# Nielsen & Chuang Exercise 2.2 - “Matrix representations: example” [closed]

Reproduced from Exercise 2.2 of Nielsen & Chuang's Quantum Computation and Quantum Information (10th Anniversary Edition):

Suppose $$V$$ is a vector space with basis vectors $$|0\rangle$$ and $$|1\rangle$$, and $$A$$ is a linear operator from $$V$$ to $$V$$ such that $$A|0\rangle = |1\rangle$$ and $$A|1\rangle = |0\rangle$$. Give a matrix representation for $$A$$, with respect to the input basis $$|0\rangle, |1\rangle$$, and the output basis $$|0\rangle, |1\rangle$$. Find input and output bases which give rise to a different matrix representation of $$A$$.

Note: This question is part of a series attempting to provide worked solutions to the exercises provided in the above book.

Immediately, we can see that $$A = |1\rangle\langle0| + |0\rangle\langle1|.$$ If the input and out bases are $\{|0\rangle, |1\rangle\}$, then $$|0\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \quad |1\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \quad\textrm{and}\quad \langle0| = \begin{pmatrix} 1 & 0 \end{pmatrix}, \quad \langle1| = \begin{pmatrix} 0 & 1 \end{pmatrix},$$ so we can write the first equation as \begin{align} A &= \begin{pmatrix} 1 \\ 0 \end{pmatrix} \otimes \begin{pmatrix} 0 & 1 \end{pmatrix} + \begin{pmatrix} 0 \\ 1 \end{pmatrix} \otimes \begin{pmatrix} 1 & 0 \end{pmatrix} \\ &= \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} + \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}\\ &= \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \end{align} to solve the first question. Note that in this case, $A$ is the equal to the bitflip or Pauli $X$ operation.
Secondly, for fun, let us choose to write $A$ in input basis $\{|+\rangle, |-\rangle\}$, where $$|+\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle) \quad\textrm{and}\quad |-\rangle = \frac{1}{\sqrt{2}}(|0\rangle - |1\rangle)$$ and output basis $\{|L\rangle, |R\rangle\}$, where $$|L\rangle = \frac{1}{\sqrt{2}}(|0\rangle + i|1\rangle) \quad\textrm{and}\quad |R\rangle = \frac{1}{\sqrt{2}}(|0\rangle - i|1\rangle).$$ Rewriting our original $\{|0\rangle, |1\rangle\}$ bases vectors in terms of the above bases, we find \begin{align} |0\rangle &= \frac{1}{\sqrt{2}}(|+\rangle + |-\rangle) = \frac{1}{\sqrt{2}}(|L\rangle + |R\rangle) \quad \textrm{and}\\ |1\rangle &= \frac{1}{\sqrt{2}}(|+\rangle - |-\rangle) = \frac{i}{\sqrt{2}}(|R\rangle - |L\rangle), \end{align} with $\langle0| = (|0\rangle)^\dagger$ and $\langle1| = (|1\rangle)^\dagger$.
From this we can rewrite $A$ as \begin{align} A &= \frac{i}{2}(|R\rangle - |L\rangle)(\langle+| + \langle-|) + \frac{1}{2}(|L\rangle + |R\rangle)(\langle+| - \langle-|) \\ &= \frac{1}{2}\big[ (1+i)|R\rangle\langle+| + (1-i)|L\rangle\langle+| + (-1+i)|R\rangle\langle-| + (-1-i)|L\rangle\langle-|\big] \\ &= \frac{1+i}{2}(|R\rangle\langle+| - i|L\rangle\langle+| + i|R\rangle\langle-| - |L\rangle\langle-|) \end{align} To get $A$ in the desired matrix form, we then set $$|L\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \quad |R\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \quad\textrm{and}\quad \langle+| = \begin{pmatrix} 1 & 0 \end{pmatrix}, \quad \langle-| = \begin{pmatrix} 0 & 1 \end{pmatrix},$$ such that $$A = \frac{1+i}{2} \begin{pmatrix} -i & -1 \\ 1 & i \end{pmatrix}.$$
• How do you know that $\left|0\right> = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \quad \left|1\right> = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$ ? Do the vectors always have these components? – Turkhan Badalov Aug 15 '18 at 20:20
• Just wondering, if $\left|0\right> = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$, why is it that you can also set $\left|L\right> = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$? – wei2912 Nov 4 '18 at 8:28