I am seeking clarification on the comparative aspects of two prominent measures of entanglement: Von Neumann entropy and concurrence. My goal is to understand the key differences in how these measures quantify entanglement and their appropriate applications.
Von Neumann Entropy:
- I understand that the Von Neumann entropy $S(\rho) = -\text{Tr}(\rho \log \rho)$ measures the mixedness of a quantum state described by a density matrix $\rho$. For a bipartite system, the entropy of the reduced density matrix can indicate the entanglement between the subsystems. Could someone elaborate on how the Von Neumann entropy reflects the entanglement in both pure and mixed states? What are its advantages and limitations in the context of entanglement measurement?
Concurrence:
- Concurrence is defined for two-qubit systems and, for a pure state $|\psi\rangle$, is given by $C(|\psi\rangle) = |\langle \psi | \tilde{\psi} \rangle|$, where $\tilde{\psi}$ is the spin-flipped state. For mixed states, the calculation involves a more complex process. How does concurrence directly quantify entanglement for two-qubit systems, and what is its significance compared to Von Neumann entropy? What are the main steps in calculating concurrence for mixed states?
Comparative Insights:
- In what scenarios is Von Neumann entropy more appropriate than concurrence, and vice versa? How do these measures differ in terms of practical application and interpretability?
I appreciate any insights or examples that can help clarify these points and enhance my understanding of when and why to use each measure.
Thank you!
