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A two-qubit system was originally in the state $ \frac{3}{4}|00\rangle-\frac{\sqrt{5}}{4}|01\rangle+\frac{1}{4}|10\rangle-\frac{1}{4}|11\rangle $ , and then we measured the first qubit to be 0 . Now, if we measure the second qubit in the standard basis, what is the probability that the outcome is 0 ?

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If you measure the first qubit to be $0$, you are left only with two possible outcomes, i.e. $|00\rangle $ and $|01\rangle$ with probability amplitudes $\frac{3}{4}$ and $-\frac{\sqrt5}{4}$.

Since they have to be normalized, you divide both amplitudes by $\sqrt{(\frac{3}{4})^2 + (-\frac{\sqrt5}{4})^2}$.

Then you will have a new state with two state vectors and all you have to do is calculate the probability of measuring $|00\rangle $, which should be pretty trivial!

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