Question

I'm learning about the definition of von Neumann entanglement entropy

S(rho₁)=−Tr[rho₁lnrho₁]where rho1 is the reduced density matrix rho₁=Tr₂(rho).

I was confused that the entanglement entropy of a bipartite pure state is different if it is defined using the von Neumann entropy of the reduced density matrix of subsystem 2? i.e., is the following equation correct?
S(rho1)=−Tr[rho1lnrho₁]=S(rho₂)=−Tr[rho₂lnrho₂]

If it's true, how to prove this?

Answer 1

For a pure bipartite quantum system, the entanglement entropy of subsystem 1 is equal to that of subsystem 2. This is due to the Schmidt decomposition which implies that the non-zero eigenvalues of the reduced density matrices of both subsystems are identical, leading to the same entropies.

Step-by-step explanation:

In the context of quantum mechanics and information theory, the entanglement entropy is a measure of bipartite entanglement for a composite quantum system. Considering a pure bipartite state represented by a density matrix ρ, one can define the reduced density matrices ρ₁ and ρ₂ after tracing out the other subsystem. The von Neumann entropy of these reduced density matrices is used to quantify the entanglement.

Importantly, for a pure state, the entanglement entropy of subsystem 1, S(ρ₁), is indeed equal to that of subsystem 2, S(ρ₂). This equivalence arises due to the Schmidt decomposition, which implies that the non-zero eigenvalues of ρ₁ and ρ₂ are the same, leading to equal entropies - Tr[ρ₁ ln ρ₁] = - Tr[ρ₂ ln ρ₂].

Therefore, regardless of whether we take the subsystem 1 or 2, the entanglement entropy will be the same, reflecting that they are intrinsically linked and share the same entanglement properties.