma}
\begin{proof}
For $x>0$, put $s=f(x)^2\in(0,1)$. Differentiating $s=f(x)^2$ and
$B(s)=b(x)$ gives
\[
\frac{ds}{dx}=\frac{2s(1-B)}{x},\qquad
\frac{db}{dx}=\frac{B-s}{x}.
\]
Here $B(s)=x\coth x>1$, so we may set
\[
R(s)=\frac{1-s}{s(B(s)-1)}.
\]
The derivatives give
\begin{equation}\label{int:B-derivative}
-B'(s)=\frac{B(s)-s}{2s(B(s)-1)}
=\frac12\left(\frac1s+R(s)\right).
\end{equation}
The term $1/s$ is already a Stieltjes kernel, so it remains to obtain
such a representation for $R$. We first construct the extension of $B$
using
\begin{equation}\label{int:u-parametrization}
s=\left(\frac{u}{\sin u}\right)^2,\qquad B(s)=u\cot u.
\end{equation}
For $q>0$ let $p_*(q)\in(\pi/2,\pi)$ be the unique solution of
\[
p_*(q)\tan p_*(q)=-q\tanh q,
\]
and put
\[
D=\{p+iq:q>0,\ 0<p<p_*(q)\},\qquad
\mathcal Q=\{z:\operatorname{Re}z>0,\ \operatorname{Im}z<0\}.
\]
Existence, uniqueness and continuity of $p_*$ follow because
$p\tan p$ increases strictly from $-\infty$ to $0$ on $(\pi/2,\pi)$:
its derivative is $(p+\sin p\cos p)/\cos^2p>0$.
Thus $D$ is connected. The map $a(u)=\sin u/u$ takes $D$ into
$\mathcal Q$, as follows from
\begin{align*}
\operatorname{Re}a(p+iq)
&=\frac{p\sin p\cosh q+q\cos p\sinh q}{p^2+q^2}>0,\\
\operatorname{Im}a(p+iq)
&=\frac{p\cos p\sinh q-q\sin p\cosh q}{p^2+q^2}<0.
\end{align*}
The first sign is precisely the boundary equation when $p>\pi/2$,
and is immediate for $p\le\pi/2$. For the second, when $p<\pi/2$ use
$\tan p/p>1>\tanh q/q$; for $p\ge\pi/2$ the sign is immediate.
This map is proper into $\mathcal Q$. Its finite boundary pieces map
to the coordinate axes: $p=0$ and $q=0$ map to the positive real axis,
and $p=p_*(q)$ to the negative imaginary axis. The limiting corners
$u=0,\pi$ map to $1,0$, respectively. Moreover,
\[
|a(p+iq)|\ge\frac{\sinh q}{\sqrt{\pi^2+q^2}}\longrightarrow\infty
\qquad(q\longrightarrow\infty)
\]
uniformly for $0\le p\le\pi$. Thus the inverse image of a compact
subset of $\mathcal Q$ cannot approach the boundary or infinity.
There are no critical points in $D$, because
\begin{equation}\label{int:anti-pick-sign}
\operatorname{Im}(u\cot u)
=\frac{q\sin(2p)-p\sinh(2q)}{\cosh(2q)-\cos(2p)}<0,
\end{equation}
whereas $a'(u)=0$ would imply $u\cot u=1$.
The strict sign follows from $\sin(2p)\le2p$ and $\sinh(2q)>2q$.
For completeness, properness and the inverse function theorem imply
that $a(D)$ is both open and relatively closed in $\mathcal Q$, hence
is all of $\mathcal Q$. Each fiber is finite, and local inverse
neighborhoods at its points give an evenly covered neighborhood: an
additional inverse point arbitrarily close to the chosen image point
would, by properness, accumulate at that fiber. Thus $a$ is a covering
map. A connected covering of the simply connected quadrant has one
sheet: inverse branches continue along paths by successive evenly
covered neighborhoods, and homotopies make the continuations
independent of the path. Therefore $a:D\to\mathcal Q$ has a
holomorphic inverse.
For $\operatorname{Im}s>0$, take the branch of $s^{-1/2}$ in
$\mathcal Q$ and define $u=a^{-1}(s^{-1/2})$. Equations
\eqref{int:u-parametrization} and \eqref{int:anti-pick-sign} give a
holomorphic $B$ with $\operatorname{Im}B<0$.
To check its continuation at $s_0>0$, the preceding bound on $|a|$
bounds $q$ as $s\to s_0$ from above. Every cluster point of $u$ lies
on $p=0$ or $q=0$, since the curved boundary has purely imaginary
image. On these pieces $\sinh q/q$ increases strictly from $1$ to
$\infty$, and $\sin p/p$ decreases strictly from $1$ to $0$ for
$0<p<\pi$. These observations identify a unique boundary limit:
$u=iq$ if $0<s_0<1$, and $u=p\in(0,\pi)$ if $s_0>1$.
The derivatives there are nonzero, so local analytic inversion gives
continuation across $s_0$. At $s_0=1$ use $u^2$ as parameter:
\[
\left(\frac{u}{\sin u}\right)^2=1+\frac{u^2}{3}+O(u^4),
\qquad u\cot u=1-\frac{u^2}{3}+O(u^4).
\]
This gives $B(1)=1$, $B'(1)=-1$. The two real parametrizations
also show that $B$ is strictly decreasing on $(0,\infty)$ and agrees
with \eqref{int:B-definition}.
We next recall the classical Herglotz representation
\cite{Herglotz1911}, with a positive-harmonic proof in the form needed
below. If $P$ is holomorphic with $\operatorname{Im}P\ge0$ in
the upper half-plane and extends holomorphically with real values
across $(0,\infty)$, then
\begin{equation}\label{int:herglotz}
P(s)=a_0+\beta s+\int_{(-\infty,0]}
\left(\frac1{r-s}-\frac{r}{1+r^2}\right)d\mu(r),
\qquad \beta\ge0,
\end{equation}
where $a_0\in\R$, $\mu\ge0$ and
$\int(1+r^2)^{-1}d\mu(r)<\infty$.
Indeed, under $s=i(1+z)/(1-z)$ its imaginary part becomes a
nonnegative harmonic function $v$ on the disk. The measures
$v(\varrho e^{i\theta})d\theta/(2\pi)$ have the fixed mass $v(0)$.
A weak limit as $\varrho\uparrow1$, together with the Poisson formula
on each smaller disk, represents $v$ by a positive boundary measure.
The atom at $z=1$ contributes $\beta\operatorname{Im}s$; at the
other boundary points the Poisson kernel becomes
$(1+r^2)\operatorname{Im}s/|r-s|^2$. Incorporate the factor
$1+r^2$ into $d\mu(r)$. The measure has no mass on the arcs
corresponding to $r>0$, because $v$ tends uniformly to zero on their
compact subarcs, by the assumed continuation. The analytic
integral in \eqref{int:herglotz} converges locally uniformly; its
imaginary part is the one just obtained. The remaining holomorphic
function has zero imaginary part and is a real constant. This
proves \eqref{int:herglotz}.
Apply \eqref{int:herglotz} to $P=-B$ and subtract the value at $1$.
The function
\begin{equation}\label{int:G}
G(s)=\frac{B(s)-1}{1-s}
=\beta+\int_{(-\infty,0]}\frac{d\mu(r)}{(1-r)(s-r)}
\end{equation}
has a removable value $G(1)=1$. It is positive on the positive real
axis, and $H(s)=sG(s)$ has nonnegative imaginary part in the upper
half-plane: this follows termwise from $\beta\ge0$ and
$\operatorname{Im}(s/(s-r))\ge0$ for $r\le0$.
The function $H$ has no zero there. Otherwise the nonnegative
harmonic function $\operatorname{Im}H$ would attain zero inside,
and the minimum principle would force $H$ to be a real constant;
its positive real-axis values rule out the constant zero.
Thus
\[
R(s)=\frac1{sG(s)}
\]
continues the real function $R$ above holomorphically to the upper
half-plane and across $(0,\infty)$,
is positive on that interval, and satisfies
$\operatorname{Im}(-R)\ge0$.
Apply \eqref{int:herglotz} once more, now to $-R$, with coefficients
$\widetilde\beta,\widetilde\mu$. For real $s>1$ it gives
\begin{align*}
0\le R(1)-R(s)
&=\widetilde\beta(s-1)
+\int_{(-\infty,0]}
\left(\frac1{1-r}-\frac1{s-r}\right)d\widetilde\mu(r)\\
&\le R(1).
\end{align*}
Consequently $\widetilde\beta=0$, and monotone convergence gives
$\int(1-r)^{-1}d\widetilde\mu(r)\le R(1)$. We may therefore write
\begin{equation}\label{int:R-stieltjes}
R(s)=c_0+\int_{(-\infty,0]}\frac{d\widetilde\mu(r)}{s-r},
\qquad
c_0=R(1)-\int_{(-\infty,0]}\frac{d\widetilde\mu(r)}{1-r}\ge0.
\end{equation}
Initially this identity follows on the positive axis; local uniform
convergence and analytic continuation give it in the upper half-plane
as well.
The holomorphic functions $-B'$ and $\tfrac12(1/s+R(s))$ agree on
$0<s<1$ by \eqref{int:B-derivative}. The identity theorem extends their
equality to the upper half-plane and the positive real axis. Equation
\eqref{int:R-stieltjes}, after replacing $r$ by $-r$, now proves
\eqref{int:stieltjes-form}; the term $1/(2s)$ contributes an atom
of mass $1/2$ at zero and ensures strict positivity.
\end{proof}
\subsection{The additional weight and the final minimization}
The lemma extends the same integral definition \eqref{int:E-integral}
continuously and positively to all $s,t>0$, including the values used in
\eqref{int:basic-weights}. The fundamental theorem of calculus gives
\begin{equation}\label{int:E}
E=
\begin{cases}
\displaystyle\frac{m(t-s)}{2(B(s)-B(t))},&s\ne t,\\[6pt]
\displaystyle-\frac{m}{2B'(s)},&s=t.
\end{cases}
\end{equation}
Lemma~\ref{int:stieltjes} shows that $E$ is obtainable by the
comparison-preserving operations already established. Indeed, after
substitution of \eqref{int:stieltjes-form}, every reciprocal term is
of the form
\[
\frac1{m(\theta s+(1-\theta)t+r)}
=\frac1{\theta(ms)+(1-\theta)(mt)+rm},
\]
and the constant term is a multiple of $1/m$.
