The Local Orthogonality between Quantum States and Entanglement Decomposition

Abstract

For a quantum state $\rho$, let $E_{f}(\rho)$ be the entanglement of formation. Professors Horodecki proved the following important results: If $\rho$ is composed of the locally orthogonal pure state ensemble \{\out{\psi_{i}}{\psi_{i}}\}_{i=1}^K with probability distribution $p=(p_i)$ such that \rho =\sum_{i=1}^{K}p_{i}\out{\psi_{i}}{\psi_{i}}, then E_{f}(\rho) = \sum_{i}p_{i}E_{f}(\out{\psi_{i}}{\psi_{i}}). In this paper, we generalize the conclusion to quantum state $\rho$ which is composed of locally orthogonal quantum state ensemble $\{\rho_{a}\}_{a\in\Sigma}$. Finally, we present an interesting example to show that the conditions of these conclusions are existence