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

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.

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Occurred: Sep 24, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The entropy photon-number inequality (Guha-Erkmen-Shapiro) · Entropy photon-number inequality

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Unreleased internal OpenAI model
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Lean proof (OAI.EntropyPhotonNumber.entropy_photon_number_inequality)

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This event attributed to Unreleased internal OpenAI model

The entropy photon-number inequality (Guha-Erkmen-Shapiro) parent of this event

The entropy photon-number inequality (Guha-Erkmen-Shapiro) evidence for this event

VibeMathed record: The entropy photon-number inequality (Guha-Erkmen-Shapiro) evidence for this event

Guha, Erkmen, Shapiro, The entropy photon-number inequality and its consequences (2007) evidence for this event

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. parent of this event

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The entropy photon-number inequality (Guha-Erkmen-Shapiro) — Mathematical Frontier Network