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

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?

Your mistake is computing exponent of matrix; use the formula

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

• I haven't known about this "hack". Could you suggest a site where I can read about it? Commented Dec 6, 2019 at 21:12
• This "hack" should be in any good QM or Quantum Information course; see wikipedia; I substituted $\hat{n}=(0,1,0)$ Commented Dec 6, 2019 at 21:19
• This is an application of a matrix function definition. See my answer below. Commented Dec 6, 2019 at 23:43

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. Commented Dec 7, 2019 at 0:29
• For more information see Nielsen and Chuang, Quantum Computation and Quantum Information, pg. 75 (chapter 2.1.8 - Operator functions) Commented Dec 7, 2019 at 9:07
• @Martin, thanks for your note. One quick question, in this case what is 𝑣𝑖? Commented Jan 16, 2020 at 9:25
• please use comments for additional questions. $v_{i}$ is ith eigenvector of matrix $A$. Commented Jan 16, 2020 at 12:16
• Please don't add "thank you" as an answer. Instead, vote up the answers that you find helpful. - From Review Commented Jan 16, 2020 at 14:21