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The entropy photon-number inequality (Guha-Erkmen-Shapiro)

For an nn-mode bosonic state ρ\rho let N(ρ)=g−1(S(ρ)/n)N(\rho)=g^{-1}(S(\rho)/n), where g(t)=(t+1)log⁡(t+1)−tlog⁡tg(t)=(t+1)\log(t+1)-t\log t is the entropy per mode of a thermal state; NN is the mean photon number per mode of the product thermal state with the same entropy. Guha, Erkmen and Shapiro (2007) conjectured the entropy photon-number inequality, a quantum analogue of the entropy power inequality with consequences for minimum output entropy and broadcast capacities. Konig-Smith and De Palma-Mari-Giovannetti proved weaker entropy power inequalities. If independent nn-mode states ρA,ρB\rho_A,\rho_B are mixed on a beam splitter of transmissivity η∈[0,1]\eta\in[0,1] with output ρC\rho_C, is N(ρC)≥ηN(ρA)+(1−η)N(ρB)N(\rho_C)\ge\eta N(\rho_A)+(1-\eta)N(\rho_B)?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Quantum Shannon theory; bosonic channels
Posed by
Saikat Guha, Baris I. Erkmen and Jeffrey H. Shapiro ('The entropy photon-number inequality and its consequences', arXiv 0710.5666, 2007)
Year posed
2007
Years open
19y
Solved
2026-09-24
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
36 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: for finite n≥1n\ge1, η∈[0,1]\eta\in[0,1] and nn-mode states ρA,ρB\rho_A,\rho_B of finite mean photon number, the output of passive mixing satisfies g−1(S(ρC)/n)≥ηg−1(S(ρA)/n)+(1−η)g−1(S(ρB)/n)g^{-1}(S(\rho_C)/n)\ge\eta g^{-1}(S(\rho_A)/n)+(1-\eta)g^{-1}(S(\rho_B)/n), with equality for product thermal inputs. Corollaries: the exact minimum output entropy at fixed input entropy for tensor powers of the thermal attenuator, and the capacity region of the degraded pure-loss bosonic broadcast channel. Not covered: infinite-energy finite-entropy inputs, amplifiers and additive-noise channels, and uniqueness of optimizers.

What the AI did

The release README says every result in openai/math was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not one of the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family has a single manuscript.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the conjecture: the inequality for every finite nn, every η∈[0,1]\eta\in[0,1] and independent inputs of finite mean photon number, with arbitrary entanglement among modes inside each input. The finite-energy hypothesis is a scope limit the paper states. The challenge is not in lean/formalization.yaml; lean/ComparatorChallenges/EntropyPhotonNumber.json exists with solution_module OAI.InformationTheory.PhotonNumber.Inequality, whose file exists at the pinned commit. The statement EntropyPhotonNumber.lean was read here; not rebuilt here. It models states as positive trace-one operators on untruncated ℓ2(Nn)\ell^2(\mathbb N^n), entropy by functional calculus, the beam-splitter output through explicit number-basis coefficients and a partial-trace HasSum condition, and states ηg−1(SA/n)+(1−η)g−1(SB/n)≤g−1(SC/n)\eta g^{-1}(S_A/n)+(1-\eta)g^{-1}(S_B/n)\le g^{-1}(S_C/n) for finite-energy inputs. This states the headline. Permitted axioms: propext, Quot.sound, Classical.choice.

Sources

Changelog1 change

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