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Suppose there're two systems $A$ and $B$, if we're in a pure state $|\psi\rangle\in\mathbb{H}_a\otimes\mathbb{H}_b$, let $\hat A$ be an operator acts on $\mathbb{H}_a$, and $|\psi\rangle=\sum_{i,j}\psi_{i,j}|a_i\rangle|b_j\rangle$. Then, the expectation value of A with respect to the state $|\psi\rangle$ can be calculated as: $$ \langle A\rangle_{\psi}= Tr_{\mathbb{H}_a\otimes\mathbb{H}_b}(A|\psi\rangle\langle\psi|)=\sum_{i,j}\langle a_i|\langle b_j|A\otimes\mathbb{1}_B|\psi\rangle\langle\psi|a_i\rangle|b_j\rangle=... $$

I'm very confused how those two equalities coming from? Especially for the term $|\psi\rangle\langle\psi|$ in the sum, can I understand their function as generating a $\psi$ matrix with different constants $\psi_{i,j}$? Thanks!!

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I’m not sure I understand your question:

That is the definition of the expectation value of an operator $A$ using the trace

$$\langle A \rangle = Tr (\rho A)$$

where the density matrix is given by

$$\rho = \mid \psi \rangle \langle \psi \mid $$

The density matrix is ubiquitous in quantum mechanics.

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