The entropy photon-number inequality

Follow-up works

• Zixuan He, The active entropy photon-number inequality: exact minima, energy budgets and input rigidity (Zenodo preprint, doi:10.5281/zenodo.23352232, v1.3, 13 October 2026). Treats the amplifier-type ("active") entropy photon-number inequality for independent multimode inputs with arbitrary internal correlations, with constrained output-entropy minima for thermal amplifiers and stability bounds for near-optimal inputs; the v1.2 PDF cites "OpenAI" and links to the openai/math repository at commit adc7f12.

Formalization

• OpenAI math release, Lean scope note for family 273 (lean/docs/273.md, 6 October 2026). Says the Lean formalization covers the two-input inequality (Theorem 1.1) for finite-energy inputs in any finite positive number of modes, allowing internal entanglement, and excludes the thermal-attenuator minimum-output-entropy and broadcast-capacity consequences.
• OpenAI, Inequality.lean (openai/math, commit adc7f12). Lean source containing the theorem entropy_photon_number_inequality, which states the beam-splitter inequality $\eta g^{-1}(S(\rho_A)/n)+(1-\eta)g^{-1}(S(\rho_B)/n)\le g^{-1}(S(\rho_C)/n)$.

Related prior literature mentioned by others

• Giacomo De Palma, Andrea Mari and Vittorio Giovannetti, A generalization of the Entropy Power Inequality to Bosonic Quantum Systems (arXiv:1402.0404; Nature Photonics 8, 958–964, 2014). The QIQCOP Zoo's discussion of the OpenAI claim identifies this paper's exponential quantum entropy power inequality as a weaker predecessor to the claimed entropy photon-number inequality.
• Giacomo De Palma, Dario Trevisan and Vittorio Giovannetti, Gaussian states minimize the output entropy of one-mode quantum Gaussian channels (arXiv:1610.09970; Physical Review Letters 118, 160503, 2017). The QIQCOP Zoo's discussion of the OpenAI claim cites Theorem 4 as the sharp one-mode thermal-environment case preceding the manuscript's arbitrary two-input, multimode claim.

Possibly relevant literature (found by the agent, not checked)

• S. Guha, J. H. Shapiro, B. I. Erkmen, Capacity of the Bosonic Wiretap Channel and the Entropy Photon-Number Inequality (arXiv:0801.0841, January 2008). Derives the wiretap-channel capacity, and with it the quantum capacity of the lossy bosonic channel, from a minimum output entropy conjecture that follows from the EPnI; the paper discusses only the attenuator and broadcast consequences.
• M. M. Wilde, P. Hayden, S. Guha, Quantum trade-off coding for bosonic communication (Phys. Rev. A 86, 062306 (2012); arXiv:1105.0119). Its arXiv comments say the optimality of its pure-loss trade-off region is proven if the EPnI holds, a further channel consequence of Theorem 1.1 beyond the paper's stated corollaries.
• H. Qi, M. M. Wilde, S. Guha, On the minimum output entropy of single-mode phase-insensitive Gaussian channels (arXiv:1607.05262, 2016/2017). Notes that a multimode version of the De Palma–Trevisan–Giovannetti constrained minimum-output-entropy result would give an important special case of the EPnI, which relates to the paper's Corollary 1.2 (constrained minimum for identical thermal attenuators).
• G. De Palma, D. Trevisan, V. Giovannetti, L. Ambrosio, Gaussian optimizers for entropic inequalities in quantum information (J. Math. Phys. 59, 081101 (2018); arXiv:1803.02360). A survey of the quantum Gaussian optimizer conjectures, including quantum counterparts of the entropy power inequality, giving background for the status of the conjecture summarised in the paper's introduction.

Digests and expositions

• Gradient Damage, Spacetime Penrose Inequality, Strong Cosmic Censorship Near Kerr and 8 More (video compilation, 15 October 2026). The chapter beginning at 2:49 introduces the entropy photon-number inequality through beam-splitter mixing, with the accompanying page attributing the claim to OpenAI and distinguishing the animations from proofs. [family-level]