On the continuous Zauner conjecture

Abstract

In a recent paper by S. Pandey, V. Paulsen, J. Prakash, and M. Rahaman, the authors studied the entanglement breaking quantum channels ${\Phi }_t:{\mathbb{C}}^{d\times d}\to {\mathbb{C}}^{d\times d}$ for $t\in [-\frac{1}{d^2-1},\frac{1}{d+1}]$ defined by ${\Phi }_t(X)=tX+(1-t)\text{Tr}(X)\frac{1}{d}I$. They proved that Zauner's conjecture is equivalent to the statement that entanglement breaking rank of ${\Phi }_{\frac{1}{d+1}}$ is $d^2$. The authors made the extended conjecture that $\text{ebr}({\Phi }_t)=d^2$ for every $t\in [0,\frac{1}{d+1}]$ and proved it in dimensions 2 and 3. In this paper we prove that for any $t\in [-\frac{1}{d^2-1},\frac{1}{d+1}]\setminus \{0\}$ the equality $\text{ebr}({\Phi }_t)=d^2$ is equivalent to the existence of a pair of informationally complete unit norm tight frames $\{|x_i⟩{\}}_{i=1}^{d^2},\{|y_i⟩{\}}_{i=1}^{d^2}$ in ${\mathbb{C}}^d$ which are mutually unbiased in a certain sense. That is, for any $i\ne j$ it holds that $|⟨x_i|y_j⟩{|}^2=\frac{1-t}{d}$ and $|⟨x_i|y_i⟩{|}^2=\frac{t(d^2-1)+1}{d}$ (also it follows that $|⟨x_i|x_j⟩⟨y_i|y_j⟩|=|t|$). Though, our numerical searches for solutions were not successful in dimensions 4 and 5 for values of $t$ other than $0$ or $\frac{1}{d+1}$.