Result 273, Mathematical physics

The entropy photon-number inequality

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.

Lean formalization Proof

The bigger picture

Why it matters

When two independent quantum light fields mix at a beam splitter, how little disorder can the output have? The manuscript claims a sharp lower bound, linking that question to fundamental limits on quantum communication.

What changes?

The unreviewed manuscript reports the entropy photon-number inequality for two independent, finite-mean-energy bosonic inputs in any finite number of modes, or field components. Entropy photon number expresses disorder as the average photon count per mode of a thermal state with matching entropy. The output's value is at least the inputs' average weighted by the beam splitter's transmissivity. Modes within either input may be arbitrarily entangled. Product thermal inputs attain equality, even when their entropies differ.

What does that help mathematicians do?

The bound yields an exact minimum output entropy at fixed input entropy for every finite collection of identical thermal attenuators, channels that mix a signal with thermal noise. The claimed minimum covers finite-energy inputs entangled across those channels. Researchers could therefore rule out using such entanglement to push the output entropy below this limit, rather than relying on calculations restricted to separate inputs.

Are there practical applications?

Its immediate value is a foundational communication limit: the manuscript also claims the exact classical capacity region for a degraded, two-receiver pure-loss bosonic broadcast channel under a mean photon constraint. This region specifies which pairs of information rates the model permits. It is a theoretical consequence for the stated channel, not a demonstrated device improvement or deployment.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

The entropy photon-number inequality

September 24, 2026 32 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/273.md.

The entropy photon-number inequality

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 nn of modes. Write N(ρ)=g−1(S(ρ)/n)N(\rho)=g^{-1}(S(\rho)/n), where SS is von Neumann entropy and g(x)=(x+1)log⁡(x+1)−xlog⁡xg(x)=(x+1)\log(x+1)-x\log x is the thermal entropy per mode. For a beam splitter of transmissivity 0≤η≤10\le\eta\le1, its output satisfies N(ρC)≥ηN(ρA)+(1−η)N(ρB)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

Result Comparator statement
Entropy photon-number inequality for finite-energy bosonic inputs EntropyPhotonNumber.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.