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 $\Phi_t:\mbb{C}^{d\times d} \to \mbb{C}^{d \times d}$ for $t \in [-\frac{1}{d^2-1}, \frac{1}{d+1}]$ defined by $\Phi_t(X) = tX+ (1-t)\tr(X) \frac{1}{d}I$. 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 $\ebr(\Phi_t)=d^2$ 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 $\ebr(\Phi_t)=d^2$ is equivalent to the existence of a pair of informationally-complete unit-norm tight frames $\{|x_i\ra\}_{i=1}^{d^2}, \{|y_i\ra\}_{i=1}^{d^2}$ in $\mbb{C}^d $ which are mutually unbiased in the following sense: for any $i\neq j$ it holds that $|\la x_i|y_j\ra|^2 = \frac{1-t}{d}$ and $|\la x_i|y_i\ra|^2 = \frac{t(d^2-1)+1}{d}$. Moreover, it follows that $|\la x_i|x_j\ra\la y_i|y_j\ra|=|t|$ for $i\neq j$. However, our numerical searches for solutions were not successful in dimensions 4 and 5 for values of $t$ other than $0$ or $\frac{1}{d+1}$.