Context and earlier results

Shannon introduced entropy power in his theory of communication [28]; the Fisher-information proofs of Stam and Blachman [30, 4] 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 1. The formulation of Guha, Erkmen, and Shapiro [20] 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 [18] and connected noisy-channel capacity questions with minimum output entropy [17]. The constrained minimum-output problem also appears in the bosonic broadcast-channel analysis of Guha, Shapiro, and Erkmen [21]. Early EPnI special cases included one-mode number-diagonal inputs with two nonzero probabilities and a vacuum second port [6].

König and Smith [24] 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 [10] proved the exponential bound for every transmissivity, \[ e^{S(\rho_C)/n} \ge\eta e^{S(\rho_A)/n}+(1-\eta)e^{S(\rho_B)/n}. \tag{5}\] 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 [11] 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 (5) for arbitrary independent finite-energy inputs [12].

The entropy photon-number inequality requires a sharper comparison than (5) 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 [7].

The Gaussian optimizer theorem of Giovannetti, Holevo, and García-Patrón [19] 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 [13], then the sharp entropy bound for the quantum-limited attenuator [14], and finally the sharp bound for one-mode phase-covariant Gaussian channels with thermal noise [15]. Their multimode extension [16] covers inputs diagonal in some product basis, including classical correlations in that basis.

Later advances address further parts of this landscape. De Palma [8] 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 [3] 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 [2] established subset-convolution inequalities for exponential entropy power and entropy monotonicity under repeated symmetric quantum convolution of an identical input.

Theorem 1 supplies the thermal comparison for two freely varying inputs, including arbitrary entanglement within either port. Corollary 2 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 1 as reference states and as limits of regularized minimizers. The original inputs need not be Gaussian, number-diagonal, or independent across modes.

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 3. 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 4 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 at those minimizers. It is not asserted as a universal channel inequality.

Section 5 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 6, 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 2 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.

Entropy estimates and compactness

We first record the entropy facts needed in the infinite-dimensional argument. The relative entropy introduced by Umegaki [31] 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 [29, 32]; we include the elementary cutoff proof used here.

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 \[ D(\rho\Vert\tau_r)=-S(\rho)+n\log(r+1)+h(r)\mathop{\mathrm{Tr}}\rho N. \tag{9}\]

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.

This thermal representation and its relation to entropy photon number are also the starting point of the original formulation [20]. In the argument below it is only the auxiliary reference states that have this special form.

Operations that preserve comparison

The operations below belong to the parallel-sum and operator-mean calculus of Anderson–Duffin and Kubo–Ando [1, 25]. 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 (11). 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 parallel sum \[ X:Y=(X^{-1}+Y^{-1})^{-1}. \tag{14}\] 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 \[ \frac1{\ell(X,Y)}=\int_0^1\frac{dt}{(1-t)X+tY}, \qquad \frac{XY}{\ell(X,Y)}=\int_0^1\frac{dt}{t/X+(1-t)/Y}. \tag{15}\] 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 (12). Since their logarithmic mean is \(m\), we obtain comparison for the three families \[ m,\qquad ms,\qquad mt, \qquad s=f(x)^2,\quad t=f(y)^2. \tag{16}\] The next step constructs one further weight from these three.

A Stieltjes representation

The function \(x\mapsto f(x)^2\) decreases from \(1\) to \(0\) on \([0,\infty)\). Define \(B\) on \((0,1]\) by \[ B\bigl(f(x)^2\bigr)=b(x). \tag{17}\] The scalar quadratic (27), 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 \(0<s,t<1\), \[ \frac1E=\frac2m\int_0^1 -B'\bigl(\theta s+(1-\theta)t\bigr)\,d\theta, \qquad s=f(x)^2,\quad t=f(y)^2. \tag{18}\] 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.

