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I'm currently working through a tensor product example for a two qubit system.

For the expression:

$$

\rho_A = \sum_{J=0}^{1}\langle J | \Psi \rangle \langle \Psi | J \rangle

$$

Which has been defined as from going to a global state to a local state.

Here

$$ |\Psi \rangle = |\Psi^+ \rangle = \frac{1}{\sqrt{2}}(|01\rangle + |10\rangle ) $$

I'm not sure how to get from here to here: $$

\rho_A = \frac{1}{2} |1\rangle\langle 1| + \frac{1}{2} |0\rangle\langle 0|

$$

I'm finding that the terms in the summation reduce to dot products rather than the form of the answer. i.e

$$

\rho_A = \frac{1}{2}(\langle 0 | 01 \rangle + \langle 0 | 10 \rangle + \langle 1| 01 \rangle + \langle 1 | 10 \rangle + \langle 01 | 0 \rangle + \langle 10 | 0 \rangle + \langle 10 | 1 \rangle + \langle 01 | 1 \rangle )

$$

Which is where I'm thinking of going wrong?

Thanks for any help.