# Reduced density operator of a maximally entangled state

Is the reduced density operator of a maximally entangled pure state always maximally mixed (trace being half)? I test it on 4 bell state and this claim is true. I wonder why and can the degree of entanglement of pure state be revealed by the trace of the reduced density operator?

For your first question, if you don't take it as a definiton (which is often done), you can prove it by noting that a maximally entangled (pure) state has the Schmidt decomposition $\sum_{i=1}^d 1/\sqrt{d} |i\rangle\otimes |i\rangle$ is an appropriate basis. The partial trace is the identity.