The additional weight and the final minimization

The lemma extends the same integral definition (18) continuously and positively to all \(s,t>0\), including the values used in (16). The fundamental theorem of calculus gives \[ E= \begin{cases} \displaystyle\frac{m(t-s)}{2(B(s)-B(t))},&s\ne t,\\[6pt] \displaystyle-\frac{m}{2B'(s)},&s=t. \end{cases} \tag{26}\] Lemma 7 shows that \(E\) is obtainable by the comparison-preserving operations already established. Indeed, after substitution of (19), 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 6 by an exact scalar minimization. For \(\xi,\phi\in\mathbb C\) put \[ \mathcal Q(\xi,\phi)= \frac14\sum_{\pm}mf(x)e^{\pm x}|\xi\mp\phi|^2 +\frac14\sum_{\pm}mf(y)e^{\pm y}|\xi\pm\phi|^2 +E|\phi|^2. \tag{27}\] 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 \[ \phi=\frac{x-y}{a+2e}\xi, \qquad \min_\phi\mathcal Q(\xi,\phi) =\frac m2\left(a-\frac{(x-y)^2}{a+2e}\right)|\xi|^2. \tag{28}\] If \(x^2\ne y^2\), the identity \(b(z)^2=f(z)^2+z^2\) and (26) give \[(a+2e)(b(x)-b(y))=x^2-y^2.\] Hence the minimum in (28) is \[ m\frac{y b(x)+x b(y)}{x+y}|\xi|^2 =m\frac{f(x)f(y)}{f(x+y)}|\xi|^2, \tag{29}\] 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 (26) and (20) yield \[e=\frac{s(b-1)}{b-s},\qquad b+e=\frac{x^2}{b-s}.\] Thus (28) 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 (29) in every case.

For a source test \(Z\), let \(\Phi\) be its coordinatewise minimizer in (28). 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 (29), 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 6.

The auxiliary ports and the functional

Suppose Theorem 1 fails. The endpoint transmissions give equality, so \(0<\eta<1\). By Lemma 4, 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 \[ \begin{split} J_\zeta(\rho_1,\rho_2) ={}&\frac{S(\rho_0)-ng(r_0)}{h_0} -\sum_i w_i\frac{S(\rho_i)-ng(r_i)}{h_i}\\ &+\varepsilon\sum_i\frac{w_i}{h_i} D(\rho_i\Vert\tau_{r_i}) +\zeta\sum_i\mathop{\mathrm{Tr}}\rho_iW_i^4. \end{split} \tag{30}\]

The normalization of the relative-entropy penalty also matches the closing stationarity calculation. The thermal entropy formula and (9) give \[\frac{D(\rho\Vert\tau_r)}{h(r)} =-\frac{S(\rho)-ng(r)}{h(r)}+\bigl(\mathop{\mathrm{Tr}}\rho N-nr\bigr).\] 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 4.

Next choose \(\varepsilon>0\) small enough to retain negativity and to have \[ (1+\varepsilon)\sum_i\lambda_ir_i\le r_0, \qquad (1+\varepsilon)\sum_i\lambda_i(r_i+1)\le r_0+1. \tag{31}\] 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 5 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 \[ J_\zeta(\rho_1^{\mathrm{w}},\rho_2^{\mathrm{w}})\le -c \qquad(0<\zeta\le\zeta_0). \tag{32}\] All parameters except \(\zeta\) remain fixed from now on.

A Gibbs regularity lemma

The fourth-moment penalty yields more regularity than its definition alone suggests. We record the needed form argument explicitly.

Attainment and the exact logarithm identity

The component entropy-Hessian estimate

Work at one of the minima in Proposition 9. 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 \[ \|Z\|_s^2 =\sum_{l,r}h_s\ell(p_{s,l},p_{s,r})|Z_{lr}|^2, \qquad v_s(l,r)=\frac{\log(p_{s,l}/p_{s,r})}{h_s}. \tag{37}\] 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 [27, 26]. 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 \(\langle Z\rangle_s=\mathop{\mathrm{Tr}}\rho_sZ\). Set \[ L_iZ=\frac{T_iZ-\langle T_iZ\rangle_i I}{\sqrt{\lambda_i}}. \tag{38}\] 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.

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 \[ h_s\ell(p_l,p_r)R_{s,+}(v_s(l,r))=p_l, \qquad h_s\ell(p_l,p_r)R_{s,-}(v_s(l,r))=p_r. \tag{43}\] Thus the corresponding squared norms of a centered observable are \(\mathop{\mathrm{Tr}}\rho_sZZ^\dagger\) and \(\mathop{\mathrm{Tr}}\rho_sZ^\dagger Z\).

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 \(T_iZ-\mathop{\mathrm{Tr}}\gamma Z\,I\) 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 (42) 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 10 and (31) 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 \(r_s+1\).

