The Exercise

A qubit is prepared in $|0\rangle$ with probability $3/4$ and in $|1\rangle$ with probability $1/4$, so that its state is $$\rho = \tfrac34\, |0\rangle\langle 0| + \tfrac14\, |1\rangle\langle 1| .$$(1)^✦ A second source prepares the same two states with probabilities $2/3$ and $1/3$, and a third prepares the states $|+\rangle$ and $|-\rangle$, defined below, with those probabilities: $$\sigma = \tfrac23\, |0\rangle\langle 0| + \tfrac13\, |1\rangle\langle 1| ,$$(2) $$\sigma' = \tfrac23\, |+\rangle\langle +| + \tfrac13\, |-\rangle\langle -| .$$(3) The exercise asks for the trace distance from $\rho$ to $\sigma$ and from $\rho$ to $\sigma'$.

A Qubit and Its States

A quantum system $S$ is given by its state space, a finite dimensional complex Hilbert space.^✦ The qubit is the quantum system whose state space is $\mathbb{C}^2$, the two dimensional complex Euclidean space.^✦

A state $\rho$ of $S$ is a positive operator of unit trace on the state space of $S$, also called a density operator of $S$.^✦ A pure state $\psi$ of $S$ is a vector $\psi$ of unit norm in the state space of $S$; its projector $|\psi\rangle\langle\psi|$ is a state of $S$.

For $i \in \{0, 1\}$, the basis state $|i\rangle$ of the qubit is the pure state whose vector is the column of $\mathbb{C}^2$ with entry $1$ in position $i$ and $0$ in the other.^✦ The states $|+\rangle$ and $|-\rangle$ of the qubit are the pure states with vectors $$|\pm\rangle = \bigl(1/\sqrt{2},\, \pm 1/\sqrt{2}\bigr) = \frac{|0\rangle \pm |1\rangle}{\sqrt{2}} ,$$(4) the eigenvectors of the Pauli operator $X = |0\rangle\langle 1| + |1\rangle\langle 0|$ of the qubit, with eigenvalues $+1$ and $-1$.

Mixtures and Trace Distance

The mixture of states $\rho_i$ of $S$, indexed by $i$ in a finite set, with weights $p_i \geq 0$ that sum to one, is the state of $S$ with density operator $$\sum_i p_i\, \rho_i .$$(5)^✦

The trace distance $D(\rho, \sigma)$ between two states $\rho$ and $\sigma$ of $S$, with $|A|$ the positive square root of $A^\dagger A$, is $$D(\rho, \sigma) = \tfrac12 \operatorname{tr} |\rho - \sigma| ,$$(6)^✦

The Claim

What Is Checked

The claim is stated over the trace distance (6) between states of the qubit, and over the states $\rho$, $\sigma$ and $\sigma'$, each a mixture (5) of the projectors onto $|0\rangle$ and $|1\rangle$, or onto $|+\rangle$ and $|-\rangle$, with the weights of (1), (2) and (3).

The library proves that the projectors onto $|0\rangle$, $|1\rangle$, $|+\rangle$ and $|-\rangle$ are states, so $\rho$, $\sigma$ and $\sigma'$ are states of the qubit. It proves as well that the vectors of $|0\rangle$, $|1\rangle$, $|+\rangle$ and $|-\rangle$ have unit norm.