Photon-number entropy and bosonic communication

Source and verification

This article explains family 273 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

🏷️ Bosonic modes and thermal entropy

A bosonic mode is a quantum harmonic oscillator. For $n$ modes, the Hilbert space is ${\ell }^2({\mathbb{N}}_0^n)$, with basis vectors $|k_1,\dots ,k_n⟩$ recording occupation numbers. The annihilation operator $d_j$ lowers $k_j$ by one with coefficient $\sqrt{k_j}$; its adjoint raises it. The number operator $N={\sum }_j{d}_j^{\ast }{d}_j$ measures total photon number. A state $\rho$ is a positive trace-class operator of trace one; finite energy means $\mathrm{Tr}\rho N<\infty$.

Its von Neumann entropy is $S(\rho )=-\mathrm{Tr}\rho \log \rho$, with natural logarithms throughout. A useful reference state is the product thermal state with mean $r$ in every mode:

$$\begin{array}{c}h(r)=\log (1+1/r),\\ {\tau }_r=(r+1{)}^{-n}{e}^{-h(r)N},\\ g(r)=(r+1)\log (r+1)-r\log r,\\ S({\tau }_r)=ng(r),\mathrm{Tr}{\tau }_{r}N=nr.\end{array}$$

At $r=0$, use the vacuum state and $g(0)=0$. The strictly increasing function $g$ maps $[0,\infty )$ onto itself. The entropy photon number $\mathcal{N}(\rho )=g^{-1}(S(\rho )/n)$ is therefore the thermal mean that reproduces the state’s entropy. It need not equal its actual mean photon number: a pure state has zero entropy photon number even when its energy is large.

The relative entropy $D(\rho \Vert \sigma )=\mathrm{Tr}\rho (\log \rho -\log \sigma )$ is nonnegative whenever the expression is finite. Comparing with a thermal state gives $S(\rho )\le ng(E/n)$ when $\mathrm{Tr}\rho N\le E$. This explains why thermal states represent the largest entropy compatible with a given energy.

🏷️ Independent ports and the sharp inequality

A beam splitter mixes two independent groups of $n$ modes. If their annihilation operators are $a_j,b_j$, the retained output modes are $c_j=\sqrt{\eta }{a}_j+\sqrt{1-\eta }{b}_j$, where $0\le \eta \le 1$. This is a unitary rotation on the combined system followed by discarding the other output. The joint input is ${\rho }_A\otimes {\rho }_B$. Modes within $A$, or within $B$, may be correlated or entangled.

Entropy photon-number inequality, as stated in the manuscript

For every finite positive number of modes and independent finite-energy input states, the retained state satisfies

$$\begin{array}{rl}\mathcal{N}({\rho }_C)& \ge \eta \mathcal{N}({\rho }_A)\\ & +(1-\eta )\mathcal{N}({\rho }_B).\end{array}$$

The family manuscript claims a proof of this inequality in the stated generality. (OpenAI, 2026) Independent thermal inputs ${\tau }_{r_A},{\tau }_{r_B}$ give output ${\tau }_{\eta r_A+(1-\eta )r_B}$, so they attain equality even when their temperatures differ. The endpoints $\eta =0,1$ simply retain one input. These facts do not classify every equality case.

For comparison, an entropy bound linear in $S({\rho }_A),S({\rho }_B)$ gives the same conclusion when their entropies coincide. Unequal thermal means demand the nonlinear function $g^{-1}$: matching the exact thermal output is essential for an entropy-constrained optimization problem.

🏷️ Compactness in an infinite-dimensional state space

The proof must control unbounded number operators and entropy without treating a finite-dimensional calculation as automatically valid in Fock space. The starting lemma says that a fixed energy ball is compact in trace norm and entropy is continuous on it.

Let $P_K$ project onto total number at most $K$, and put $q=\mathrm{Tr}\rho (I-P_K)$. Then $q\le E/(K+1)$. The off-diagonal blocks have trace norm at most $\sqrt{q}$ by Hilbert–Schmidt Cauchy–Schwarz, so finite-dimensional compressions approximate the whole energy ball uniformly. Lower semicontinuity of the energy makes the ball closed.

For entropy, pinching into the low- and high-number blocks changes entropy by at most the binary entropy $H_2(q)$. The weighted tail entropy is at most $qng(E/(qn))$, which tends to zero with $q$. Entropy on the remaining finite-dimensional block is continuous. These estimates give the needed uniform approximation. They also show that a hypothetical strict violation survives normalized finite-number cutoffs. No convergence of the unbounded energy expectation is required later.

