Codes and Expansions (CodEx) Seminar

Danylo Yakymenko (Institute of Mathematics of NAS of Ukraine)
On the continuous Zauner conjecture

It was discovered by S. Pandey, V. Paulsen, J. Prakash, and M. Rahaman that Zauner's conjecture about SIC existence is equivalent to the statement that the entanglement breaking rank of the quantum channel \(\Phi_{\frac{1}{d+1}}\) is \(d^2\), where \(\Phi_t:\mathbb{C}^{d\times d} \to \mathbb{C}^{d \times d}\) is defined by \(\Phi_t(X) = tX+ (1-t){\rm tr}(X) \frac{1}{d}I\) for all \(t \in [-\frac{1}{d^2-1}, \frac{1}{d+1}]\). They've made an extended version of the conjecture that states that the entanglement breaking rank of \(\Phi_{t}\) is also \(d^2\). In this talk we explain this connection from a different perspective, which concerns the length of separability of separable Werner states. It turns out that the extended conjecture is equivalent to the existence of a pair of informationally-complete unit-norm tight frames \(\{|x_i\rangle\}_{i=1}^{d^2}, \{|y_i\rangle\}_{i=1}^{d^2}\) in \(\mathbb{C}^d\) which are mutually unbiased in the following sense: for any \(i\neq j\) it holds that \(|\langle x_i|y_j\rangle|^2 = \frac{1-t}{d}\) and \(|\langle x_i|y_i\rangle|^2 = \frac{t(d^2-1)+1}{d}\). We'll argue why this conjecture might not be true in general, as numerical search suggests.