Questions from Quantum Measurements

In summary: It's just that there is no canonical way of writing the equation in terms of eigenvectors.I apologise.This is the Pauli Y matrix.Few Information:1. X and Y are written in the {∣0⟩,∣1⟩}2. ∣+⟩=(1/√2)*(∣0⟩+∣1⟩)3. Pauli expressed these matrices in terms of eigenvectors in an important paper.4. The eigenvalues and eigenvectors can be found by diagonalization of the matrix.By definition, an eigen
  • #1
ARoyC
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11
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.
 

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  • #2
Hi, everyone.

Please check the following two questions.

1.
Screenshot 2023-07-08 122050.jpg

2.
Screenshot 2023-07-08 122200.jpg

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
 
  • #3
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
$$\begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 &-1 \end{pmatrix}.$$
 
  • #4
vanhees71 said:
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?
 
  • #5
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.
 
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  • #6
vanhees71 said:
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.

Screenshot 2023-07-10 204848.jpg
This is the Pauli Y matrix.
 
  • #7
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⟩)
 
  • #8
ARoyC said:
I apologise.

View attachment 329075This is the Pauli Y matrix.
The usual notation for that is ##\sigma_y##.
 
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  • #9
PeroK said:
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.
 
  • #10
ARoyC said:
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##.
 
  • #11
vanhees71 said:
What is "Pauli Y matrix".

PeroK said:
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.
 
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  • #12
George Jones said:
As compensation, my mother-in-law made some wonderful aloo paratha for me.
I'm quite partial to saag aloo myself.
 
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  • #13
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.
 
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Related to Questions from Quantum Measurements

What is the role of the observer in quantum measurements?

In quantum mechanics, the observer plays a crucial role in the measurement process. According to the Copenhagen interpretation, the act of measurement causes the wave function to collapse from a superposition of states to a single eigenstate. This means that the properties of a quantum system are not determined until they are observed.

What is wave function collapse?

Wave function collapse is a phenomenon where a quantum system transitions from a superposition of multiple states to a single eigenstate due to measurement. Before measurement, a quantum system exists in a superposition, representing multiple possibilities. The act of measurement forces the system to 'choose' one of these possibilities, collapsing the wave function to a definite state.

What is the Schrödinger's cat thought experiment?

Schrödinger's cat is a thought experiment that illustrates the concept of superposition and wave function collapse in quantum mechanics. It describes a scenario where a cat is placed in a sealed box with a radioactive atom, a Geiger counter, poison, and a hammer. If the atom decays, the Geiger counter triggers the hammer to release the poison, killing the cat. Until the box is opened and observed, the cat is considered to be both alive and dead simultaneously, representing a superposition of states.

What is the difference between classical and quantum measurements?

Classical measurements do not affect the system being measured; the properties of classical objects are well-defined and can be measured without altering the system. In contrast, quantum measurements fundamentally alter the state of the system. The act of measuring a quantum system forces it into a specific state, a phenomenon not seen in classical physics.

What is quantum entanglement and how does it relate to measurements?

Quantum entanglement is a phenomenon where two or more particles become interconnected in such a way that the state of one particle instantaneously affects the state of the other, regardless of the distance between them. When a measurement is made on one entangled particle, the state of the other particle is immediately determined. This non-local property challenges classical intuitions about separability and locality in measurements.

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