Positive finite quadratures followed by inversion are parallel sums
of positive multiples of these positive linear combinations of
$m,ms,mt$. Truncating the $r$-integral and refining partitions gives
pointwise convergence for every $s,t>0$: on each bounded interval the
integrands are continuous, and the tail is controlled by
$\int(1+r)^{-1}d\nu(r)<\infty$.
The source finite-parameter hypothesis and target Fatou argument
therefore prove comparison for $E$.
We complete the proof of Theorem~\ref{thm:interpolation} by an exact
scalar minimization. For $\xi,\phi\in\C$ put
\begin{equation}\label{int:quadratic}
\mathcal Q(\xi,\phi)=
\frac14\sum_{\pm}mf(x)e^{\pm x}|\xi\mp\phi|^2
+\frac14\sum_{\pm}mf(y)e^{\pm y}|\xi\pm\phi|^2
+E|\phi|^2.
\end{equation}
Write $a=b(x)+b(y)$ and $e=E/m$. Since
$f(x)e^{\pm x}=b(x)\pm x$, expansion gives
\[
\mathcal Q(\xi,\phi)
=\frac{ma}{2}|\xi|^2
+m\left(\frac a2+e\right)|\phi|^2
-m(x-y)\operatorname{Re}(\overline\xi\phi).
\]
The unique minimizer and its value are
\begin{equation}\label{int:minimizer}
\phi=\frac{x-y}{a+2e}\xi,
\qquad
\min_\phi\mathcal Q(\xi,\phi)
=\frac m2\left(a-\frac{(x-y)^2}{a+2e}\right)|\xi|^2.
\end{equation}
If $x^2\ne y^2$, the identity $b(z)^2=f(z)^2+z^2$ and
\eqref{int:E} give
\[
(a+2e)(b(x)-b(y))=x^2-y^2.
\]
Hence the minimum in \eqref{int:minimizer} is
\begin{equation}\label{int:minimum-value}
m\frac{y b(x)+x b(y)}{x+y}|\xi|^2
=m\frac{f(x)f(y)}{f(x+y)}|\xi|^2,
\end{equation}
where the second equality is the addition formula for $\sinh$,
with continuous values when $x$ or $y$ is zero.
No singular minimization is hidden when $x^2=y^2$.
If $y=x$, the minimizer is $\phi=0$ and the value is
$mb(x)|\xi|^2=mf(x)^2|\xi|^2/f(2x)$.
If $y=-x\ne0$, write $s=f(x)^2$ and $b=b(x)$.
Equations \eqref{int:E} and \eqref{int:B-derivative} yield
\[
e=\frac{s(b-1)}{b-s},\qquad
b+e=\frac{x^2}{b-s}.
\]
Thus \eqref{int:minimizer} gives
$m(b-x^2/(b+e))|\xi|^2=ms|\xi|^2$, as required.
At $x=y=0$, we have $E=m/2$, $\phi=0$, and the value is $m|\xi|^2$.
This proves \eqref{int:minimum-value} in every case.
For a source test $Z$, let $\Phi$ be its coordinatewise minimizer in
\eqref{int:minimizer}. It belongs to $\mathcal T$ by the invariance
assumption. Apply the four assumed comparisons to $Z\mp\Phi$ and
$Z\pm\Phi$, and the comparison for $E$ to $\Phi$.
After summing, the source side is $Q_{F,0}(Z)$ by
\eqref{int:minimum-value}, whereas the target side is
\[
\sum_i w_i\sum_\alpha
\mathcal Q_i\bigl((L_iZ)_\alpha,(L_i\Phi)_\alpha\bigr)
\ge\sum_i w_i Q_{F,i}(L_iZ).
\]
This is the claimed comparison, and proves
Theorem~\ref{thm:interpolation}.
==== END The-entropy-photon-number-inequality//sections/02-interpolation.tex ====
==== BEGIN The-entropy-photon-number-inequality//sections/03-minimization.tex ====
\section{A regularized minimum and its Hessian}\label{sec:minimization}
We argue by contradiction. This section constructs a strictly negative
minimum with enough regularity to support the later operator identities.
It then extracts an entropy-Hessian estimate at that minimum.
\subsection{The auxiliary ports and the functional}
Suppose Theorem~\ref{thm:epni} fails. The endpoint transmissions give
equality, so $0<\eta<1$. By Lemma~\ref{lem:energy-compactness}, normalized
number cutoffs of the inputs converge with bounded energies and convergent
entropies, as do their outputs. Thus there is a fixed strictly violating
pair $(\rho_1^{\mathrm{w}},\rho_2^{\mathrm{w}})$ with finite number support.
Write $a_i=g^{-1}(S(\rho_i^{\mathrm{w}})/n)$. At least one $a_i$ is positive.
Besides the two variable ports $1,2$, introduce a fixed thermal port
$D$, in state $\rho_D=\tau_{r_D}$ with $r_D>0$. Its mixing weight and those
of the variable inputs are
\[
0<\lambda_D<1,\qquad
\lambda_1=(1-\lambda_D)\eta,\qquad
\lambda_2=(1-\lambda_D)(1-\eta).
\]
The retained port, denoted by $0$, has annihilators
$d_{0,j}=\sum_{s=1,2,D}\sqrt{\lambda_s}\,d_{s,j}$ before reduction.
For positive reference parameters $r_1,r_2$, set
\[
r_0=\lambda_1r_1+\lambda_2r_2+\lambda_Dr_D,
\qquad h_s=h(r_s)\quad(s=0,1,2,D).
\]
Let $\gamma$ be the retained state obtained from
$\rho_1\otimes\rho_2\otimes\rho_D$. We also replace a small fraction of
that output by its reference thermal:
\[
\rho_0=(1-\kappa)\gamma+\kappa\tau_{r_0},\qquad
0<\kappa<1,\qquad w_i=(1-\kappa)\lambda_i\quad(i=1,2).
\]
Unless otherwise indicated, sums indexed by $i$ run over $1,2$.
Since $h(r)=g'(r)$, the quotient $(S(\rho)-ng(r))/h(r)$ expresses the
entropy deviation in the linearized scale of total photon number at the
thermal reference.
For $\varepsilon,\zeta>0$ we minimize, over input states of finite fourth
moment, the functional
\begin{equation}\label{eq:functional}
\begin{split}
J_\zeta(\rho_1,\rho_2)
={}&\frac{S(\rho_0)-ng(r_0)}{h_0}
-\sum_i w_i\frac{S(\rho_i)-ng(r_i)}{h_i}\\
&+\varepsilon\sum_i\frac{w_i}{h_i}
D(\rho_i\Vert\tau_{r_i})
+\zeta\sum_i\Tr\rho_iW_i^4.
\end{split}
\end{equation}
The normalization of the relative-entropy penalty also matches the closing
stationarity calculation. The thermal entropy formula and
\eqref{eq:thermal-relative-entropy} give
\[
\frac{D(\rho\Vert\tau_r)}{h(r)}
=-\frac{S(\rho)-ng(r)}{h(r)}+\bigl(\Tr\rho N-nr\bigr).
\]
Thus the input terms pair an entropy coefficient $1+\varepsilon$ with an
energy coefficient $\varepsilon$. Their centered-energy coefficients
differ by one in stationarity, matching the output centered-energy
relation while retaining the strict defect and mean terms.
The parameters are chosen in the following order. First choose $r_i>0$
equal or sufficiently close to $a_i$. At
$\lambda_D=\kappa=\varepsilon=\zeta=0$, the value at the fixed witness
approaches
\[
\frac{S(\rho_C^{\mathrm{w}})
-ng(\eta a_1+(1-\eta)a_2)}
{h(\eta a_1+(1-\eta)a_2)}<0.
\]
The input differences vanish in this limit. If $a_i=0$, this follows
from $g(r)/h(r)\to0$ as $r\downarrow0$; the output denominator has a
strictly positive, finite limit. Fix $r_D>0$ and then choose
$\lambda_D>0$ sufficiently small to retain negativity. Indeed, the
additional mixing can be performed after the original beam splitter;
as $\lambda_D\downarrow0$, its unitary converges strongly on each total
number sector. The resulting states converge in trace norm with bounded
energy, and their entropies converge by Lemma~\ref{lem:energy-compactness}.
Next choose $\varepsilon>0$ small enough to retain negativity and to have
\begin{equation}\label{eq:slack}
(1+\varepsilon)\sum_i\lambda_ir_i\le r_0,
\qquad
(1+\varepsilon)\sum_i\lambda_i(r_i+1)\le r_0+1.
\end{equation}
At $\varepsilon=0$ the gaps are respectively $\lambda_Dr_D$ and
$\lambda_D(r_D+1)$, both positive. Finally take $\kappa>0$ sufficiently
small. The additional bound on $\kappa$ used in Section~\ref{sec:metric}
depends only on the parameters already fixed, so it is compatible with
this choice. The witness has finite fourth moments and finite thermal
relative entropies. Consequently there are constants $c>0$ and
$\zeta_0>0$ such that
\begin{equation}\label{eq:negative-witness}
J_\zeta(\rho_1^{\mathrm{w}},\rho_2^{\mathrm{w}})\le -c
\qquad(0<\zeta\le\zeta_0).
\end{equation}
All parameters except $\zeta$ remain fixed from now on.
\subsection{A Gibbs regularity lemma}
The fourth-moment penalty yields more regularity than its definition
alone suggests. We record the needed form argument explicitly.
\begin{lemma}\label{lem:gibbs-regularity}
Let $W=1+N$ on the $n$-mode Fock space. Suppose $k>0$ and $B$ is a
bounded positive operator. The positive form
\[
K=kW^4+W^{1/2}BW^{1/2},\qquad \mathcal D(K^{1/2})=\mathcal D(W^2),
\]
defines a positive self-adjoint operator with compact inverse. Its
normalized Gibbs state $\omega=e^{-K}/\Tr e^{-K}$ is faithful and
satisfies $\Tr\omega W^7<\infty$. It uniquely minimizes
$\Tr\sigma K-S(\sigma)$ among states with finite fourth moment.
