Measuring multiqubit systems

You measure the first qubit of the following state in the X basis and obtain |+>.

$\frac{|00\rangle+|11\rangle}{\sqrt{2}}$

What is the state of the second qubit?

The given state can be rewritten in the X basis as:

$\frac{|++\rangle+|--\rangle}{\sqrt{2}}$

Therefore, if |+> was measured for the first qubit, the second qubit will be in the |+> state too.

What does the following gate sequence simplify to (U_i means apply gate U to qubit i)?

$H_1 \otimes H_2 \;CNOT_{12} \; H_1 \otimes H_2$

It simplifies to:

$CNOT_{21}$

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