Entropy photon-number inequality

Problem

Does the entropy photon-number inequality hold for every pair of independent finite-energy bosonic inputs? Let \(n\geq1\) and \(0\leq\eta\leq1\). The input is \(\rho_A\otimes\rho_B\), where each factor is an \(n\)-mode state with finite total mean photon number. Correlations among the modes within either factor are allowed. Mix corresponding input modes on beam splitters and retain the outputs

\begin{equation} c_j=\sqrt\eta\,a_j+\sqrt{1-\eta}\,b_j,\qquad j=1,\ldots,n. \tag{1} \end{equation}

Let \(\rho_C\) be the joint state of the modes in Eq. (1). Use natural logarithms, \(S(\rho)=-\operatorname{Tr}\rho\ln\rho\), and \(g(x)=(x+1)\ln(x+1)-x\ln x\) for \(x\geq0\), with \(0\ln0=0\). The proposed inequality is

\begin{equation} g^{-1}\!\left(\frac{S(\rho_C)}n\right) \geq\eta g^{-1}\!\left(\frac{S(\rho_A)}n\right) +(1-\eta)g^{-1}\!\left(\frac{S(\rho_B)}n\right). \tag{2} \end{equation}

Prove Eq. (2) in this full domain or provide a physical input pair that violates it.

Source

The entropy photon-number conjecture is stated explicitly in De Palma, Mari, and Giovannetti, Sec. II.3, Eq. (34), with attribution there to the earlier Guha–Shapiro–Erkmen work [DMG14].

Progress

Reports do not certify correctness or automatically change the problem's status. Progress policy.

The quantum entropy power inequality proves \(e^{S(\rho_C)/n}\geq\eta e^{S(\rho_A)/n}+(1-\eta)e^{S(\rho_B)/n}\). This is weaker than Eq. (2). The same paper bounds the possible deficit in Eq. (2) by \(1/2-1/e\); see Eqs. (5) and (35) [DMG14].
For \(n=1\) and a thermal second input with mean photon number \(b\geq0\), Theorem 4, Eq. (24), proves \(S(\rho_C)\geq g(\eta g^{-1}(S(\rho_A))+(1-\eta)b)\). Thermal first inputs attain equality. This permits arbitrary first inputs but does not permit an arbitrary nonthermal second input [DTG17].
Reported progress: Issue #103 (withdrawn).
OpenAI reports a proof of Eq. (2) for two independent finite-energy bosonic inputs and any finite number of modes (Theorem 1.1, p. 2) [OAI26]. Arbitrary entanglement among modes within either input is allowed. Product thermal inputs attain equality, including when the two input entropies differ.

Comment

The preprint [OAI26] reports a proof in the full finite-energy domain stated above; the catalog retains the Unsolved designation while documenting this report. Independence of the two input systems is essential. Taking one input to be vacuum yields the finite-energy case of the catalog’s multimode constrained pure-loss output-entropy question.

References

[DMG14]: G. De Palma, A. Mari, and V. Giovannetti, “A Generalization of the Entropy Power Inequality to Bosonic Quantum Systems,” Nature Photonics 8, 958–964 (2014).DOI arXiv
[DTG17]: G. De Palma, D. Trevisan, and V. Giovannetti, “Gaussian States Minimize the Output Entropy of One-Mode Quantum Gaussian Channels,” Physical Review Letters 118, 160503 (2017).DOI arXiv
[OAI26]: OpenAI, “The entropy photon-number inequality,” OpenAI Math Release preprint, 24 September 2026. manuscript (release version).link

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