\end{lemma}
\begin{proof}
Put $C=kI+W^{-3/2}BW^{-3/2}$. The form is
$\|C^{1/2}W^2\phi\|^2$ on $\mathcal D(W^2)$, and is closed because
$kI\le C\le\|C\|I$. The operator
\[
U=W^{-2}C^{-1}W^{-2}
\]
is positive, compact, and injective. Choose a complete orthonormal
eigenbasis $\psi_j$ with $U\psi_j=u_j^{-1}\psi_j$ and $u_j>0$.
Every $\psi_j$ belongs to $\mathcal D(W^2)$ and satisfies the form
eigenvalue equation for $K$ with eigenvalue $u_j$.
For completeness, the vectors
$\sqrt{u_j}\,C^{-1/2}W^{-2}\psi_j$ are orthonormal and complete:
orthonormality follows from $U$, and completeness from the dense range
of $C^{-1/2}W^{-2}$. Parseval's identity therefore gives
\[
\|C^{1/2}W^2\phi\|^2
=\sum_j u_j|\langle\psi_j,\phi\rangle|^2
\qquad(\phi\in\mathcal D(W^2)).
\]
Closedness and finite spectral sums show that the domain of this
diagonal form is exactly $\mathcal D(W^2)$. This constructs $K=U^{-1}$.
The bound $U\le k^{-1}W^{-4}$ gives
\[
\#\{j:u_j\le a\}
\le\operatorname{rank}\mathbf1_{\{W^4\le a/k\}},
\]
by comparing dimensions with the complementary number subspace.
The right side grows polynomially in $a$. Also
$\|W^2\psi_j\|^2\le u_j/k$. Test the form eigenvalue equation against
each number vector and multiply that coordinate equation by $W^{-1/2}$.
It gives
\[
kW^{7/2}\psi_j
=u_jW^{-1/2}\psi_j-BW^{1/2}\psi_j.
\]
The right side is square summable, so $\psi_j\in\mathcal D(W^{7/2})$,
with the explicit bound
\[
\|W^{7/2}\psi_j\|
\le k^{-1}\bigl(u_j+\|B\|\sqrt{u_j/k}\bigr).
\]
Polynomial eigenvalue counting and the exponential weights $e^{-u_j}$
now prove both $0<\Tr e^{-K}<\infty$ and
$\Tr(e^{-K}W^7)<\infty$. Finally, for a state $\sigma$ with finite
fourth moment, all terms in
\[
\Tr\sigma K-S(\sigma)
=D(\sigma\Vert\omega)-\log\Tr e^{-K}
\]
are finite. Nonnegativity and the equality case of relative entropy
give the asserted unique minimum.
\end{proof}
\subsection{Attainment and the exact logarithm identity}
\begin{proposition}\label{prop:regularized-minimum}
For every $0<\zeta\le\zeta_0$, the functional $J_\zeta$ attains a
minimum of value at most $-c$. The input energies at these minima are
bounded uniformly in $\zeta$. At each minimum all four states
$\rho_s$, $s=0,1,2,D$, are faithful, have finite seventh moment, and
satisfy $0\le H_s:=-\log\rho_s\le C_sW_s^4$ as forms. In fact
$H_0\le CW_0$. The inputs obey the exact identities
\begin{equation}\label{eq:log-identity}
\frac{(1+\varepsilon)w_i}{h_i}H_i
=\frac{1-\kappa}{h_0}T_iH_0
+\varepsilon w_iN_i+\zeta W_i^4+c_iI,
\qquad i=1,2,
\end{equation}
where $T_i$ is the output-observable slice with the other two inputs
fixed, and $c_i$ is a real scalar. The moment and logarithm bounds need
not be uniform as $\zeta\downarrow0$.
\end{proposition}
\begin{proof}
Write $E_i=\Tr\rho_iN_i$. Expanding the thermal relative entropy
expresses $J_\zeta$, up to constants depending only on the fixed
parameters, as
\begin{equation}\label{eq:coercive-functional}
\frac{S(\rho_0)}{h_0}
-\sum_i\frac{(1+\varepsilon)w_i}{h_i}S(\rho_i)
+\varepsilon\sum_i w_iE_i
+\zeta\sum_i\Tr\rho_iW_i^4.
\end{equation}
The first term is nonnegative. The bound
$S(\rho_i)\le ng(E_i/n)=O(\log(1+E_i))$ shows that the positive linear
energy terms dominate every negative entropy term. Thus bounded upper
sublevels have bounded input energies, uniformly in
$0<\zeta\le\zeta_0$. For fixed $\zeta>0$ their fourth moments are
bounded as well. Take a minimizing sequence in the sublevel supplied
by \eqref{eq:negative-witness}. Lemma~\ref{lem:energy-compactness}
gives a trace-norm convergent subsequence of each input. The product
and channel maps are trace-norm continuous, and output energies are
bounded by the combined input energy plus the fixed replacement
energy. Entropy is continuous along this subsequence. The energy and
fourth-moment expectations are lower semicontinuous, by bounded
spectral truncation. Equation~\eqref{eq:coercive-functional} therefore
gives an admissible minimum, and the same sublevel estimate gives its
uniform energy bound.
Fix such a minimum. The replacement gives $\rho_0\ge\kappa\tau_{r_0}$.
The resolvent representation of the logarithm, together with inverse
order for positive operators, implies
\[
0\le H_0\le-\log(\kappa\tau_{r_0})\le C W_0
\]
as positive forms. To define $T_i$, pull an output observable back
through the passive unitary and take its partial expectation in the
states of the other two ports. For bounded $Z$, this is a bounded
unital positive slice satisfying
\begin{equation}\label{eq:slice-expectation}
\Tr\rho_iT_iZ=\Tr\gamma Z.
\end{equation}
For $H_0$, use increasing bounded spectral truncations to define the
positive form $T_iH_0$. Since the retained number is bounded by total
number before reduction, slicing $H_0\le CW_0$ yields
\[
0\le T_iH_0\le C_iW_i.
\]
In particular, its expectation is the output cross entropy for every
varied input of finite energy.
Vary only input $i$ and replace the output entropy in $J_\zeta$ by
the cross entropy $\Tr\widetilde\rho_0H_0$. Nonnegativity of relative
entropy makes the resulting functional $\overline J_i$ a majorant,
with equality at the chosen minimum. Thus
\[
\overline J_i(\widetilde\rho_i)
\ge J_\zeta(\widetilde\rho_i)
\ge J_\zeta(\rho_i)=\overline J_i(\rho_i).
\]
After division by its entropy coefficient
$(1+\varepsilon)w_i/h_i$, this majorant is
$\Tr\widetilde\rho_iK_i-S(\widetilde\rho_i)$ plus a constant, where
\begin{equation}\label{eq:gibbs-hamiltonian}
K_i=\frac{h_i}{(1+\varepsilon)w_i}
\left(\frac{1-\kappa}{h_0}T_iH_0
+\varepsilon w_iN_i+\zeta W_i^4\right).
\end{equation}
This is a form $kW_i^4+R_i$ with $k>0$ and
$0\le R_i\le C_iW_i$. Hence
$R_i=W_i^{1/2}B_iW_i^{1/2}$ for a bounded positive $B_i$.
Lemma~\ref{lem:gibbs-regularity} identifies the input uniquely as
\[
\rho_i=\frac{e^{-K_i}}{\Tr e^{-K_i}}.
\]
It is faithful, has finite seventh moment, and satisfies
$H_i=K_i+\log\Tr e^{-K_i}$, proving \eqref{eq:log-identity} and the
claimed input logarithm bounds.
The fixed thermal port has every number moment. Total-number
preservation and
$(x_1+x_2+x_D)^7\le3^6(x_1^7+x_2^7+x_D^7)$ for nonnegative numbers
give finite seventh moment of the raw output; the thermal replacement
preserves this property. This proves all remaining assertions.
\end{proof}
\subsection{The component entropy-Hessian estimate}
Work at one of the minima in Proposition~\ref{prop:regularized-minimum}.
For each label $s$, fix an eigenbasis
$\rho_s=\operatorname{diag}(p_{s,l})$, with all $p_{s,l}>0$. These orthonormal eigenbases generally differ from the number bases used to define the energy, and they depend on the minimizing states. With
$\ell(a,b)=(a-b)/(\log a-\log b)$ and $\ell(a,a)=a$, define
\begin{equation}\label{eq:logmean-metric}
\|Z\|_s^2
=\sum_{l,r}h_s\ell(p_{s,l},p_{s,r})|Z_{lr}|^2,
\qquad
v_s(l,r)=\frac{\log(p_{s,l}/p_{s,r})}{h_s}.
\end{equation}
Up to the factor $h_s$, this is the Bogoliubov inner product on
observables. Its dual form on state perturbations is the
Bogoliubov--Kubo--Mori metric, obtained from the Hessian of relative
entropy \cite{PetzToth1993,LesniewskiRuskai1999}. We establish the needed
differentiation formulas on finite eigenblocks and then use density;
no bounded inverse for the full state is assumed.
These weighted matrix spaces are Hilbert spaces, with inner product
conjugate-linear in the first variable. Bounded observables belong to
them because the logarithmic mean is at most the arithmetic mean.
The centered subspace is the orthogonal complement of $I$; the
orthogonal projection subtracts
$\langle Z\rangle_sI$, where $\langle Z\rangle_s=\Tr\rho_sZ$.
Set
\begin{equation}\label{eq:slice-map}
L_iZ=\frac{T_iZ-\langle T_iZ\rangle_i I}{\sqrt{\lambda_i}}.
\end{equation}
Initially the centered output tests are finite eigenblock matrices
with their expectation times $I$ subtracted. They form a dense
subspace. Such a test can have an infinite identity tail, but has
only finitely many frequencies $v_0$. Multiplication by any function
of $v_0$ preserves this class and its centering: every diagonal
entry, including that tail, has frequency zero.
\begin{proposition}\label{prop:hessian}
For $i=1,2$, the slice map on the centered output space satisfies
\begin{equation}\label{eq:hessian}
\|L_iZ\|_i^2
\le\frac{1+\varepsilon}{1-\kappa}\,\|Z\|_0^2.
\end{equation}
It consequently extends continuously to the completed centered
spaces. For bounded observables this extension agrees with
\eqref{eq:slice-map}.
