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.