The Exercise

A qubit is prepared in one of three states, with density matrices in the computational basis $\{|0\rangle, |1\rangle\}$ $$\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \qquad \frac12 \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}, \qquad \frac13 \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}.$$(1) The first two are pure, the projectors $|0\rangle\langle 0|$ and $|+\rangle\langle +|$ with $|+\rangle = (|0\rangle + |1\rangle)/\sqrt2$; the third is mixed. The exercise asks for the von Neumann entropy of each.

States of a Qubit

A quantum system $Q$ is a physical object whose state space is a finite dimensional complex Hilbert space.^✦ The qubit is the two level system whose state space is $\mathbb{C}^2$.^✦ A state $\rho$ of $Q$ is a positive operator of unit trace on the state space of $Q$.^✦

States of the qubit are written with the Pauli 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) Let $\vec\sigma = (X, Y, Z)$ be the Pauli vector and $I$ the identity matrix. For a real vector $\vec r = (r_1, r_2, r_3)$ the matrix $\vec r \cdot \vec\sigma$ is $r_1 X + r_2 Y + r_3 Z$^✦, and the Bloch matrix of $\vec r$ is $$\frac12 \left( I + \vec r \cdot \vec\sigma \right) = \frac12 \begin{pmatrix} 1 + r_3 & r_1 - i r_2 \\ r_1 + i r_2 & 1 - r_3 \end{pmatrix}.$$(3)^✦ The Bloch state of a real vector $\vec r$ with $r_1^2 + r_2^2 + r_3^2 \le 1$ is the state of the qubit whose density operator is its Bloch matrix (3).^✦

The three states of (1) are the Bloch states of $(0, 0, 1)$, $(1, 0, 0)$ and $(\tfrac23, 0, \tfrac13)$.

The Entropy

The von Neumann entropy of a state $\rho$ of a system $Q$ is $$S(\rho) = -\operatorname{tr}(\rho \ln \rho),$$(4)^✦ the sum of $-\lambda \ln \lambda$ over the eigenvalues $\lambda$ of $\rho$, taken with the natural logarithm and so measured in nats. The Shannon entropy of two probabilities $p_1$ and $p_2$ is $H(p_1, p_2) = -p_1 \ln p_1 - p_2 \ln p_2$, with $0 \ln 0 = 0$; for a state with two eigenvalues $p_1$ and $p_2$, $S(\rho) = H(p_1, p_2)$.

The eigenvalues of the Bloch matrix (3) are $(1 \pm |\vec r|)/2$, with $|\vec r|$ the length of $\vec r$.

The Claim

What Is Checked

The claim is stated over the von Neumann entropy (4) of three Bloch states of the qubit, those of the vectors $(0, 0, 1)$, $(1, 0, 0)$ and $(\tfrac23, 0, \tfrac13)$, and over the Shannon entropy $H$ of two numbers.

The Bloch matrix of a real vector $\vec r$ is positive exactly when $|\vec r| \le 1$, and its trace is one. The Bloch state of a vector in the closed unit ball is therefore a density operator, a state of the qubit.