\end{proposition}
\begin{proof}
For a single varied input of finite fourth moment, denote the
resulting modified output by $\widetilde\rho_0$. The logarithm
identity \eqref{eq:log-identity} cancels all linear terms in the
objective difference, giving exactly
\begin{equation}\label{eq:relative-entropy-difference}
J_\zeta(\widetilde\rho_i)-J_\zeta(\rho_i)
=\frac{(1+\varepsilon)w_i}{h_i}
D(\widetilde\rho_i\Vert\rho_i)
-\frac1{h_0}D(\widetilde\rho_0\Vert\rho_0)\ge0.
\end{equation}
All cross entropies are finite by the fourth-moment assumptions and
the bounds on $H_i,H_0$.
Let $A$ be a traceless Hermitian matrix on a finite input eigenblock.
Then $\rho_i+tA$ is a state for all sufficiently small positive and
negative $t$. Its fourth moment is finite: each selected eigenvector
has finite seventh moment by the construction in
Lemma~\ref{lem:gibbs-regularity}. Finite-dimensional differentiation
on that block gives
\[
D(\rho_i+tA\Vert\rho_i)
=\frac{t^2}{2}\mathcal Q_i(A)+o(t^2),\qquad
\mathcal Q_i(A)=\sum_{l,r}
\frac{|A_{lr}|^2}{\ell(p_{i,l},p_{i,r})}.
\]
For example, the derivative of the logarithm is the integral of
$(\rho_i+uI)^{-1}A(\rho_i+uI)^{-1}$ over $u>0$ on this block.
To extract a dual lower bound without restricting the actual output
perturbation to a finite eigenblock, we test the Gibbs variational formula
with a centered output observable. Let $Z$ be a Hermitian centered output
test as above. The Gibbs
inequality gives
\[
D(\widetilde\rho_0\Vert\rho_0)
\ge t\Tr\widetilde\rho_0Z
-\log\Tr\exp(\log\rho_0+tZ).
\]
The logarithm of this partition function has first derivative zero
and second derivative $\|Z\|_0^2/h_0$ at zero. This follows from
finite-block exponential differentiation: before centering, only a
finite eigenblock changes, and centering merely multiplies the
exponential by a scalar. Writing $Z_t=\Tr e^{\log\rho_0+tZ}$ and
$\sigma_t=Z_t^{-1}e^{\log\rho_0+tZ}$, the same block decomposition
gives $0<Z_t<\infty$ and makes $\sigma_t$ faithful.
Since $-\log\sigma_t=H_0-tZ+\log Z_t$, the finite energy of
$\widetilde\rho_0$, the bound $H_0\le CW_0$, and boundedness of $Z$
give finite entropy and finite cross entropy against $\sigma_t$.
Lemma~\ref{lem:gibbs-variational} therefore gives the displayed lower bound.
Since the output perturbation induced by
$tA$ is $(1-\kappa)t$ times the raw channel perturbation, the lower
bound has expansion
\[
t^2\left((1-\kappa)\Tr A T_iZ
-\frac{\|Z\|_0^2}{2h_0}\right)+o(t^2).
\]
Put $\alpha=(1+\varepsilon)w_i h_0/h_i$. Comparing with
\eqref{eq:relative-entropy-difference} yields
\begin{equation}\label{eq:hessian-duality}
2(1-\kappa)\Tr A T_iZ-\alpha\mathcal Q_i(A)
\le\frac{\|Z\|_0^2}{h_0}.
\end{equation}
We can optimize the left side over the completed traceless Hermitian
space for $\mathcal Q_i$. Indeed, trace is continuous in this norm,
since
\[
|\Tr A|^2\le\left(\sum_l p_{i,l}\right)
\sum_l\frac{|A_{ll}|^2}{p_{i,l}}
\le\mathcal Q_i(A).
\]
Finite eigenblock truncation followed by correcting the trace at one
fixed diagonal entry proves density in the trace-zero subspace.
The constrained dual of $\mathcal Q_i$ is the logarithmic-mean norm
of the observable after subtracting its $\rho_i$ expectation. Thus
the supremum in \eqref{eq:hessian-duality} equals
\[
\frac{(1-\kappa)^2}{\alpha h_i}
\|T_iZ-\langle T_iZ\rangle_iI\|_i^2.
\]
It follows that
\[
(1-\kappa)^2
\|T_iZ-\langle T_iZ\rangle_iI\|_i^2
\le(1+\varepsilon)w_i\|Z\|_0^2.
\]
Dividing by $\lambda_i$ gives \eqref{eq:hessian}. The estimate
extends to complex tests by writing real and imaginary Hermitian
parts: symmetry of the logarithmic-mean weights and preservation
of adjoints make each squared norm the sum of the corresponding
two squared norms.
Density now gives the continuous extension. To identify it on a
bounded $Z$, approximate by $P_mZP_m$, where $P_m$ are increasing
finite spectral projections of $\rho_0$, and center each approximant.
They converge in both output arithmetic norms and hence in the
logarithmic-mean norm. The slice is completely positive and unital,
so its Schwarz inequality bounds the corresponding input arithmetic
norms by the raw output norms. Those in turn are at most
$(1-\kappa)^{-1}$ times the modified output norms, because
$\rho_0\ge(1-\kappa)\gamma$. The sliced approximants therefore
converge to the bounded slice in \eqref{eq:slice-map}, as required.
\end{proof}
==== END The-entropy-photon-number-inequality//sections/03-minimization.tex ====
==== BEGIN The-entropy-photon-number-inequality//sections/04-metric.tex ====
\section{The metric adapted to thermal relaxation}\label{sec:metric}
Fix a minimizing pair from Proposition~\ref{prop:regularized-minimum}.
All comparisons in this section concern these states. We use the centered
observable spaces, modular frequencies $v_s$, and maps $L_i$ introduced in
Section~\ref{sec:minimization}; sums over $i$ run over $1,2$.
For a positive scalar weight $X_s(v_s)$, write
\[
\|Z\|_{X_s,s}^2=\langle Z,X_sZ\rangle_s.
\]
We say that a family of weights satisfies comparison when
\begin{equation}\label{eq:weight-comparison}
\sum_i w_i\|L_iZ\|_{X_i,i}^2\le \|Z\|_{X_0,0}^2.
\end{equation}
Initially, $Z$ is a finite eigenblock matrix centered by a scalar multiple
of the identity. Such tests have finitely many frequencies, including zero;
they need not have finite matrix support after centering.
\subsection{Four elementary comparisons}
Define, with continuous values at zero,
\[
b_s(v)=\frac v2\coth\frac{h_sv}{2},\qquad
R_{s,\pm}(v)=b_s(v)\pm\frac v2,
\qquad b_s(0)=\frac1{h_s}.
\]
If $\rho_s=\operatorname{diag}(p_l)$, then
\begin{equation}\label{eq:arithmetic-weights}
h_s\ell(p_l,p_r)R_{s,+}(v_s(l,r))=p_l,
\qquad
h_s\ell(p_l,p_r)R_{s,-}(v_s(l,r))=p_r.
\end{equation}
Thus the corresponding squared norms of a centered observable are
$\Tr\rho_sZZ^\dagger$ and $\Tr\rho_sZ^\dagger Z$.
Both families $R_+$ and $R_-$ satisfy comparison. To see this, embed $Z$
as an observable $\widetilde Z$ on the three input ports and equip the
joint observable space with either arithmetic inner product. The centered
one-port subspaces are mutually orthogonal because the joint state is
$\rho_1\otimes\rho_2\otimes\rho_D$. The centered slice
$T_iZ-\Tr\gamma Z\,I$ is the orthogonal projection onto the $i$th such
subspace: its pairing with a bounded one-port observable is the defining
partial-expectation identity. Consequently the sum of the two slice
variances is at most the corresponding variance in $\gamma$.
The variance in $\rho_0=(1-\kappa)\gamma+\kappa\tau_{r_0}$ is at least
$(1-\kappa)$ times that variance in $\gamma$. Since
$w_i/\lambda_i=1-\kappa$, these statements give
\eqref{eq:weight-comparison} for both multiplication orders.
Only independence across ports has been used.
The two constant families $r_s$ and $r_s+1$ also satisfy comparison.
Indeed, Proposition~\ref{prop:hessian} and \eqref{eq:slack} give
\[
\sum_i w_i r_i\|L_iZ\|_i^2
\le (1+\varepsilon)\sum_i\lambda_i r_i\|Z\|_0^2
\le r_0\|Z\|_0^2,
\]
and the same argument applies with $r_s+1$.
\begin{proposition}[Interpolated comparison]\label{prop:metric}
The positive weights
\begin{equation}\label{eq:thermal-metric}
F_s(v)=\frac1{h_s}
\frac{f(h_sv/2)f(-h_s/2)}{f(h_s(v-1)/2)}
=\frac{b_s(v)-(r_s+\tfrac12)v}{1-v}
\end{equation}
satisfy \eqref{eq:weight-comparison}. The quotient at $v=1$ is understood
continuously. For each fixed $r_s>0$ there are constants
$0<c_s\le C_s<\infty$, independent of the minimizing states, such that
$c_s\le F_s(v)\le C_s$ for every $v\in\R$.
\end{proposition}
\begin{proof}
Apply Theorem~\ref{thm:interpolation} with
\[
m=1/h_s,\qquad x=h_sv_s/2,\qquad y=-h_s/2.
\]
The ambient coordinate spaces in Theorem~\ref{thm:interpolation} are the full weighted matrix spaces; its source test space consists of the centered finite-eigenblock tests described above. Multipliers depending on the frequency preserve that test space, because every diagonal entry has frequency zero. The first two weights are $R_{s,+},R_{s,-}$; the other two are
$r_s,r_s+1$. The conclusion is the first expression in
\eqref{eq:thermal-metric}. The second follows from the hyperbolic addition
formula and $\coth(h_s/2)=2r_s+1$.
The first expression is continuous and strictly positive, while the second
gives the limits $r_s$ at $+\infty$ and $r_s+1$ at $-\infty$.
This proves both bounds.
\end{proof}
The thermal weight has a useful logarithmic-mean interpretation:
\begin{equation}\label{eq:tilted-logmean}
\ell(p_l,p_r)F_s(v_s(l,r))
=\ell\bigl(r_sp_l,(r_s+1)p_r\bigr).