🏷️ A regularized negative minimum

The manuscript argues by contradiction. Start with a finite-support violating pair. Introduce a small independent thermal port $D$ and weights ${\lambda }_1=(1-{\lambda }_D)\eta$, ${\lambda }_2=(1-{\lambda }_D)(1-\eta )$. Choose positive reference means $r_1,r_2,r_D$, and put $r_0={\sum }_{s=1,2,D}{\lambda }_s{r}_s$. If $\gamma$ is the raw retained output, replace a small fraction by its reference thermal:

$$\begin{array}{c}{\rho }_0=(1-\kappa )\gamma +\kappa {\tau }_{r_0},\\ w_i=(1-\kappa ){\lambda }_i.\end{array}$$

Write $h_s=h(r_s)$, $W_i=I+N_i$, and introduce the thermal entropy deviations $A_s=[S({\rho }_s)-ng(r_s)]/h_s$. The auxiliary objective is

$$\begin{array}{rl}{J}_{\zeta }=& A_0-\sum _{i=1}^2{w}_i{A}_i\\ & +\epsilon \sum _{i=1}^2\frac{w_i}{h_i}D({\rho }_i\Vert {\tau }_{r_i})\\ & +\zeta \sum _{i=1}^2\mathrm{Tr}{\rho }_i{W}_i^4.\end{array}$$

The references, thermal-port weight, $\epsilon >0$, and replacement fraction $\kappa >0$ are chosen in that order, sufficiently small where needed to preserve a fixed negative witness value. They are then held fixed while $\zeta \downarrow 0$. The thermal port supplies strict slack for two weighted comparisons used below.

Each term has a separate purpose. The relative-entropy penalty contributes a positive linear energy term, which dominates the logarithmic growth of entropy and gives a uniform energy bound. The fourth-moment penalty gives existence and sufficient regularity at each fixed $\zeta$. The replacement makes the output faithful and bounds its negative logarithm by a constant times $W_0$.

At a minimum, the inputs have Gibbs representations and finite seventh number moments. These higher moments need not be uniform as $\zeta$ vanishes. The Gibbs identity is obtained by replacing the output entropy with its cross-entropy majorant at the minimizing output, then minimizing the resulting input Gibbs functional. This avoids differentiating entropy at an arbitrary singular infinite-dimensional state.

🏷️ Entropy curvature and the thermal defect

For positive numbers, the logarithmic mean is $\ell (a,b)=(a-b)/(\log a-\log b)$, extended by $\ell (a,a)=a$. In an eigenbasis of a faithful minimizing state, with eigenvalues $p_{s,l}$, define

$$\begin{array}{c}\Vert Z{\Vert }_s^2=\sum _{l,r}{h}_s\ell (p_{s,l},p_{s,r})|Z_{lr}{|}^2,\\ v_s(l,r)=h_s^{-1}\log (p_{s,l}/p_{s,r}).\end{array}$$

These eigenbases depend on the state and generally differ from the number basis. The dual quadratic form is $1/h_s$ times the Hessian of relative entropy. If $T_i$ is the observable slice of the raw mixing channel with other inputs fixed, set $L_{i}Z=(T_{i}Z-\mathrm{Tr}({\rho }_i{T}_{i}Z)I)/\sqrt{{\lambda }_i}$. At the minimum, the manuscript obtains $\Vert L_{i}Z{\Vert }_i^2\le (1+\epsilon )(1-\kappa {)}^{-1}\Vert Z{\Vert }_0^2$ on centered observables.

The reason is concrete: varying one input cancels every linear term in the objective difference, leaving a positive multiple of input relative entropy minus output relative entropy. Differentiate on finite input eigenblocks; use the Gibbs variational formula to test the actual output perturbation against finite eigenblock observables; then optimize the resulting quadratic inequality. Density extends it to completed weighted spaces. This is an estimate at the auxiliary minima, not a universal contraction assertion for arbitrary input states.

A four-weight interpolation lemma combines this entropy-Hessian estimate with the two arithmetic variance comparisons. Its proof uses positive sums, limits, and variational parallel sums of quadratic forms. The scalar interpolation function is $f(x)=x/\sinh x$, with $f(0)=1$. Its resulting positive frequency weight can be characterized by

$$\begin{array}{rl}& \ell (p_{s,l},p_{s,r})F_s(v_s(l,r))\\ & =\ell (r_s{p}_{s,l},(r_s+1)p_{s,r}).\end{array}$$

For fixed $r_s>0$, $F_s$ is bounded above and below by positive constants. Let $\Vert Z{\Vert }_{F_s}^2$ be the preceding observable norm with the additional factor $F_s(v_s(l,r))$ in each summand, and let the star denote the dual norm of a functional. The interpolation lemma is a substantive analytic input; the few norm formulas here do not replace its proof.

