How to get specific state applying $e^{-i\phi \sigma_2/2}$ to $|0\rangle$ or $|1\rangle$?

2

I try to solve problems from Problems in Quantum Computing.

I stuck with problem #3:

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I do the following:

Because: $$ \sigma_2 = \begin{pmatrix} 0 & -i\\ i & 0 \end{pmatrix}$$ Then: $$ -i \frac{\phi}{2}\sigma_2 = \begin{pmatrix} 0 & -\frac{\phi}{2}\\ \frac{\phi}{2} & 0 \end{pmatrix} $$

$$\exp\begin{pmatrix} 0 & -\frac{\phi}{2}\\ \frac{\phi}{2} & 0 \end{pmatrix} = \begin{pmatrix} 1 & \exp(-\frac{\phi}{2})\\ \exp(\frac{\phi}{2}) & 1 \end{pmatrix}$$

If I multiply the result of the last calculation with $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$ or $\begin{pmatrix} 0 \\ 1 \end{pmatrix}$ I can't get $\psi_1(\phi)$ or $\psi_2(\phi)$. I get some unnormalised state like:

$$ \begin{pmatrix} 1 \\ \exp(\frac{\phi}{2}) \end{pmatrix} $$

Does it mean that the definition of the problem is not correct?

Kenenbek Arzymatov

Posted 2019-12-06T19:41:52.013

Reputation: 211

Answers

3

Your mistake is computing exponent of matrix; use the formula

$$\exp(i\theta\sigma_2)=\cos(\theta)\cdot I+i\cdot \sin(\theta)\cdot\sigma_2$$

kludg

Posted 2019-12-06T19:41:52.013

Reputation: 2 447

I haven't known about this "hack". Could you suggest a site where I can read about it? – Kenenbek Arzymatov – 2019-12-06T21:12:13.253

1

This "hack" should be in any good QM or Quantum Information course; see wikipedia; I substituted $\hat{n}=(0,1,0)$

– kludg – 2019-12-06T21:19:32.137

This is an application of a matrix function definition. See my answer below. – Martin Vesely – 2019-12-06T23:43:49.447

5

A matrix function $f(A)$ for normal matrix $A$ is defined as follows \begin{equation} f(A)=\sum_{i=1}^{n}f(\lambda_i)v_iv_i^T \end{equation} where $\lambda_{i}$ is an eigenvalue and $v_{i}$ is coresponding eigenvector (note: transposed vector $v_{i}$ is a row vector).

In your case: $f(A) = \mathrm{e}^A$ and $A = -i\frac{\phi}{2}\sigma_{2}$.

Martin Vesely

Posted 2019-12-06T19:41:52.013

Reputation: 7 763

+1, though a reference stating the formula along with a proof would be appreciated. – Sanchayan Dutta – 2019-12-07T00:29:31.450

For more information see Nielsen and Chuang, Quantum Computation and Quantum Information, pg. 75 (chapter 2.1.8 - Operator functions) – Martin Vesely – 2019-12-07T09:07:48.520

@Martin, thanks for your note. One quick question, in this case what is ? – Parfait Atchadé – 2020-01-16T09:25:31.503

please use comments for additional questions. $v_{i}$ is ith eigenvector of matrix $A$. – Martin Vesely – 2020-01-16T12:16:00.017

Please don't add "thank you" as an answer. Instead, vote up the answers that you find helpful. - From Review

– met927 – 2020-01-16T14:21:38.117