\end{equation}
Indeed the logarithmic difference on the right is
$h_s(v_s(l,r)-1)$; substitution in \eqref{eq:thermal-metric} proves the
identity, including its continuous values. Up to a fixed normalization,
this is the frequency-dependent logarithmic mean used in the
detailed-balance entropy geometry of Carlen and Maas
\cite[Definition~5.7 and Lemma~5.8]{CarlenMaas2017}.
Proposition~\ref{prop:metric} establishes the particular comparison
needed at our regularized minima.
Write $\|Z\|_{F_s}=\|Z\|_{F_s,s}$, and let
$\|q\|_{F_s,*}$ denote the dual norm of a linear functional on the centered
space. For a list of $n$ functionals, its squared norm is the sum of the
$n$ squared dual norms. Density and Proposition~\ref{prop:hessian} extend
Proposition~\ref{prop:metric} to the completed centered spaces.
On bounded observables the extension of $L_i$ is the actual centered
slice. In fact, bounded spectral compressions of $Z$ converge strongly
with their adjoints, as do their slices. Dominated convergence in either
arithmetic norm gives convergence in the logarithmic-mean norm, since
$\ell(a,b)\le(a+b)/2$. This also explains why the comparison applies to
fixed number-basis tests later, although the minimizing eigenbases vary.
\subsection{The thermal defect functionals}
Let $d_{s,j}$ be the annihilator of mode $j$ on port $s$, and put
\[
\mu_s=(\Tr\rho_s d_{s,j})_{j=1}^n,\qquad
d'_{s,j}=d_{s,j}-(\mu_s)_jI,\qquad
e'_s=\Tr\rho_sN_s-|\mu_s|^2.
\]
For bounded $Z$, define
\begin{equation}\label{eq:thermal-defect}
q_{s,j}(Z)=(r_s+1)\Tr\rho_s(d'_{s,j})^\dagger Z
-r_s\Tr\rho_sZ(d'_{s,j})^\dagger.
\end{equation}
A coherent displacement shifts $d_{s,j}$ and $(\mu_s)_j$ by the same
constant, so these centered functionals also vanish on displaced thermal
states. The final argument must control $\mu_s$ separately to recover the
fixed, undisplaced thermal reference.
These are well-defined weak trace pairings and vanish on $I$.
For example, $d\rho=(d\rho^{1/2})\rho^{1/2}$ is trace class by
Hilbert--Schmidt factorization whenever $\Tr\rho N<\infty$; the same
holds for $d^\dagger\rho$, $\rho d$, and $\rho d^\dagger$.
Thus no domain invariance for an arbitrary bounded $Z$ is being assumed.
We record the summability needed for both the metric estimate and the
subsequent generator calculation.
\begin{lemma}[Dual norm and logarithmic sums]\label{lem:dual-finiteness}
At each minimizing state, $q_s$ has finite dual norm. Set
\[
K_{s,lr}=(r_s+1)p_r-r_sp_l.
\]
Then
\begin{equation}\label{eq:dual-norm-sum}
\|q_s\|_{F_s,*}^2
=\sum_{j,l,r}(1-v_s(l,r))K_{s,lr}|(d'_{s,j})_{lr}|^2.
\end{equation}
The sums obtained by replacing the coefficient on the right by
$|K_{s,lr}|$ or $|v_s(l,r)K_{s,lr}|$ are finite.
\end{lemma}
\begin{proof}
Suppress the label $s$ and write $H=-\log\rho$.
By Proposition~\ref{prop:regularized-minimum},
$\Tr\rho W^7<\infty$ and $0\le H\le CW^4$ as forms.
Every eigenvector $\psi_r$ of $\rho$ belongs to $D(W^{7/2})$, since
$p_r>0$. Number shifts give
$\|W^2d'\psi\|+\|W^2(d')^\dagger\psi\|
\le C'\|W^{5/2}\psi\|$.
Consequently the crossed logarithmic sums, such as
\[
\sum_{l,r}p_r(-\log p_l)|d'_{lr}|^2
=\sum_r p_r\langle d'\psi_r,Hd'\psi_r\rangle,
\]
are bounded by a constant times $\Tr\rho W^5$.
For the uncrossed sums, use
\[
-\log p_r\le C\langle\psi_r,W^4\psi_r\rangle,
\qquad
\langle W^4\rangle_{\psi_r}\langle W\rangle_{\psi_r}
\le\langle W^5\rangle_{\psi_r}.
\]
The last inequality is the elementary nonnegative covariance inequality
for the increasing functions $t^4,t$ of the same positive random variable.
It bounds $(-\log p_r)\|d'\psi_r\|^2$ and its creation-operator analogue.
The corresponding sums without logarithms are finite by finite energy.
These bounds imply the claimed absolute summability of $K$ and $vK$.
In matrix coordinates, \eqref{eq:thermal-defect} is
\[
q_j(Z)=\sum_{l,r}K_{lr}\overline{d'_{j,lr}}Z_{lr}.
\]
By \eqref{eq:arithmetic-weights} and \eqref{eq:thermal-metric},
\begin{equation}\label{eq:defect-multiplier}
\frac{K_{lr}}{h\ell(p_l,p_r)}
=b(v)-(r+\tfrac12)v=(1-v)F(v).
\end{equation}
Hence the representing vector in the $F$ inner product is
$(1-v)d'_j$, and its squared norm is the sum in
\eqref{eq:dual-norm-sum}. That sum is finite by the preceding estimates.
Finally, $F(0)=1/h$, so orthogonality to $I$ in the $F$ inner product is
exactly zero $\rho$-mean. The representing vector is orthogonal to $I$
because $q_j(I)=0$. Restricting to the centered space therefore does not
change the dual norm.
\end{proof}
\begin{lemma}[Weak convolution identity]\label{lem:weak-traces}
Let $\mu_\gamma=\sum_i\sqrt{\lambda_i}\mu_i$ be the raw output mean.
For every bounded output observable $Z$,
\begin{equation}\label{eq:defect-convolution}
q_{0,j}(Z)=\sum_i w_iq_{i,j}(L_iZ)
+\kappa(1-\kappa)\overline{(\mu_\gamma)_j}
\Tr[(\gamma-\tau_{r_0})Z].
\end{equation}
\end{lemma}
\begin{proof}
The thermal defect is zero on $\tau_r$, since
$(r+1)d\tau_r=r\tau_rd$ by its geometric number weights.
For the raw output, define $q_{\gamma,j}$ by
\eqref{eq:thermal-defect} using its mean and reference parameter $r_0$.
The terms with coefficient $1$ in the identity
\[
q_{\gamma,j}(Z)
=\sum_{s=1,2,D}\sqrt{\lambda_s}\,q_{s,j}(T_sZ)
\]
agree by $d_{0,j}=\sum_s\sqrt{\lambda_s}d_{s,j}$.
The remaining terms are expected commutators. After rotation,
$d_{s,j}^\dagger$ is $\sqrt{\lambda_s}d_{0,j}^\dagger$ plus an operator
on discarded modes. The weak commutator of that discarded operator with
a retained observable has zero trace. Thus its retained coefficient is
$\sum_s\lambda_sr_s=r_0$, as required.
These statements use only finite energy: all one-mode-factor products
with the state are trace class as above. A discarded operator can first
be cut off in its own number basis; its bounded cutoff commutes with
$Z\otimes I$, so trace cyclicity applies. The cutoff products converge
in trace norm by the same Hilbert--Schmidt factorization. This justifies
the weak commutator and passage between joint, reduced, and sliced traces,
without assigning an operator domain to $[d_{0,j}^\dagger,Z]$.
The $D$ term vanishes. Since the mean of the replaced output is
$(1-\kappa)\mu_\gamma$, direct centering gives
\[
q_{0,j}=(1-\kappa)q_{\gamma,j}
+\kappa(1-\kappa)\overline{(\mu_\gamma)_j}
\Tr[(\gamma-\tau_{r_0})\,\cdot\,].
\]
The first term is $\sum_iw_iq_{i,j}\circ L_i$, since $q_{i,j}(I)=0$.
\end{proof}
\begin{proposition}[Defect estimate]\label{prop:defect-estimate}
Set
\[
A_*^2=\sum_iw_i\|q_i\|_{F_i,*}^2,
\qquad M_*^2=\sum_iw_i|\mu_i|^2,
\qquad C_0=(h_0\inf_{v\in\R}F_0(v))^{-1/2}.
\]
Then
\begin{equation}\label{eq:defect-triangle}
\|q_0\|_{F_0,*}\le A_*+\sqrt{2\kappa}\,C_0M_*.
\end{equation}
In particular, the previously reserved choice of $\kappa$ may be made
small enough that
\begin{equation}\label{eq:defect-estimate}
\|q_0\|_{F_0,*}^2
\le A_*^2+\frac\varepsilon2(A_*^2+M_*^2)
\end{equation}
holds uniformly in $\zeta$ and the minimizing states.
\end{proposition}
\begin{proof}
By Proposition~\ref{prop:metric} and Cauchy--Schwarz, the dual norm of the
first term in \eqref{eq:defect-convolution}, summed over the modes, is at
most $A_*$. For the second term put
\[
J(Z)=\sqrt{\kappa(1-\kappa)}\Tr[(\gamma-\tau_{r_0})Z].
\]
The variance between the two branches of the mixture gives
$|J(Z)|^2\le\|Z\|_{R_{0,+},0}^2$ and
$|J(Z)|^2\le\|Z\|_{R_{0,-},0}^2$ on centered tests.
The two-weight logarithmic-mean operation proved in
Section~\ref{int:operations} applies to this scalar-valued map,
with both target weights equal to $1$.
Since $\ell(R_{0,+},R_{0,-})=1/h_0$, it yields
\[
|J(Z)|^2\le\|Z\|_0^2/h_0
\le C_0^2\|Z\|_{F_0}^2.
\]
Moreover,
$(1-\kappa)|\mu_\gamma|^2\le2\sum_iw_i|\mu_i|^2=2M_*^2$.
The triangle inequality now proves \eqref{eq:defect-triangle}.
Using Young's inequality with coefficient $\varepsilon/2$, it is enough
to impose
\[
0<\kappa\le\frac{\varepsilon^2}{4(\varepsilon+2)C_0^2}
\]
to obtain \eqref{eq:defect-estimate}. The right-hand side depends only on
the reference parameters and $\varepsilon$, which were fixed before
$\kappa$; this requirement is therefore compatible with the parameter
choice in Proposition~\ref{prop:regularized-minimum}.
