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Mathematical physics · Result 273

The entropy photon-number inequality

1 paperLean

Proves the entropy photon-number inequality for beam-splitter mixing of two independent finite-energy bosonic inputs in any finite number of modes: the output's entropy photon number is at least the transmissivity-weighted average of the inputs'. Arbitrary entanglement within each input is allowed, and product thermal inputs attain equality even when their entropies differ.

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▶Introduction: the problem and why it mattersin preparation
▶The presented solutionin preparation

Papers 1

First page of The entropy photon-number inequality

The entropy photon-number inequality

OpenAI · September 24, 2026

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.

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@misc{OAI:The-entropy-photon-number-inequality-September-24-2026,
  author = {{OpenAI}},
  title = {{The entropy photon-number inequality}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-entropy-photon-number-inequality-September-24-2026/paper.pdf}{OAI:The-entropy-photon-number-inequality-September-24-2026}},
  year = {2026}
}

What was formalized in Lean

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalization proves the entropy photon-number inequality for two independent bosonic inputs with finite mean energy and any finite positive number \(n\) of modes. Write \(N(\rho)=g^{-1}(S(\rho)/n)\), where \(S\) is von Neumann entropy and \(g(x)=(x+1)\log(x+1)-x\log x\) is the thermal entropy per mode. For a beam splitter of transmissivity \(0\le\eta\le1\), its output satisfies \(N(\rho_C)\ge\eta N(\rho_A)+(1-\eta)N(\rho_B)\).

Entanglement among modes within either input is allowed. The thermal-attenuator minimum-output-entropy and broadcast-capacity consequences in the paper are outside this selected inequality.

Comparator links

ResultComparator statement
Entropy photon-number inequality for finite-energy bosonic inputsEntropyPhotonNumber.lean

Scope note on GitHub →