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This is one of the problems from Assignment 2 from CS191x at edx.org, so please do not post explicit answers.

We have two qubits in the state |0+⟩ and we want to entangle them by performing a partial measurement on the first qubit. Which of the following measurements should we perform on the first qubit?

(I am not including possible answers as I want to understand it in general, not only this specific case)

I am not sure how to determine the state after measuring one of the qubits. For the standard basis my reasoning goes like this: if we measure the first qubit in this base, its resulting state will be |0⟩ with the probability of 1. Because the state of two qubits is |0+⟩, we know for sure that the second qubit will then be in the state |+⟩ and therefore there is no entanglement.

Firstly, I do not know if my reasoning above is correct at all. Secondly, how to determine the resulting state in a more general case, e.g. in the sign basis? (|+⟩, |-⟩).

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I am not sure how to determine the state after measuring one of the qubits.

I believe there are two parts to this question: Using measurements and using partial measurements.

What you've said about making a projective measurement (that is a measurement in the standard bassis) on the first qubit is correct. The resulting state of the system will be |0+⟩ with 100% probability.

As another example, if you make a projective measurement on the second qubit, do you know what state will be returned?

If you look at the measurement postulate, you will see that it will return the state |0⟩|1⟩ with 50% probability and the state |0⟩|0⟩ with 50% probability.

Clearly, neither of these are entangled.

The second part is how you perform a "partial measurement" which should depend on the context of the assignment/text.

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