\end{proof}
==== END The-entropy-photon-number-inequality//sections/04-metric.tex ====
==== BEGIN The-entropy-photon-number-inequality//sections/05-stationarity.tex ====
\section{Stationarity and the thermal limit}\label{sec:stationarity}
For each label $s$, consider the thermal relaxation expression
\begin{equation}\label{eq:thermal-generator}
\delta_s=\mathcal L_s\rho_s,\qquad
\mathcal L_s\rho=
\sum_{j=1}^n\bigl[(r_s+1)\mathcal D[d_{s,j}]\rho
+r_s\mathcal D[d_{s,j}^\dagger]\rho\bigr],
\qquad
\mathcal D[d]\rho=d\rho d^\dagger-\tfrac12\{d^\dagger d,\rho\}.
\end{equation}
This is the standard bosonic thermal-relaxation, or quantum
Ornstein--Uhlenbeck, generator; its entropy production has been studied
in \cite{HuberKoenigVershynina2017,CarlenMaas2017,DePalmaHuber2018}.
Here we use it as a trace-class expression at the regularized minima.
The proof requires the weighted trace identities established below,
rather than a log-Sobolev inequality or differentiation of entropy along
an unbounded semigroup. The geometric weights
give $\mathcal L_s\tau_{r_s}=0$.
\subsection{Weighted traces and the exact stationarity identity}
\begin{lemma}[Generator covariance and logarithmic pairings]
\label{lem:generator-covariance}
At a minimizing pair, $\delta_s$ is traceless and
$W_s^2\delta_sW_s^2$ is trace class. If $\Phi_i$ is the raw channel with
the other two input states fixed, extended linearly to trace-class
operators, then
\begin{equation}\label{eq:generator-covariance}
\delta_0=(1-\kappa)\sum_i\Phi_i(\delta_i).
\end{equation}
The traces against the forms $H_s$ and $W_s^4$ are well defined, and
\begin{equation}\label{eq:logarithmic-slicing}
\Tr\delta_0H_0=(1-\kappa)\sum_i\Tr\delta_iT_iH_0.
\end{equation}
\end{lemma}
\begin{proof}
First consider finite total-number cutoffs, where all products in the
calculation are finite rank. This covariance calculation is linear in
the joint input and applies to arbitrary finite-number-supported joint
operators, without a product assumption. The sum over the three ports of the
coefficient-$1$ terms $\mathcal D[d_{s,j}]$ is invariant under the real
orthogonal mode rotation. Terms acting only on discarded modes vanish
under partial trace. The remaining heat expression is
\[
\mathcal D[d]\rho+\mathcal D[d^\dagger]\rho
=-\tfrac12\bigl([d,[d^\dagger,\rho]]
+[d^\dagger,[d,\rho]]\bigr).
\]
Writing an original mode as its retained component plus discarded
components, every commutator involving a discarded component disappears
under partial trace. The retained part has coefficient $\lambda_s$.
The resulting generator on the raw output therefore has parameter
$\sum_s\lambda_sr_s=r_0$.
We next justify this covariance calculation for the full input state.
Here is a weighted trace argument extending the finite-cutoff calculation.
Suppress a label and put $T=W^3\rho W^3$, a positive trace-class operator
by the finite seventh moment. Each term of $W^2\mathcal L\rho W^2$ is a
finite sum of bounded-factor products around $T$. For example,
\[
W^2d\rho d^\dagger W^2
=(W^2dW^{-3})T(W^{-3}d^\dagger W^2),
\qquad
W^2d^\dagger d\rho W^2=(d^\dagger dW^{-1})TW^{-1}.
\]
The outer factors are bounded because a mode shifts number by one and
has coefficients growing as the square root of number. The creation
terms satisfy the same estimates. For the number cutoff $P_K$,
$W^3P_K\rho P_KW^3=P_KTP_K\to T$ in trace norm. Hence the cutoff
generator expressions converge in the $W^2$-sandwiched trace norm.
In particular they have trace-zero limits.
The same estimates hold on the joint input using
$W_{\mathrm{tot}}=I+N_1+N_2+N_D$. The joint input has finite seventh
moment, and the rotation preserves total-number sectors. Partial trace
is continuous in the required weighted trace norms: for $k=3$ on cutoff
states and $k=2$ on generator expressions, insert the bounded factor
$(W_0^k\otimes I)W_{\mathrm{tot}}^{-k}$, of norm at most one, on both sides
before taking the partial trace. On the full product input the fixed
thermal port has zero generator, as does the replacement thermal.
Consequently the covariance just proved reduces to
\eqref{eq:generator-covariance}.
The form bound $0\le H_s\le C_sW_s^4$ makes
$W_s^{-2}H_sW_s^{-2}$ bounded. Its pairing with
$W_s^2\delta_sW_s^2$ defines the logarithmic trace; $W_s^4$ is treated
the same way. For the slicing assertion, decompose the self-adjoint
trace-class operator $W_i^2\delta_iW_i^2$ into its positive and negative
parts and conjugate back by $W_i^{-2}$. This expresses $\delta_i$ as a
difference of positive operators with finite fourth moment. The positive
slicing identity for $T_iH_0$ applies to each part, proving
\eqref{eq:logarithmic-slicing}.
\end{proof}
Define $\sigma_s=\Tr\delta_sH_s/h_s$. Pair the exact Hamiltonian identity
\eqref{eq:log-identity} with $\delta_i$ and sum over $i$. The scalar terms
vanish by tracelessness, and Lemma~\ref{lem:generator-covariance} gives
\begin{equation}\label{eq:stationarity}
0=\sigma_0-(1+\varepsilon)\sum_iw_i\sigma_i
+\varepsilon\sum_iw_i\Tr\delta_iN_i
+\zeta\sum_i\Tr\delta_iW_i^4.
\end{equation}
This is an algebraic consequence of the Gibbs identity. It requires
neither that $\rho_i+t\delta_i$ remain positive for two-sided $t$, nor
that entropy be differentiated along this unbounded generator.
\begin{lemma}[Entropy production identity]\label{lem:entropy-production}
The quantities in \eqref{eq:stationarity} satisfy
\begin{equation}\label{eq:entropy-production}
\sigma_s=\|q_s\|_{F_s,*}^2-(e'_s-nr_s).
\end{equation}
\end{lemma}
\begin{proof}
In an eigenbasis of $\rho_s$, direct evaluation of the two dissipators
gives
\begin{equation}\label{eq:production-sum}
\sigma_s=-\sum_{j,l,r}v_s(l,r)
\bigl((r_s+1)p_r-r_sp_l\bigr)|(d'_{s,j})_{lr}|^2.
\end{equation}
For example, the contribution of $\mathcal D[d]\rho$ to the trace
against $H$ is
$\sum_{l,r}p_r(-\log p_l+\log p_r)|d_{lr}|^2$.
The creation term is obtained by interchanging $l,r$; centering $d$
changes only diagonal entries, where $v_s(l,l)=0$.
All these expansions are legitimate. Lemma~\ref{lem:dual-finiteness}
controls the crossed and uncrossed logarithmic sums absolutely. In the
anticommutator terms, $d^\dagger d\psi_r$ and $dd^\dagger\psi_r$ belong
to $D(W^2)$ because $\psi_r\in D(W^{7/2})$. The form pairing with
$H\psi_r=(-\log p_r)\psi_r$ is therefore valid. For the eigenbasis truncations $\rho^{(m)}=\sum_{r\le m}p_r
|\psi_r\rangle\langle\psi_r|$, we have
\[
W^3\rho^{(m)}W^3=\sum_{r\le m}p_r
|W^3\psi_r\rangle\langle W^3\psi_r|
\longrightarrow W^3\rho W^3
\quad\text{in trace norm}.
\]
The bounded-factor expansions in
Lemma~\ref{lem:generator-covariance} therefore justify the eigenvector
expansions in the required weighted trace norm as well.
Subtracting \eqref{eq:production-sum} from
\eqref{eq:dual-norm-sum} leaves
\[
\sum_{j,l,r}\bigl((r_s+1)p_r-r_sp_l\bigr)|(d'_{s,j})_{lr}|^2
=(r_s+1)e'_s-r_s(e'_s+n)=e'_s-nr_s,
\]
where the canonical commutation relations give the additional $n$.
This proves \eqref{eq:entropy-production}.
\end{proof}
\subsection{Cancellation and convergence to the thermal inputs}
The number shifts in \eqref{eq:thermal-generator} give
\begin{equation}\label{eq:number-drift}
\Tr\delta_iN_i=nr_i-\Tr\rho_iN_i
=-(e'_i-nr_i)-|\mu_i|^2.
\end{equation}
They also give
\begin{align}
\Tr\delta_iW_i^4&=\Tr\rho_iP_i(N_i),\label{eq:moment-drift}\\
P_i(N_i)&=(r_i+1)N_i\bigl((W_i-1)^4-W_i^4\bigr)
+r_i(N_i+n)\bigl((W_i+1)^4-W_i^4\bigr).\notag
\end{align}
This polynomial has leading term $-4N_i^4$; hence its supremum on
$N_i\in\{0,1,\ldots\}$ is finite. In particular,
\begin{equation}\label{eq:moment-drift-bound}
P:=\sum_i\Tr\delta_iW_i^4\le C,
\end{equation}
with $C$ independent of $\zeta$ and the minimizing states.
Independence across the input ports and $\mu_D=0$ imply
\[
e'_\gamma=\sum_i\lambda_ie'_i+\lambda_Dnr_D,
\qquad \mu_\gamma=\sum_i\sqrt{\lambda_i}\mu_i.
\]
Taking account of the two branches of the replacement gives the exact
centered-energy identity
\begin{equation}\label{eq:centered-energy}
e'_0-nr_0=\sum_iw_i(e'_i-nr_i)
+\kappa(1-\kappa)|\mu_\gamma|^2.
\end{equation}
Insert \eqref{eq:entropy-production}, \eqref{eq:number-drift}, and
\eqref{eq:centered-energy} into \eqref{eq:stationarity}.
