The Local Orthogonality between Quantum States and Entanglement Decomposition

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

Partial coherence versus entanglement

We study partial coherence and its connections with entanglement. First, we provide a sufficient and necessary condition for bipartite pure state transformation under partial incoherent operations: A bipartite pure state can be transformed to another one if and only if a majorization relationship holds between their partial coherence vectors. As a consequence, we introduce the concept of maximal partial coherent states in the sense that they can be used to construct any bipartite state of the same system via partial incoherent operations. Second, we provide a strategy to construct measures of partial coherence by the use of symmetric concave functions. Third, we establish some relationships between partial coherence and entanglement. We show that the minimal partial coherence under local unitary operations is a measure of entanglement for bipartite pure states, which can be extended to all mixed states by convex-roof. We also show that partial coherence measures are induced through maximal entanglement under partial incoherent operations for bipartite pure states. There is a one-to-one correspondence between entanglement and partial coherence measures.Comment: 12 pages, 2 figures, 1 tabl