To measure deviation from a displaced thermal state, put ${\mu }_{s,j}=\mathrm{Tr}{\rho }_s{d}_{s,j}$ and $d_{s,j}^{\prime }=d_{s,j}-{\mu }_{s,j}I$. Define the functionals

$$\begin{array}{rl}{q}_{s,j}(Z)=& (r_s+1)\mathrm{Tr}{\rho }_s{d}_{s,j}^{\prime }{}^{\ast }Z\\ & -r_s\mathrm{Tr}{\rho }_{s}Z{d}_{s,j}^{\prime }{}^{\ast }.\end{array}$$

They vanish on the identity. Their vanishing alone allows displacement, which is why the means must also be controlled. Write $\Vert q_s{\Vert }_{F_s,\ast }^2={\sum }_j\Vert q_{s,j}{\Vert }_{F_s,\ast }^2$, and set $A_{\ast }^2={\sum }_i{w}_i\Vert q_i{\Vert }_{F_i,\ast }^2$, $M_{\ast }^2={\sum }_i{w}_i|{\mu }_i{|}^2$.

The defect of the mixed output is the weighted sum of the pulled-back input defects plus an explicit replacement error proportional to $\kappa (1-\kappa )\overline{{\mu }_{\gamma ,j}}\mathrm{Tr}[(\gamma -{\tau }_{r_0})Z]$. Interpolation controls the sum; a mixture variance bound controls that error. Choosing the already fixed $\kappa$ sufficiently small gives

$$\Vert q_0{\Vert }_{F_0,\ast }^2\le A_{\ast }^2+\frac{\epsilon }{2}(A_{\ast }^2+M_{\ast }^2).$$

🏷️ Stationarity, cancellation, and rigidity

The thermal evolution with reference mean $r$ has generator $(r+1){\sum }_j\mathcal{D}[d_j]+r{\sum }_j\mathcal{D}[d_j^{\ast }]$, where $\mathcal{D}[a](\rho )=a\rho a^{\ast }-\frac{1}{2}(a^{\ast }a\rho +\rho a^{\ast }a)$. Let ${\delta }_s$ be its value at ${\rho }_s$. With centered energy $e_s^{\prime }=\mathrm{Tr}{\rho }_s{N}_s-|{\mu }_s{|}^2$, the manuscript proves

$$\begin{array}{rl}& \frac{1}{h_s}\mathrm{Tr}{\delta }_s(-\log {\rho }_s)\\ & =\Vert q_s{\Vert }_{F_s,\ast }^2-(e_s^{\prime }-n{r}_s),\\ & \mathrm{Tr}{\delta }_i{N}_i=-(e_i^{\prime }-n{r}_i)-|{\mu }_i{|}^2.\end{array}$$

The generator commutes with mixing in the required weighted trace sense. Pairing it with the Gibbs identities gives stationarity for $J_{\zeta }$. Independence between ports and the output replacement yield the exact energy relation

$$\begin{array}{rl}{e}_0^{\prime }-n{r}_0=& \sum _i{w}_i(e_i^{\prime }-n{r}_i)\\ & +\kappa (1-\kappa )|{\mu }_{\gamma }{|}^2,\end{array}$$

where ${\mu }_{\gamma }={\sum }_i\sqrt{{\lambda }_i}{\mu }_i$. Thus every centered-energy term cancels. If $P={\sum }_i\mathrm{Tr}{\delta }_i{W}_i^4$, the remaining identity and consequence are

$$\begin{array}{c}0=\Vert q_0{\Vert }_{F_0,\ast }^2-(1+\epsilon )A_{\ast }^2-\epsilon M_{\ast }^2\\ -\kappa (1-\kappa )|{\mu }_{\gamma }{|}^2+\zeta P,\\ \frac{\epsilon }{2}(A_{\ast }^2+M_{\ast }^2)\le C\zeta .\end{array}$$

Here $P\le C$ uniformly: its number-operator polynomial has leading term $-4N^4$, so its supremum is finite. Both defects and means therefore vanish as $\zeta \downarrow 0$.

Take a trace-norm convergent subsequence using the uniform energy bound. For a fixed finite number-basis test $Z$, both $d_j^{\ast }Z$ and $Z{d}_j^{\ast }$ are bounded finite-rank operators. The vanishing defects and means pass to the limit through these tests and give $(r_i+1)d_j{\hat{\rho }}_i=r_i{\hat{\rho }}_i{d}_j$ as number-basis matrices.