All centered-energy terms cancel, yielding, before discarding any term,
\begin{equation}\label{eq:exact-cancellation}
0=\|q_0\|_{F_0,*}^2-(1+\varepsilon)A_*^2
-\varepsilon M_*^2
-\kappa(1-\kappa)|\mu_\gamma|^2+\zeta P.
\end{equation}
Now Proposition~\ref{prop:defect-estimate} and
\eqref{eq:moment-drift-bound} imply
\begin{equation}\label{eq:vanishing-defects}
0\le-\frac\varepsilon2(A_*^2+M_*^2)+C\zeta.
\end{equation}
Thus, as $\zeta\downarrow0$, both input means tend to zero and both
lists of input defect functionals tend to zero in their dual norms.
The uniform energy bound in Proposition~\ref{prop:regularized-minimum}
and Lemma~\ref{lem:energy-compactness} provide a subsequence along which
both inputs converge in trace norm, say to $\widehat\rho_i$.
To identify these limits, fix a matrix $Z$ supported on a finite block of
the number basis. By Proposition~\ref{prop:metric} and the arithmetic
bound for the logarithmic mean, the centered version of $Z$ has
$F_i$ norm at most a fixed constant times $\|Z\|$, uniformly in the
minimizing state. Hence $q_{i,j}(Z)\to0$ by
\eqref{eq:vanishing-defects}. Replacing $d'_{i,j}$ by $d_{i,j}$ changes
this functional by $\overline{(\mu_i)_j}\Tr\rho_iZ$, which also tends
to zero. Both $d_{i,j}^\dagger Z$ and $Zd_{i,j}^\dagger$ are bounded
finite-rank operators for this fixed test. Trace-norm convergence
therefore passes the uncentered trace identity to the limit, giving
\begin{equation}\label{eq:thermal-recurrence}
(r_i+1)d_{i,j}\widehat\rho_i=r_i\widehat\rho_id_{i,j}
\quad\text{as number-basis matrices, for every }j.
\end{equation}
No convergence of energy or of unbounded first moments is used here.
For completeness, \eqref{eq:thermal-recurrence} determines the whole
$n$-mode state. If multi-indices $\mathbf p,\mathbf b$ satisfy $p_j>0$,
its matrix entries obey
\[
(r_i+1)\sqrt{p_j}\,
(\widehat\rho_i)_{\mathbf p,\mathbf b}
=r_i\sqrt{b_j}\,
(\widehat\rho_i)_{\mathbf p-\mathbf e_j,\mathbf b-\mathbf e_j},
\]
with zero right-hand side when $b_j=0$.
If $p_j>b_j$, iteration kills the entry; if $p_j<b_j$, self-adjointness
does the same. Thus all off-diagonal entries vanish.
Diagonal entries have ratio $r_i/(r_i+1)$ whenever any one coordinate
is increased by one. Trace one fixes the normalization, so
$\widehat\rho_i=\tau_{r_i}$.
This conclusion excludes internal correlations as well as off-diagonal
coherences in the limit; no product assumption within an input was made.
The raw output now converges to $\tau_{r_0}$ by continuity of the channel
and Proposition~\ref{prop:thermal-mixing}; the replaced output has the
same limit. Uniformly bounded energies give convergence of all three
entropies by Lemma~\ref{lem:energy-compactness}. The entropy portion of
$J_\zeta$ in \eqref{eq:functional} consequently tends to zero.
Its relative-entropy and fourth-moment penalties are nonnegative, so
\[
\liminf_{\zeta\downarrow0}J_\zeta\ge0
\]
along the selected subsequence. This contradicts the uniform strict
negative upper bound for the minima. It is unnecessary to prove that
either penalty tends to zero, or to send any of the other auxiliary
parameters to zero.
The strict violation assumed in Section~\ref{sec:minimization} is therefore
impossible. Since the reduction there preserved every arbitrary
finite-energy input pair at each fixed finite $n$, this proves
Theorem~\ref{thm:epni} for $0<\eta<1$. At $\eta=0$ or $1$ the retained
state is the corresponding input and the asserted inequality is equality.
==== END The-entropy-photon-number-inequality//sections/05-stationarity.tex ====
==== BEGIN The-entropy-photon-number-inequality//sections/06-broadcast.tex ====
\section{A degraded pure-loss broadcast capacity region}\label{sec:broadcast}
Write \(\mathcal L_\xi=\mathcal E_{\xi,0}\) for the pure-loss attenuator.
Consider the memoryless broadcast channel with one input mode per use,
passively split with vacuum auxiliary inputs between receivers \(B,C\)
and an inaccessible loss output. Let \(\eta_B,\eta_C\) be the power
transmissivities to the two receivers, so the marginal channels are
\(\mathcal L_{\eta_B}\) and \(\mathcal L_{\eta_C}\). We assume
\[
0\le\eta_C<\eta_B,\qquad \eta_B+\eta_C\le1.
\]
For an \(n\)-use code, the messages \(m_B\in\{1,\ldots,M_B\}\) and
\(m_C\in\{1,\ldots,M_C\}\) are independent and uniform. The encoder
may assign to each pair an arbitrary finite-energy \(n\)-mode state
\(\rho_{m_B,m_C}^{(n)}\), including states entangled across uses, subject to
\begin{equation}\label{eq:broadcast-energy}
\frac1{M_BM_C}\sum_{m_B,m_C}
\Tr\!\left(\rho_{m_B,m_C}^{(n)}
\sum_{j=1}^n a_j^\dagger a_j\right)
\le n\bar N.
\end{equation}
Thus the photon constraint is averaged over uses and the entire
codebook, not imposed separately on each codeword. Each receiver uses
an arbitrary collective measurement on its own \(n\) output modes to
recover its intended message, with vanishing average error. The
receivers do not cooperate, and there is no feedback.
\begin{corollary}[Degraded two-receiver pure-loss broadcast capacity]
\label{cor:broadcast-capacity}
Let \(0\le\bar N<\infty\), \(0\le\eta_C<\eta_B\), and
\(\eta_B+\eta_C\le1\). The classical capacity region for the channel
and coding model above, with rates in nats per use, is the closed
convex hull of the nonnegative pairs \((R_B,R_C)\) satisfying, for
some \(0\le\beta\le1\),
\begin{equation}\label{eq:broadcast-region}
\begin{aligned}
R_B&\le g(\eta_B\beta\bar N),\\
R_C&\le g(\eta_C\bar N)-g(\eta_C\beta\bar N),
\end{aligned}
\end{equation}
where \(g\) is the natural-log function in~\eqref{eq:g-definition}.
\end{corollary}
\begin{proof}
\emph{Converse.}
Set \(\lambda=\eta_C/\eta_B\in[0,1)\). Composition of pure-loss
attenuators gives
\(\mathcal L_{\eta_C}=\mathcal L_\lambda\circ\mathcal L_{\eta_B}\).
This is a degrading channel for the receiver marginals, implemented
with a new independent vacuum port; it does not assert independence
of the physical \(B\) and \(C\) outputs.
For every finite \(n\) and every finite-energy \(n\)-mode state
\(\sigma\), Theorem~\ref{thm:epni} with an independent vacuum port,
equivalently Corollary~\ref{cor:thermal-attenuator} at \(N_B=0\), gives
\begin{equation}\label{eq:broadcast-vacuum-bound}
S\!\left(\mathcal L_\lambda^{\otimes n}(\sigma)\right)
\ge n g\!\left(\lambda g^{-1}\!\left(\frac{S(\sigma)}n\right)\right).
\end{equation}
This supplies the finite-block vacuum-port inequality required from
Strong Conjecture~2 in Guha, Shapiro, and Erkmen's broadcast analysis
\cite[Appendix~A]{GSE2007} on all states needed here. It does not
require \(\sigma\) to be a product across modes.
We follow the converse of Guha, Shapiro, and Erkmen~\cite{GSE2007},
giving the averaging and information argument for the present coding
model. For a fixed \(n\)-use code, let \(\sigma_{mk}^Y\) be the output
at receiver \(Y\in\{B,C\}\) for messages \(m=m_B\), \(k=m_C\), and set
\[
\sigma_k^Y=\frac1{M_B}\sum_m\sigma_{mk}^Y,\qquad
\bar\sigma^C=\frac1{M_C}\sum_k\sigma_k^C,\qquad
\bar N_k=\frac1{nM_B}\sum_m\Tr\rho_{m,k}^{(n)}N.
\]
The codebook constraint gives \(M_C^{-1}\sum_k\bar N_k\le\bar N\).
For every finite-energy input, a vacuum attenuator satisfies
\[
\Tr\bigl(\mathcal L_\xi^{\otimes n}(\rho)N\bigr)=\xi\Tr\rho N:
\]
expanding each output number operator, the independent vacuum has zero
number and first moments, so its number term and both cross terms vanish.
This calculation allows arbitrary entanglement across input modes.
Thus \(\sigma_k^B\) has energy \(n\eta_B\bar N_k\), while
\(\bar\sigma^C\) has energy at most \(n\eta_C\bar N\). All conditional
states have finite energy and entropy, and linearity and degradation give
\(\sigma_k^C=\mathcal L_\lambda^{\otimes n}(\sigma_k^B)\).
Let \(p_{B,n},p_{C,n}\) be the intended-message average error
probabilities, and write \(R_{Y,n}=n^{-1}\log M_Y\). Fano's inequality
and the Holevo bounds used in the cited converse give
\[
\begin{aligned}
(1-p_{B,n})R_{B,n}
&\le \frac1{nM_C}\sum_kS(\sigma_k^B)+\frac{H_2(p_{B,n})}{n},\\
(1-p_{C,n})R_{C,n}
&\le \frac{S(\bar\sigma^C)}n-\frac1{nM_C}\sum_kS(\sigma_k^C)
+\frac{H_2(p_{C,n})}{n},
\end{aligned}
\]
where \(H_2\) is the binary entropy. For the first bound, reveal the
independent weak message as side information to the strong decoder and
discard the nonnegative entropies of the individual output states in the
conditional Holevo quantity. This only enlarges the information used for
the upper bound; it does not change the decoder required by the model.
The second bound is the Holevo bound for the weak message.
Assume \(\bar N>0\) for now, and put
\[
u_{k,n}=\frac{S(\sigma_k^B)}n,\qquad
\bar u_n=\frac1{M_C}\sum_k u_{k,n}.
