Measurement of a qubit in the computational basis - Phase estimation

In summary, the conversation discusses the measurement of a qubit in the computational basis and how it relates to the probability of obtaining a measurement outcome. The question is then raised about a specific equation and its derivation, and the conversation ends with a potential explanation for the equation.
  • #1
Peter_Newman
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Hello,

I have a question about the measurement of a qubit in the computational basis. I would like to first state what I know so far and then ask my actual question at the end.What I know:
Let's say we have a qubit in the general state of . Now we can define the following measurement operators depending on whether we want to measure the qubit in state or . Let's say I am interested in the state .

The corresponding operator would then be defined as follows . The probability of obtaining a measurement outcome is then defined by:

.My Question:
I read the following in the Wikipedia article on Quantum Phase Estimation (Wiki, section measurement). We have now given there the following quantum state:



Now it is said that a measurement in the computational basis on the first register yields the result with probability;



I am interested in the last equation here (), how do you arrive at it? With what I know so far, I can't really derive the last equation, so I would be interested in knowing how the derivation is. Also the simplification does not open up to me. Maybe someone here can demystify it.
 
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  • #2
It seems to me like there is a term (x-a) missing in the exponential function. Might that be the case?
 
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  • #3
Yes that is unfortunately correct!
I would like to improve my first post regarding this error. Unfortunately, I can no longer edit this one...

Correct it is:





Based on this, I would now assert the following as to why one come up with .
So the scalar product of is only 1 if , if this is the case, everything reduces to , where we put out the constant and note that one of the exp terms is 1 since iff . Right? For all other , the scalar product is . Therefore, then follows.
 
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  • #4
That looks reasonable to me.
 
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