Complete the beam-splitter sharp Young inequality conjecture

Establish the full beam-splitter conjecture proposed in de Pal­ma’s survey for the quantum convolution operation, extending the partial resolution obtained from the sharp Hausdorff–Young inequality and the associated dilation argument.

Background

The paper defines quantum convolution on quantum Euclidean spaces through pointwise multiplication of the coefficient functions, equivalently relating it to the beam-splitter channel in quantum information theory. Using the sharp Hausdorff–Young inequality proved earlier, it derives a sharp Young-type inequality for quantum convolutions when the input exponents satisfy 1 ≤ p,q ≤ 2 and the output exponent satisfies r ≥ 2.

The authors explicitly state that their result only partially solves a conjecture from de Palma’s survey concerning the beam-splitter channel. Thus, the unresolved problem is to prove the conjecture in its entirety, including the cases not covered by the theorem and dilation argument presented in the paper.

References

Up to a dilation argument, this result partially solves the conjecture proposed in Conjecture V.16, which was proposed for beam-splitter; also see Remark~\ref{bs}.

— Noncommutative sharp Hausdorff-Young inequality  (2609.10424 - Qiu et al., 9 Sep 2026) in Section 5, subsection “Sharp Young's inequality,” immediately after Theorem 5.1

This equality question is distinct from the entropy photon-number inequality (EPnI), which asks whether product thermal inputs minimize output entropy for prescribed input entropies. The EPnI is a stronger conjectured bound; seeSecs.~V.2 and~VII.1.

— Equality in the Bosonic Quantum Entropy Power Inequality  (2609.40061 - Li, 30 Sep 2026) in Section 1, subsection "Main results"; discussion distinguishing the qEPI from the EPnI