Improve lower bounds on maximal Choi ranks of extreme UCPT maps

Develop improved lower bounds on the maximal possible Choi rank of extreme unital completely positive and trace-preserving (UCPT) maps on finite-dimensional Hilbert spaces beyond dimension 4, sufficient to determine for which dimensions the tensor product preserves extremality in the class UCPT. Specifically, ascertain lower bounds stronger than the current bound that the maximal Choi rank is at least the dimension, so that the preservation or failure of extremality under tensor products can be decisively characterized across dimensions.

Background

The paper studies whether extremality of quantum channels is preserved under tensor products for CPT, UCP, and UCPT maps. For UCPT maps, the authors show preservation in dimension 2 and provide counterexamples in dimensions 3 and 4 using known high-rank extreme maps.

They note that prior work computed maximal Choi ranks of extreme UCPT maps only up to dimension 4 and provided lower bounds (at least the dimension) for higher dimensions without improvement in later constructions. The authors state that better lower bounds may suffice to resolve the dimension ranges where tensor-product extremality is preserved for UCPT maps but that such bounds have not yet been obtained.