Questions from Quantum Measurements

[ARoyC]

Homework Statement

Please check the attached screenshots for the questions.

Relevant Equations

Spectral Decomposition of Y, Probability of a Measurement Outcome and Posterior State Formula

[Mentor Note: Two similar thread starts merged]

The questions are from MIT OCW. First of all, I cannot understand what is the meaning of the measurement outcome being 0. How can an eigenvalue be 0? I tried doing the problems guessing that by 0 they mean the posterior state will be |0>. The only correct answer I got was for the first part of the second problem. I first wrote the state of the first qubit after passing through the first Hadamard gate. Then I wrote the tensor product state of the two qubits. After that, I applied the cNOT gate on the second qubit and then again the Hadamard gate on the first qubit. Then I used Born Rule to find the probability of the first qubit being in |0>. But I cannot get the answer for the other two parts using this method. Same problem for the first question.

It would be great if someone can help me out. Thank you in advance.

 

[ARoyC]

Hi, everyone.

Please check the following two questions.

1.

2.

The questions are from MIT OCW. First of all, I cannot understand what is the meaning of the measurement outcome being 0. How can an eigenvalue be 0? I tried doing the problems guessing that by 0 they mean the posterior state will be |0>. The only correct answer I got was for the first part of the second problem. I first wrote the state of the first qubit after passing through the first Hadamard gate. Then I wrote the tensor product state of the two qubits. After that, I applied the cNOT gate on the second qubit and then again the Hadamard gate on the first qubit. Then I used Born Rule to find the probability of the first qubit being in |0>. But I cannot get the answer for the other two parts using this method. Same problem for the first question.

It would be great if someone can help me out. Thank you in advance.

Regards

Annwoy Roy Choudhury

 

[vanhees71]

Please give a complete problem formulation. If we don't know, what $Y$ is, we can't answer any of the questions!

What do you think is a problem of a eigenvalue 0? Take, e.g., $\hat{S}_z$ of a spin-1 particle. It has the eigenvalues 1, 0, -1. In the corresponding eigenbasis the corresponding matrix is
$$\begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 &-1 \end{pmatrix}.$$

 

[ARoyC]

Please give a complete problem formulation. If we don't know, what $Y$ is, we can't answer any of the questions!

What do you think is a problem of a eigenvalue 0? Take, e.g., $\hat{S}_z$ of a spin-1 particle. It has the eigenvalues 1, 0, -1. In the corresponding eigenbasis the corresponding matrix is
$$\begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 &-1 \end{pmatrix}.$$

Y is Pauli Y matrix.

0 eigenvalue is common for every linear transformation for (0,0,0). So it is a trivial case. And in these questions, even if they mean that the eigenvalue is 0, then what is the method to do the problems?

 

[vanhees71]

What is "Pauli Y matrix". You HAVE TO clearly define symbols in physics, that are not standard! Otherwise nobody can understand, what's the problem.

 

[ARoyC]

What is "Pauli Y matrix". You HAVE TO clearly define symbols in physics, that are not standard! Otherwise nobody can understand, what's the problem.

I apologise.

This is the Pauli Y matrix.

 

[ARoyC]

Few Information:

1. X and Y are written in the {∣0⟩,∣1⟩} X = |0><1| + |1><0| and Y = |0><1| - |1><0|

2. ∣+⟩=(1/√2)*(∣0⟩+∣1⟩)

 

[PeroK]

I apologise.

View attachment 329075This is the Pauli Y matrix.

The usual notation for that is $\sigma_y$.

 

[ARoyC]

The usual notation for that is $\sigma_y$.

I am sorry. The screenshots of the questions are directly from MIT OCW. So, I had nothing to do with it.

 

[vela]

0 eigenvalue is common for every linear transformation for (0,0,0). So it is a trivial case. And in these questions, even if they mean that the eigenvalue is 0, then what is the method to do the problems?

By definition, an eigenvector is a non-zero vector satisfying $T \vec x = \lambda \vec x$. While $\vec x$ has to be non-zero, there's no condition on $\lambda$.

 

[George Jones]

What is "Pauli Y matrix".

The usual notation for that is $\sigma_y$.

I think this depends on context, and that it is not unusual to see X, Y, Z in quantum computing books, or quantum books that have substantial sections on quantum computing. I am not entirely sure, and I ould like to check my books, but I'm at my in-laws, and thus separated from my books by several thousand kilometres. As compensation, my mother-in-law made some wonderful aloo paratha for me.

 

[PeroK]

As compensation, my mother-in-law made some wonderful aloo paratha for me.

I'm quite partial to saag aloo myself.

 

[WWGD]

Without a Gulab Jamoón, Kuch Nahi.
Edit: But, yes, eigenvalues can be 0, eigenvectors not the 0 vector. The eigenvalues will actually be 0 when the matrix is singular, almost by definition.