bibkey: vanherstraeten2024nongaussian authors: Zacharie Van Herstraeten; Saikat Guha; Nicolas J. Cerf year: 2024 title: “Classical capacity of quantum non-Gaussian attenuator and amplifier channels” doi: 10.1142/S0219749924400033 url: https://arxiv.org/abs/2312.15623v2 claim: “The minimum output entropy of a phase-covariant attenuator channel ℳη,𝐩 is achieved by coherent states, ∀η,𝐩.” strata_touched:

• D5/S3/Quantum/QuantumChannels/FockAttenuator/BeamSplitter
• D5/S3/Quantum/QuantumChannels/FockAttenuator/DisplacementCovariance
• D5/S3/Quantum/QuantumChannels/FockAttenuator/Attenuator
• D5/S3/Quantum/QuantumChannels/PhaseCovariantAttenuatorCoherentRefutation license: citation-only triage: anchor

Classical capacity of quantum non-Gaussian attenuator and amplifier channels

Z. Van Herstraeten, S. Guha and N. J. Cerf, Int. J. Quantum Inf. 22, 2440003 (2024), arXiv:2312.15623v2. Conjecture 2, section IV.B, PDF p. 12:

The minimum output entropy of a phase-covariant attenuator channel ${\mathcal{M}}_{\eta ,\mathbf{p}}$ is achieved by coherent states, $\forall \eta ,\mathbf{p}$.

The domain is one bosonic mode, transmissivity $0\le \eta \le 1$, and every probability vector $\mathbf{p}$ on the nonnegative occupations.

The same section defines a phase-covariant attenuator by

$${\mathcal{M}}_{\eta ,\mathbf{p}}[\hat{\rho }]={\mathrm{Tr}}_2\left[{\hat{U}}_{\eta }\left(\hat{\rho }\otimes \sum _{n=0}^{\infty }{p}_n|n⟩⟨n|\right){\hat{U}}_{\eta }^{†}\right]=\sum _{n=0}^{\infty }{p}_n{\mathcal{M}}_{\eta ,n}[\hat{\rho }].$$

The channel is a convex mixture of the Fock-environment attenuators. The coherent state has coefficient $e^{-|\alpha {|}^2/2}{\alpha }^n/\sqrt{n!}$ for every complex amplitude. Beam-splitter displacement commutation implies that its output is a displacement of the vacuum output, with retained amplitude $\sqrt{\eta }\alpha$. The sign on the discarded-mode amplitude can be changed by a second-mode phase convention; it does not affect the reduced channel for a Fock-diagonal environment.

The formal result uses the mixed environment $(7/8)|1⟩⟨1|+(1/8)|5⟩⟨5|$, the balanced beam splitter, and the one-photon input. It refutes Conjecture 2. It does not settle Conjecture 1 for pure Fock environments, nor assert that the one-photon input is globally optimal. The paper states that geometric probability vectors, corresponding to thermal environments, satisfy the coherent-state minimum-output-entropy property by its cited Gaussian optimizer result.

The full Hilbert space, global beam splitter, displacement covariance and finite output entropy comparison are checked in Lean. The general infinite-dimensional spectral characterization of the chosen entropy definition is literature-attested in Wehrl’s entropy review and is not claimed here as a kernel-proved theorem.

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arXiv v2: https://arxiv.org/abs/2312.15623v2 — section IV.B, Conjecture 2, PDF p. 12, and the mixed-environment channel definition. Journal DOI: https://doi.org/10.1142/S0219749924400033. The quoted statement and channel are read from the arXiv source; the journal text has not been compared sentence by sentence.