This recurrence determines the entire state. If two multi-indices differ in some coordinate, iterate the lowering recurrence until one index reaches zero; the entry vanishes. Self-adjointness covers the reversed inequality. On the diagonal, increasing any coordinate multiplies the probability by $r_i/(r_i+1)$. Trace one fixes the product geometric normalization, so ${\hat{\rho }}_i={\tau }_{r_i}$, including the absence of correlations between modes.

The output consequently tends to ${\tau }_{r_0}$. Energy-bounded entropy continuity makes the entropy part of $J_{\zeta }$ tend to zero. The remaining penalties are nonnegative, so $\mathrm{liminf}{J}_{\zeta }\ge 0$, contradicting its uniformly negative minimum. The argument needs neither energy convergence nor vanishing of the penalty terms themselves. (OpenAI, 2026)

🏷️ Minimum output entropy

Let ${\mathcal{E}}_{\eta ,N_B}$ be a thermal attenuator: a beam splitter with an independent thermal environment of mean $N_B$. Its $n$-fold use has identical parameters in every mode and a product thermal environment. For finite-energy inputs of fixed entropy $S(\rho )=ns$, the claimed inequality gives the exact minimum per mode

$$\begin{array}{c}\underset{S(\rho )=ns}{\min }\frac{1}{n}S({\mathcal{E}}_{\eta ,N_B}^{\otimes n}(\rho ))\\ =g(\eta g^{-1}(s)+(1-\eta )N_B).\end{array}$$

The minimum is over finite-energy states, allowing internal entanglement. The product thermal input ${\tau }_{g^{-1}(s)}$ attains it by thermal mixing. This asserts attainment, not uniqueness. Vacuum inputs or environments and transmission endpoints are included; mode-dependent attenuators, amplifiers, and additive-noise channels are outside this corollary.

🏷️ A classical broadcast capacity region

Now split one input mode per use among receivers $B,C$ and a loss output, with independent vacuum auxiliary inputs. Assume $0\le {\eta }_C<{\eta }_B$ and ${\eta }_B+{\eta }_C\le 1$. Two independent uniform classical messages are encoded into finite-energy states, possibly entangled across uses. The photon constraint $\bar{N}$ is averaged over the entire codebook and all uses. Each receiver may measure its own block collectively; there is no feedback or cooperation.

The manuscript derives the closed convex hull of the nonnegative rate pairs

$$\begin{array}{rl}{R}_B& \le g({\eta }_B\beta \bar{N}),\\ R_C& \le g({\eta }_C\bar{N})-g({\eta }_C\beta \bar{N}),\\ & 0\le \beta \le 1,\end{array}$$

in nats per use. (OpenAI, 2026) The weak marginal is obtained from the strong marginal by a new vacuum attenuator of transmission $\lambda ={\eta }_C/{\eta }_B$. This degradation concerns marginal channels; the physical outputs need not be independent.

For the converse, average the strong receiver’s conditional output entropy over the weak message and express it as $ng({\eta }_B\beta \bar{N})$ for a suitable $\beta$. The energy bound guarantees $\beta \in [0,1]$. Apply the vacuum-port photon-number inequality to each conditional state, then Jensen to the convex function $u\mapsto g(\lambda g^{-1}(u))$. This bounds the weak conditional entropy below by $ng({\eta }_C\beta \bar{N})$. Fano’s inequality and the Holevo information bound give the two rate inequalities, with the weak total entropy bounded above by $ng({\eta }_C\bar{N})$.

For achievability, coherent-state superposition coding uses independent complex Gaussian displacements for the two messages, with variances $(1-\beta )N^{\prime }$ and $\beta N^{\prime }$ for $N^{\prime }<\bar{N}$. The corresponding Holevo quantities have the displayed thermal differences. The manuscript makes the continuous alphabet rigorous by finite quantization with controlled second moments, entropy continuity, and a cited finite-alphabet coding theorem. Finite coherent alphabets span finite-dimensional output spaces, so that theorem applies there.

A strict power margin also matters when selecting a deterministic code. If expected code cost is at most $N^{\prime \prime }<\bar{N}$ and expected error is $e_n\to 0$, Markov’s inequality bounds the probability of either failure by $N^{\prime \prime }/\bar{N}+\sqrt{e_n}<1$ for large $n$. Thus one code satisfies both the full-codebook energy constraint and small error. Taking closures and time sharing completes the stated region. At $\bar{N}=0$ both rates vanish; at ${\eta }_C=0$ the weak receiver has zero rate.

