How does the concept of quantum entanglement extend to many-body systems, and what are the challenges in quantifying entanglement in such systems?

In many-body systems, entanglement extends beyond pairs of particles to encompass complex, multipartite entanglement structures. Quantifying entanglement in these systems is challenging due to the exponential growth of the Hilbert space with the number of particles. Measures like entanglement entropy, mutual information, and entanglement witnesses are used, but computing these for large systems is computationally intensive. Tensor network methods, such as matrix product states and the density matrix renormalization group, help approximate and analyze entanglement in many-body systems.