Quantum Country exercise 5

This is one in a series of posts working through the exercises in the Quantum Country online introduction to quantum computing and related topics. The exercises in the original document are not numbered; I have added my own numbers for convenience in referring to them.

Exercise 5. Show that the matrix X = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} is unitary.

Answer: A matrix U is unitary if U^\dagger U = I where U^\dagger is the adjoint matrix to U (produced by taking the transpose U^T of U and then replacing all values by their complex conjugates) and I is the identity matrix.

Since X is symmetric we have X^T = X and since the values of X (and thus of X^T) are all real we have \left( X^T \right)^* = X^T. We thus have

X^\dagger X = \left( X^T \right)^* X = X^T X = XX

= \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = I

Since X^\dagger X = I the matrix X is unitary.

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