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You're right, there should be 16 terms. But since they are tracing out Alice's system in the next step only 4 terms are relevant - those that correspond to $|00\rangle \langle 00|$, $|01\rangle \langle 01|$, $|10\rangle \langle 10|$ and $|11\rangle \langle 11|$. Tracing out $|00\rangle \langle 01|\cdot |\phi\rangle\langle\psi|$ will result in 0, because $\... 2 This picture can be confusing, since it makes some assumptions that are not clear when you quickly scan through the text. The picture of the oracle$U_f$describes its effect on the basis states that go into it, not on arbitrary superposition states. If both$x$and$y$are basis states, the oracle will behave exactly as the rectangle says, but the ... 4 To answer your first question, the quantum oracles are defined by their effect on the basis states$|0\rangle$and$|1\rangle$, and if the oracle has to be computed on a superposition of basis states, its effects are expressed using the fact that the oracle is a linear transformation. This means that you never compute$f(|+\rangle)$; instead, to compute the ... 2 You've computed$\rho^{AC},\rho^{BC}$incorrectly$-$they are matrices of size$8\times 8$, also they are not equal to$I/8$. Nevertheless their entropy is equal, i.e.$S(A,C) = S(B,C)$. So$|\psi \rangle$is indeed a counterexample to the statement. The inequality$S(A,B)\geq S(A|C)-S(B|C)$, which is equivalent to $$S(A,B) + S(B,C) \geq S(A,C)$$ is ... 2 I will try to link the two concepts in the following way. It is rough, descriptive, but contains the essential idea. Here we are talking about the so-called orbital angular momentum (OAM) of an electron orbiting around the nuclei. In classical mechanics, OAM is defined as$ \vec{L} \equiv \vec{r}\times\vec{p} $where$\vec{r}\$ is the position of the ...