\]
The energy--entropy bound and concavity of \(g\), whose second derivative
is \(-1/[r(1+r)]\) for \(r>0\), imply
\[
0\le\bar u_n\le\frac1{M_C}\sum_k g(\eta_B\bar N_k)
\le g(\eta_B\bar N).
\]
There is therefore a code-dependent \(\beta_n\in[0,1]\) with
\(\bar u_n=g(\eta_B\beta_n\bar N)\).
Set \(\phi_\lambda(u)=g(\lambda g^{-1}(u))\). This function is convex.
Indeed, for \(0<\lambda<1\) and \(r>0\), differentiation using \(g\)
and \(h=g'\) gives
\[
\phi_\lambda''(g(r))
=\frac{\lambda\bigl[(1+\lambda r)h(\lambda r)-(1+r)h(r)\bigr]}
{r(1+r)(1+\lambda r)h(r)^3}\ge0,
\]
because \(t\mapsto(1+t)h(t)\) has derivative \(h(t)-1/t<0\).
Continuity includes \(u=0\), and \(\phi_0=0\).
Applying~\eqref{eq:broadcast-vacuum-bound} to each conditional state and
then Jensen's inequality gives
\[
\frac1{nM_C}\sum_k S(\sigma_k^C)
\ge\frac1{M_C}\sum_k\phi_\lambda(u_{k,n})
\ge\phi_\lambda(\bar u_n)
=g(\eta_C\beta_n\bar N).
\]
Also \(S(\bar\sigma^C)\le ng(\eta_C\bar N)\). Substituting these bounds
into the information inequalities yields
\[
\begin{aligned}
(1-p_{B,n})R_{B,n}
&\le g(\eta_B\beta_n\bar N)+\frac{H_2(p_{B,n})}{n},\\
(1-p_{C,n})R_{C,n}
&\le g(\eta_C\bar N)-g(\eta_C\beta_n\bar N)
+\frac{H_2(p_{C,n})}{n}.
\end{aligned}
\]
For any achieved rate pair, take a subsequence on which
\(\beta_n\) converges in \([0,1]\). Continuity of \(g\), the rate lower
limits, and vanishing decoding errors give the outer bound
in~\eqref{eq:broadcast-region} with the limiting \(\beta\).
\emph{Achievability.}
We use the coherent Gaussian superposition ensemble of Guha, Shapiro,
and Erkmen~\cite[Section~IV]{GSE2007}. To justify its bosonic limit and
the photon constraint, we start with the finite-alphabet superposition
construction in the proof of Yard, Hayden, and Devetak's Theorem~1
\cite[Section~II.A]{YHD2011}, expressed here in nats. For a finite
classical input alphabet, finite-dimensional quantum outputs, and a
finite law \(p(t,x)\), that construction sends a common message to both
receivers and a private message to \(B\) at the strict rates
\[
R_B<\sum_t p(t)\chi\bigl(p(x\mid t),\omega_x^B\bigr),
\qquad
R_C<\min_{Y=B,C}\chi\bigl(p(t),\omega_t^Y\bigr),
\]
where \(\omega_t^Y=\sum_xp(x\mid t)\omega_x^Y\) and
\[
\chi(p_j,\omega_j)
=S\!\left(\sum_jp_j\omega_j\right)-\sum_jp_jS(\omega_j).
\]
We use the common message as the message for \(C\); receiver \(B\) may discard
its decoded common label. Degradation and data processing make the
minimum in the common-message bound equal to the \(C\) quantity.
Assume first that \(\bar N>0\) and \(\eta_C>0\). Fix
\(0<N'<\bar N\) and \(0<\beta<1\). Let \(T,Z\) be independent centered
circular complex Gaussian variables of variances \((1-\beta)N'\) and
\(\beta N'\), respectively. Partition the disk \(\{|z|\le\ell\}\)
into finitely many sets of diameter at most \(1/\ell\), with its
complement as one further cell, and let \(T_\ell,Z_\ell\) be the
corresponding conditional expectations of \(T,Z\). For either
\(X=T,Z\) and its corresponding \(X_\ell\),
\[
\mathbb E|X-X_\ell|^2
\le\ell^{-2}+\mathbb E\!\left[|X|^2\mathbf1_{\{|X|>\ell\}}\right]
\longrightarrow0.
\]
They remain centered and independent, and
conditional expectation contracts second moments. Hence
\[
\mathbb E|T_\ell+Z_\ell|^2
=\mathbb E|T_\ell|^2+\mathbb E|Z_\ell|^2\le N'.
\]
For a symbol \(x=(t,z)\) of their finite product alphabet, transmit
the coherent state \(|t+z\rangle\). The finite law in the coding
construction is
\(p(t,(t',z))=\mathbf1_{\{t=t'\}}p_{T_\ell}(t)p_{Z_\ell}(z)\).
At receiver transmissivity
\(\xi\in\{\eta_B,\eta_C\}\), put
\[
\theta_{\ell,\xi}
=\mathbb E|\sqrt\xi Z_\ell\rangle\langle\sqrt\xi Z_\ell|,
\qquad
\omega_{\ell,\xi}
=\mathbb E|\sqrt\xi(T_\ell+Z_\ell)\rangle
\langle\sqrt\xi(T_\ell+Z_\ell)|.
\]
Conditioned on \(T_\ell=t\), the receiver state is a unitary
displacement of \(\theta_{\ell,\xi}\), and the output for each symbol
is pure. The private-message Holevo quantity at \(B\) and the
common-message Holevo quantity at \(C\) are therefore
\begin{equation}\label{eq:broadcast-holevo}
S(\theta_{\ell,\eta_B}),\qquad
S(\omega_{\ell,\eta_C})-S(\theta_{\ell,\eta_C}).
\end{equation}
The coherent-projector identity
\[
\bigl\||u\rangle\langle u|-|v\rangle\langle v|\bigr\|_1
=2\sqrt{1-e^{-|u-v|^2}}\le2|u-v|
\]
and Cauchy--Schwarz show that, in trace norm,
\(\theta_{\ell,\xi}\) tends to the one-mode thermal state of mean
photon number \(\xi\beta N'\), and \(\omega_{\ell,\xi}\) to the
one-mode thermal state of mean photon number \(\xi N'\). Their mean
photon numbers are bounded by \(\xi\beta N'\) and \(\xi N'\),
respectively. Lemma~\ref{lem:energy-compactness} thus gives convergence
of the quantities in~\eqref{eq:broadcast-holevo} to
\[
g(\eta_B\beta N'),\qquad
g(\eta_C N')-g(\eta_C\beta N').
\]
For each fixed \(\ell\), the finitely many coherent receiver vectors
span finite-dimensional spaces. The physical channel on these symbols
is therefore exactly a finite classical-input channel with such output
spaces. Decoding measurements on those spaces extend to Fock space
without changing their action on the transmitted states.
It remains to enforce~\eqref{eq:broadcast-energy} on a deterministic
code. For the fixed finite alphabet let
\[
c(x)=|t+z|^2,\qquad
c_t=\sum_xp(x\mid t)c(x),\qquad
\bar c=\sum_t p(t)c_t\le N',\qquad
c_{\max}=\max_x c(x).
\]
Fix a target rate pair strictly below the two finite Holevo bounds.
Choose the type tolerance \(\delta>0\) small enough for this rate
backoff and so that
\(\bar c+\delta c_{\max}\le N''<\bar N\) for some \(N''\).
The cited superposition construction supplies a sequence of fixed outer
codes whose words have a common type \(P_n\) at block length \(n\), with
\(\|P_n-p_T\|_1\le\delta\), where \(p_T(t)=p(t)\).
The random inner letters in the corresponding blocks have law
\(p(x\mid t)\), and the complete words are obtained by permutations.
With each outer code fixed, all expectations and probabilities below
are over these inner codebooks. Thus the photon cost \(\mathsf C_n\)
per use, averaged over the entire random codebook, satisfies
\[
\mathbb E\mathsf C_n=\sum_tP_n(t)c_t
\le\bar c+\delta c_{\max}\le N''.
\]
Before its final message expurgation, the cited construction also gives
a vanishing expected bound on its average joint decoding error over the
same inner codebooks. Let \(\mathsf E_n\) be the sum of the two
intended-message average errors, with \(B\)'s decoded common label
discarded. Each is bounded by that joint error, so, after absorbing a
factor of two, \(\mathbb E\mathsf E_n\le e_n\) for a positive sequence
\(e_n\to0\). We use this stage because its average-error estimate
suffices for the stated criterion and its full product message set is
the one over which \(\mathsf C_n\) was computed. Markov's inequality gives
\[
\Pr\{\mathsf C_n>\bar N\}
+\Pr\{\mathsf E_n>\sqrt{e_n}\}
\le\frac{N''}{\bar N}+\sqrt{e_n}<1
\]
for all sufficiently large \(n\). Hence a deterministic code satisfies
the entire-codebook photon constraint and has vanishing average error.
It retains the full uniform independent product message set. Each
codeword is a product of finitely many coherent symbols and has finite
energy, and \(\mathsf E_n\) controls each receiver's intended-message
error.
For \(0<\beta<1\), fix a rate pair strictly below the corresponding
Gaussian pair in~\eqref{eq:broadcast-region}. First choose
\(N'<\bar N\) close enough that its Gaussian pair still exceeds the
target, and then choose a finite \(\ell\) whose Holevo pair does so.
For this fixed alphabet choose the type tolerance and cost slack as
above, and only then let the block length grow. Taking closure reaches
the boundary and includes \(\beta=0,1\), and time sharing
among codes satisfying the same photon bound gives the closed convex
hull.
If \(\bar N=0\), every codeword is the vacuum and both rates are zero.
If \(\eta_C=0\), receiver \(C\) always receives the vacuum and the
region reduces to \(R_C=0\), \(R_B\le g(\eta_B\bar N)\), the
single-receiver pure-loss capacity \cite{GGLMSY2004}. Both endpoints
are included in~\eqref{eq:broadcast-region}.
\end{proof}
Corollary~\ref{cor:broadcast-capacity} concerns only the stated
two-receiver pure-loss classical coding model. It gives no
noisy-broadcast, amplifier, wiretap, quantum-capacity, feedback, or
receiver-cooperation statement.
==== END The-entropy-photon-number-inequality//sections/06-broadcast.tex ====