Extend UCPT non-extremality counterexamples to higher dimensions

Establish the existence of extreme unital completely positive and trace-preserving (UCPT) maps on Hilbert spaces of dimension greater than 4 whose Choi ranks exceed the threshold CR(ε) > 2^(1/4) dim(X), thereby enabling the construction of pairs (ε, ε') with CR(ε) CR(ε') > √2 dim(X ⊗ Y) and proving that ε ⊗ ε' is not an extreme point of UCPT(X ⊗ Y) for higher dimensions.

Background

Using Theorem 3.3, the authors show that if two extreme UCPT maps have sufficiently large Choi ranks relative to the dimensions of their spaces, then their tensor product cannot be extreme. They apply this to explicit extreme UCPT maps with high Choi ranks in dimensions 3 and 4 to produce counterexamples.

They then state that extending these counterexamples to even higher dimensions remains to be shown, which in practice requires constructing extreme UCPT maps in larger dimensions whose Choi ranks exceed the specified threshold so the theorem applies.