The Exercise

A qubit is prepared in the state $|0\rangle$ with probability $p$ and in the state $|+\rangle = (|0\rangle + |1\rangle)/\sqrt{2}$ with probability $1-p$. The two pure states are not orthogonal, $\langle 0|+\rangle = 1/\sqrt{2}$, and the density operator of the preparation is $$\rho = p\,|0\rangle\langle 0| + (1-p)\,|+\rangle\langle +| .$$(1)^✦ The exercise asks for the von Neumann entropy $S(\rho)$ of this state and for its comparison with the Shannon entropy $H(p, 1-p)$ of the weights.

A Quantum System

A quantum system $S$ is a physical system whose state space is a finite dimensional complex Hilbert space.^✦ Any such space $H$ is itself the state space of a system^✦, and the qubit is the two level system obtained in this way from $\mathbb{C}^2$^✦. A state $\rho$ of $S$ is a positive operator of unit trace on the state space of $S$.^✦

The Pauli matrices are the complex $2 \times 2$ matrices $$X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \qquad Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.$$(2) The Pauli vector is $\vec\sigma = (X, Y, Z)$, and for a real vector $\vec r = (r_1, r_2, r_3)$ the matrix $\vec r \cdot \vec\sigma$ is $r_1 X + r_2 Y + r_3 Z$^✦. Write $I$ for the $2 \times 2$ identity matrix. The Bloch matrix of $\vec r$ is $$\tfrac12 \left( I + \vec r \cdot \vec\sigma \right).$$(3)^✦ The Bloch state of a real vector $\vec r$ with $\|\vec r\| \le 1$ is the state of the qubit whose density operator is the Bloch matrix (3).^✦

The mixture of two states $\rho$ and $\sigma$ of $S$ with weight $p$, where $0 \le p \le 1$, is the state of $S$ with density operator $p\rho + (1-p)\sigma$: the state prepared by choosing $\rho$ with probability $p$ and $\sigma$ otherwise.^✦

The State of the Exercise

The state of the exercise is the mixture, with weight $p$, of the Bloch states of the unit vectors $\hat z = (0, 0, 1)$ and $\hat x = (1, 0, 0)$.^✦

For probabilities $q_1$ and $q_2$ the Shannon entropy is $H(q_1, q_2)$, and the von Neumann entropy of a state $\rho$ is the Shannon entropy of the eigenvalues of its density operator, measured in nats: $$H(q_1, q_2) = -q_1 \ln q_1 - q_2 \ln q_2 , \qquad S(\rho) = -\operatorname{tr}(\rho \ln \rho),$$(4)^✦ with $0 \ln 0 = 0$.

The Claim

The Bloch matrix depends linearly on $\vec r$, so the density operator of $\rho$ is the Bloch matrix of $\vec r = p\,\hat z + (1-p)\,\hat x = (1-p, 0, p)$. Its eigenvalues are $(1 \pm \|\vec r\|)/2$, with $\|\vec r\| = \sqrt{(1-p)^2 + p^2}$.

What Is Checked

The three statements are made over the qubit and the state $\rho$, built as the mixture with weight $p$ of the Bloch states of $\hat z$ and $\hat x$; they compare its von Neumann entropy with the Shannon entropy of two probabilities.

About these constructions the library proves that the Bloch matrix (3) is positive exactly when $\|\vec r\| \le 1$ and that its trace is one, so the Bloch state is a state of the qubit, and that the convex combination $p\rho + (1-p)\sigma$ of two density operators is again a density operator, so the mixture is a state.