OP-00680 Solution claimed

The entropy photon-number inequality

Let $\rho_A$ and $\rho_B$ be states of $n \ge 1$ bosonic modes, with annihilation operators $a_1, \dots, a_n$ and $b_1, \dots, b_n$, each with finite expected total photon number. Correlations among the $n$ modes within either input are allowed; the joint input is $\rho_A \otimes \rho_B$. For $0 \le \eta \le 1$, mix corresponding modes by a beam splitter, so that the output annihilation operators are $$c_j = \sqrt{\eta}\, a_j + \sqrt{1 - \eta}\, b_j, \qquad d_j = -\sqrt{1 - \eta}\, a_j + \sqrt{\eta}\, b_j,$$ and let $\rho_C$ be the reduced state of the $c$ modes after the $d$ modes are discarded. Write $S(\rho) = -\operatorname{Tr}(\rho \log \rho)$ and $g(t) = (t + 1)\log(t + 1) - t \log t$ for $t \ge 0$, the entropy of a one-mode thermal state with mean photon number $t$; $g$ is increasing on $[0, \infty)$, and $g^{-1}$ denotes its inverse there.

Guha, Erkmen and Shapiro proposed the inequality below as a quantum analogue of the entropy power inequality, and showed that it implies the two minimum output entropy conjectures on which capacity theorems for bosonic single-user, broadcast and wiretap channels had been conditioned. It holds when both inputs are Gaussian states. The quantum entropy power inequality $$e^{S(\rho_C)/n} \ge \eta\, e^{S(\rho_A)/n} + (1 - \eta)\, e^{S(\rho_B)/n},$$ for inputs of finite average energy, proved by König and Smith for $\eta = 1/2$ and by De Palma, Mari and Giovannetti for every $\eta$, is strictly weaker.

Problem. Is it always true that $$g^{-1}\!\left(\frac{S(\rho_C)}{n}\right) \ge \eta\, g^{-1}\!\left(\frac{S(\rho_A)}{n}\right) + (1 - \eta)\, g^{-1}\!\left(\frac{S(\rho_B)}{n}\right)?$$

Source: Guha, Erkmen and Shapiro (2008), the entropy photon-number inequality, Eq. (13); stated by AIM — Open Applied Problems for inputs of finite mean photon number, after De Palma, Trevisan, Giovannetti and Ambrosio (2018), Conjecture V.15.

Imported from github.com/MColbrook/AIM.

Classification

Primary: MP — Mathematical Physics
Secondary: IT
Keywords: AIM Open Applied Problems, bosonic channels, entropy photon-number inequality, von Neumann entropy

References

@inproceedings{guha2008epni,
  author        = {Guha, Saikat and Erkmen, Baris I. and Shapiro, Jeffrey H.},
  title         = {The entropy photon-number inequality and its consequences},
  booktitle     = {2008 Information Theory and Applications Workshop},
  publisher     = {IEEE},
  year          = {2008},
  pages         = {128--130},
  doi           = {10.1109/ITA.2008.4601037},
  eprint        = {0710.5666},
  archivePrefix = {arXiv},
}
@article{depalma2018gaussian,
  author        = {De Palma, Giacomo and Trevisan, Dario and Giovannetti, Vittorio and Ambrosio, Luigi},
  title         = {Gaussian optimizers for entropic inequalities in quantum information},
  journal       = {J. Math. Phys.},
  volume        = {59},
  number        = {8},
  year          = {2018},
  pages         = {081101},
  doi           = {10.1063/1.5038665},
  eprint        = {1803.02360},
  archivePrefix = {arXiv},
}
@misc{aim-entropy-photon-number,
  author       = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
  title        = {{The entropy photon-number inequality}},
  howpublished = {AIM --- Open Applied Problems, GitHub repository},
  year         = {2026},
  url          = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/205-entropy-photon-number.md},
  note         = {Entry 205 at commit 1b7c7fd, 2026-10-13},
}

Resolution claims

The entropy photon-number inequality v1 GitHub ↗

OpenAI

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 …

Full resolution · not verified · fidelity not established · posted on GitHub 6 Oct 2026

Status history

When	Change	By	Reason
2026-10-16	Needs verification → Solution claimed	—	Recorded here: math @ 1581d4c, build/main.tex.

Citing this problem

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OP-00680, "The entropy photon-number inequality", clopen.solutions.