🏷️ Key steps and proof outlines

The fourth-moment penalty supplies the missing operator domains

At each fixed regularization parameter, the input minimizer is a Gibbs state for a positive operator of the form

$$K=k{W}^4+W^{1/2}B{W}^{1/2},W=1+N,k>0,B\ge 0$$

with $B$ bounded. The second term need not commute with the number operator. Consequently one must prove moment bounds without diagonalizing both terms simultaneously.

Set $C=kI+W^{-3/2}B{W}^{-3/2}$. The quadratic form of $K$ is $\Vert C^{1/2}{W}^2\psi {\Vert }^2$, and

$$K^{-1}=W^{-2}{C}^{-1}{W}^{-2}.$$

This compact positive inverse constructs the self-adjoint operator and a complete eigenbasis $K{\psi }_j=u_j{\psi }_j$. Since $C\ge kI$, the number of eigenvalues below a given threshold grows at most polynomially, by comparison with $k{W}^4$.

Testing the eigenvalue equation in the number basis gives the stronger identity

$$k{W}^{7/2}{\psi }_j=u_j{W}^{-1/2}{\psi }_j-B{W}^{1/2}{\psi }_j.$$

Its right side is square summable, with norm bounded by a polynomial in $u_j$. Summing against $e^{-u_j}$ proves

$$\mathrm{Tr}(e^{-K}{W}^7)<\infty .$$

This seventh moment is then available for logarithmic and creation-annihilation trace identities. It is obtained separately at each minimum; the contradiction argument only needs the energy bound to remain uniform as the regularization vanishes (OpenAI, 2026).

Independence supplies two of the four metric comparisons

The interpolation step uses four input inequalities with distinct origins. Two are ordinary variances. In the product state of the three independent input ports, the centered subspaces of observables supported on different ports are orthogonal. The centered slice $T_{i}Z-\mathrm{Tr}(\gamma Z)I$ is the orthogonal projection of the pulled-back output observable onto the $i$-th such subspace. Bessel’s inequality therefore bounds the sum of its squared norms by the full variance.

This works for both noncommutative multiplication orders,

$$\mathrm{Tr}\rho Z{Z}^{\ast },\mathrm{Tr}\rho Z^{\ast }Z.$$

In the logarithmic-mean coordinates, their multipliers are the two weights $R_{+}$ and $R_{-}$ defined by

$$h\ell (p_l,p_r)R_{+}(v)=p_l,h\ell (p_l,p_r)R_{-}(v)=p_r.$$

Output replacement contributes the factor $1-\kappa$: the variance of a mixture is at least that fraction times the variance of its first component.

The other two comparisons use the entropy Hessian at the minimizing pair. The auxiliary thermal port supplies strict numerical slack,

$$(1+\epsilon )\sum _i{\lambda }_i{r}_i\le r_0,(1+\epsilon )\sum _i{\lambda }_i(r_i+1)\le r_0+1.$$

Multiplying the Hessian inequality by these constants gives comparison for the constant weights $r_i$ and $r_i+1$. The source’s interpolation theorem combines all four into the thermal-relaxation weight $F_s$.

The logical order matters: independence supplies the arithmetic variance inequalities, minimization supplies the entropy-curvature inequality, and the extra thermal port makes their coefficients compatible. This is not a claim that the final weighted contraction holds for arbitrary input states. At the actual minima it is exactly the estimate needed to compare thermal defects and close the stationarity cancellation already displayed above (OpenAI, 2026).

🏷️ Connections and verification limits

The preceding quantum-information family concerns secrecy and entanglement under public communication. The present family instead compares entropy under passive bosonic mixing and derives classical communication rates. Both require careful separation of a channel identity from the operational model in which it is used.

The exposition follows the manuscript’s regularization, entropy-curvature, defect-cancellation, and broadcast arguments. The weighted interpolation theorem, unbounded-operator domain estimates, and coding theorem remain substantive inputs. No numerical experiment establishes the inequality for arbitrary non-Gaussian inputs, and the selected-source review is not a complete proof verification.

🏷️ Manuscripts and verification scope

• The entropy photon-number inequality — pinned manuscript and companion source.

The family scope page describes selected formalization coverage. The selected scope is the two-input photon-number inequality for finite-energy states, allowing entanglement among modes within each input. The thermal-attenuator minimum-output-entropy and broadcast-capacity consequences are explicitly outside that selected inequality. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026. The entropy photon-number inequality.