Finding a State Vector for a Qubit with a Probability of +1 Measurement of 9/10

[RJLiberator]

Homework Statement

Find a state vector |v> ∈ ℂ^2 such that if a measurement of δx is made on a qubit in this state, then the prob. of obtaining the value of +1 in the measurement is 9/10. What is <δx> in this case?

Homework Equations

State Vectors: https://en.wikipedia.org/wiki/Quantum_state

The Attempt at a Solution

This problem seems somewhat simple to me, but I want to make sure I have the simple case down before moving forward.

We seek to find a state vector that when taking a x-hat measurement results in a prob(+1) = 9/10 and thus a prob(-1) = 1/10.

Therefore, if we let |v> = ( λ, μ ) then we know that prob(+1) = |λ+μ|^2/2 and prob(-1) = |λ-μ|^2/2
Since we know what these need to equal, it's clear to see that λ = 2/sqrt(5) and μ=1/sqrt(5) is a solution.

Thus, <δx> is calculated as |α|^2-|β|^2 where α = (λ+μ)/sqrt(2) and β = (λ-μ)/sqrt(2). So, <δx> = 8/10=4/5.

So, the state vector is thus 1/sqrt(5) (2,1).

My concern is that I am not understanding the correct definition of state vector? Does a state vector 'have' any requirements that I am missing?

Are my calculations correct?

 

[RJLiberator]

A friendly bump for this question.

My main concern is the <δx>.

Thus, <δx> is calculated as |α|^2-|β|^2 where α = (λ+μ)/sqrt(2) and β = (λ-μ)/sqrt(2). So, <δx> = 8/10=4/5.

I did this same experiment for δy and got the same exact average of 4/5.

 

[RJLiberator]

Okay, let me simplify what I am asking then:

So, the state vector is thus 1/sqrt(5) (2,1).

Is this a capable result? Does 2/sqrt(5) + 1/sqrt(5) need to equal 1? Or does it matter?

We see that if we square it, it does indeed equal 1, 4/5+1/5 = 1.

 

[DrClaude]

Find a state vector |v> ∈ ℂ^2 such that if a measurement of δx is made on a qubit in this state, then the prob. of obtaining the value of +1 in the measurement is 9/10. What is <δx> in this case?

Is that correct: measurement of δx, not x?

Is this a capable result? Does 2/sqrt(5) + 1/sqrt(5) need to equal 1? Or does it matter?

We see that if we square it, it does indeed equal 1, 4/5+1/5 = 1.

Most often, you want states to be normalized.

 

[RJLiberator]

Most often, you want states to be normalized.

Hm...
First, my state vector is not normalized, correct?
Second, since you said 'most often' does that mean I should go find a normalized state vector or is my answer fine?

Is that correct: measurement of δx, not x?

With my notation of <δx> I am looking to find the average value. It should be defined as |α|^2-|β|^2

 

[DrClaude]

Hm...
First, my state vector is not normalized, correct?
Second, since you said 'most often' does that mean I should go find a normalized state vector or is my answer fine?

Yes, your state vector is normalized, which is what you most probably want. My comment was only to clarify that state vectors don't have to be normalized, but most of the time people will expect them to be.

With my notation of <δx> I am looking to find the average value. It should be defined as |α|^2-|β|^2

Then I don't understand the original problem:

Find a state vector |v> ∈ ℂ^2 such that if a measurement of δx is made on a qubit in this state, then the prob. of obtaining the value of +1 in the measurement is 9/10.

This means that δx must be an observable. I don't see which operator it corresponds to.

 

[RJLiberator]

Yes, your state vector is normalized, which is what you most probably want. My comment was only to clarify that state vectors don't have to be normalized, but most of the time people will expect them to be.

|v>|v> = 1 is the definition of normalized and so my vector is normalized? I understand! Thank you kindly.

Then I don't understand the original problem:

I wasn't sure of my calculation, it seems a bit too obvious. |α|^2 = prob(+1) and |β|^2 = prob(-1). If this is so, then it is obvious in the first 2 seconds of this problem that it is 9/10-1/10 = 4/5 since the probabilities must add up to 1.

The next question on my assignment asks me to repeat this process only with the y version instead of the x. The same situation occurs, 9/10-1/10-4/5 for the average value.

It 'feels' like I am misinterpreting something for it to be this simple.Once again, thank you kindly for your time, this is important for me to clarify and understand and its a difficult topic to discuss over a message board with such notation!

 

[DrClaude]

|v>|v> = 1 is the definition of normalized and so my vector is normalized? I understand! Thank you kindly.

Not exactly: the norm of a state is $\langle v | v \rangle$.

As for the rest, I just figured out what you mean: my problem is that you used the wrong symbol, and that confused me. When you write δx, you mean σ_x, one of the Pauli spin matrices, right?

In that case, let's see. If the state is
$$
| \psi \rangle = \frac{2}{\sqrt{5}} | \alpha \rangle + \frac{1}{\sqrt{5}} | \beta \rangle
$$
what is the probability of measuring the system in state $| \alpha \rangle$?

 

[RJLiberator]

The probability of |α> is (2/sqrt(5)) ^2 so it would be 4/5.

The probability of |β> is thus 1/5

4/5+1/5 = 1 so that checks out.

Now using the average, we see that 4/5-1/5=3/5 as the average which is different compared to my value.

Your way seems to be more correct (via intuition), but when I look at my paper, I'm not sure how I messed up.

prob(+1) = |α|^2 = |λ+μ|^2/2 = 9/10. If we choose λ = 2/sqrt(5) and μ=1/sqrt(5) as the state vector this works out.

So, I guess, why are you using |α> to equal = 2/sqrt(5) instead of α = (λ+μ)/sqrt(2)?

This question should solve any errors.

 

[DrClaude]

Please accept my apologies. I was confused at the beginning by δx, and that lead me in the wrong direction.

You calculations in the OP are correct.

 

[RJLiberator]

Thank you for taking the time to look through this. I greatly appreciate it.

Kind regards!