## 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:

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?

3

Your mistake is computing exponent of matrix; use the formula

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

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 $$$$f(A)=\sum_{i=1}^{n}f(\lambda_i)v_iv_i^T$$$$ where $$\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 $$A = -i\frac{\phi}{2}\sigma_{2}$$.

+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