General conditional quantum entropy power inequality

Establish the general conditional quantum entropy power inequality for bosonic systems beyond the conditional-independence setting, extending the weight-[?] conditional linear inequality known for independent Gaussian inputmemory pairs.

Background

The paper uses a conditional qEPI as a key ingredient in its doubling argument. De Palma and Trevisan established such an inequality under conditional independence, while the earlier result cited in the paper treated Gaussian inputmemory pairs at the physical weight p=eta. The broader conditional inequality remains an unresolved conjectural extension and would remove the conditional-independence restriction.

This problem concerns proving a conditional entropy-power lower bound for bosonic beam-splitter outputs when the input systems are correlated with a quantum memory, rather than only in the special situations already covered by existing results.