The thermal weight has a useful logarithmic-mean interpretation: \[ \ell(p_l,p_r)F_s(v_s(l,r)) =\ell\bigl(r_sp_l,(r_s+1)p_r\bigr). \tag{45}\] Indeed the logarithmic difference on the right is \(h_s(v_s(l,r)-1)\); substitution in (44) 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 [5]. Proposition 11 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 10 extend Proposition 11 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.

The thermal defect functionals

Let \(d_{s,j}\) be the annihilator of mode \(j\) on port \(s\), and put \[\mu_s=(\mathop{\mathrm{Tr}}\rho_s d_{s,j})_{j=1}^n,\qquad d'_{s,j}=d_{s,j}-(\mu_s)_jI,\qquad e'_s=\mathop{\mathrm{Tr}}\rho_sN_s-|\mu_s|^2.\] For bounded \(Z\), define \[ q_{s,j}(Z)=(r_s+1)\mathop{\mathrm{Tr}}\rho_s(d'_{s,j})^\dagger Z -r_s\mathop{\mathrm{Tr}}\rho_sZ(d'_{s,j})^\dagger. \tag{46}\] 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 \(\mathop{\mathrm{Tr}}\rho N<\infty\); 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.

Weighted traces and the exact stationarity identity

Define \(\sigma_s=\mathop{\mathrm{Tr}}\delta_sH_s/h_s\). Pair the exact Hamiltonian identity (33) with \(\delta_i\) and sum over \(i\). The scalar terms vanish by tracelessness, and Lemma 15 gives \[ 0=\sigma_0-(1+\varepsilon)\sum_iw_i\sigma_i +\varepsilon\sum_iw_i\mathop{\mathrm{Tr}}\delta_iN_i +\zeta\sum_i\mathop{\mathrm{Tr}}\delta_iW_i^4. \tag{55}\] 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.

Cancellation and convergence to the thermal inputs

The number shifts in (52) give \[ \mathop{\mathrm{Tr}}\delta_iN_i=nr_i-\mathop{\mathrm{Tr}}\rho_iN_i =-(e'_i-nr_i)-|\mu_i|^2. \tag{58}\] They also give \[\begin{align*} \mathop{\mathrm{Tr}}\delta_iW_i^4&=\mathop{\mathrm{Tr}}\rho_iP_i(N_i),\tag{59}\\ P_i(N_i)&=(r_i+1)N_i\bigl((W_i-1)^4-W_i^4\bigr) +r_i(N_i+n)\bigl((W_i+1)^4-W_i^4\bigr). \end{align*}\] This polynomial has leading term \(-4N_i^4\); hence its supremum on \(N_i\in\{0,1,\ldots\}\) is finite. In particular, \[ P:=\sum_i\mathop{\mathrm{Tr}}\delta_iW_i^4\le C, \tag{60}\] 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 \[ e'_0-nr_0=\sum_iw_i(e'_i-nr_i) +\kappa(1-\kappa)|\mu_\gamma|^2. \tag{61}\] Insert (56), (58), and (61) into (55). All centered-energy terms cancel, yielding, before discarding any term, \[ 0=\|q_0\|_{F_0,*}^2-(1+\varepsilon)A_*^2 -\varepsilon M_*^2 -\kappa(1-\kappa)|\mu_\gamma|^2+\zeta P. \tag{62}\] Now Proposition 14 and (60) imply \[ 0\le-\frac\varepsilon2(A_*^2+M_*^2)+C\zeta. \tag{63}\] 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 9 and Lemma 4 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 11 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 (63). Replacing \(d'_{i,j}\) by \(d_{i,j}\) changes this functional by \(\overline{(\mu_i)_j}\mathop{\mathrm{Tr}}\rho_iZ\), 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 \[ (r_i+1)d_{i,j}\widehat\rho_i=r_i\widehat\rho_id_{i,j} \quad\text{as number-basis matrices, for every }j. \tag{64}\] No convergence of energy or of unbounded first moments is used here.

For completeness, (64) 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 5; the replaced output has the same limit. Uniformly bounded energies give convergence of all three entropies by Lemma 4. The entropy portion of \(J_\zeta\) in (30) 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 4 is therefore impossible. Since the reduction there preserved every arbitrary finite-energy input pair at each fixed finite \(n\), this proves Theorem 1 for \(0<\eta<1\). At \(\eta=0\) or \(1\) the retained state is the corresponding input and the asserted inequality is equality.