What does the partial trace of $|W\rangle$ states represent physically?

Question

Given the W-state $|W\rangle = |001\rangle + |010\rangle + |100\rangle$, where $|ijk \rangle $implies $|i\rangle_A \otimes |j\rangle_B \otimes |k \rangle_C$, the partial trace over first qubit turns out to be

$$Tr_A\left[|W\rangle \langle W| \right] = {}_A\langle0 |W\rangle \langle W|0\rangle_A + {}_A\langle 1 |W\rangle \langle W|1\rangle_A = |00\rangle \langle 00| + |01\rangle \langle 01| + |01\rangle \langle 10| + |10 \rangle \langle 01| + |10\rangle \langle 10|$$

What physics does this tell us?

Answer 1

The key thing that it tells you is that the W-state is partially entangled. This is perhaps a little clearer to see if you trace over two qubits: $$ \text{Tr}_{AB}(|W\rangle\langle W|)=\frac23|0\rangle\langle 0|+\frac13|1\rangle\langle 1|. $$ The state is mixed, so the overall pure state is entangled, but it's not maximally mixed, so the overall state is not maximally entangled.