On the continuous Zauner conjecture

TL;DR

This paper explores the relationship between entanglement breaking channels and mutually unbiased frames, establishing equivalences that connect Zauner's conjecture to frame theory and providing partial results in specific dimensions.

Contribution

It proves that entanglement breaking rank conditions are equivalent to the existence of special mutually unbiased frames, extending the understanding of Zauner's conjecture in quantum information theory.

Findings

Equivalence between entanglement breaking rank and mutually unbiased frames.

Validated the conjecture for dimensions 2 and 3.

Numerical searches failed for dimensions 4 and 5 beyond trivial cases.

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 $\{|x_i\rangle\}_{i=1}^{d^2}, \{|y_i\rangle\}_{i=1}^{d^2}$ in $\mathbb{C}^d $ which are mutually unbiased in a certain sense. That is, for any $i\neq j$ it holds that $|\langle x_i|y_j\rangle|^2 = \frac{1-t}{d}$ and $|\langle x_i|y_i\rangle|^2 = \frac{t(d^2-1)+1}{d}$ (also it follows that $|\langle x_i|x_j\rangle\langle y_i|y_j\rangle|=|t|$). Though, our numerical searches for solutions were not successful in dimensions 4 and 5 for values of $t$ other than $0$ or $\frac{1}{d+1}$.