The Exercise

A source emits one qubit at each use, independently from one use to the next: the state $|0\rangle$ with probability $p$ and the state $|1\rangle$ with probability $1-p$. Each emitted qubit, its preparation unknown, is in a state diagonal in the computational basis. The exercise asks for the von Neumann entropy of that state and for the rates at which long blocks of the source’s output can be compressed and recovered with fidelity tending to one.

A Quantum System

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

A state $\rho$ of $Q$ is a positive operator of unit trace on the state space of $Q$.^✦ A pure state $\psi$ of $Q$ is a unit vector $\psi$ in the state space of $Q$^✦, and it determines the state $|\psi\rangle\langle\psi|$ of $Q$, the rank one projector onto $\psi$.^✦ The basis states $|0\rangle$ and $|1\rangle$ of the qubit are the pure states whose vectors are the standard basis vectors $e_0$ and $e_1$ of $\mathbb{C}^2$.^✦

The mixture of states $\rho_i$ of $Q$, indexed by $i$ in a finite set, with weights $p_i$ that are nonnegative and sum to one, is the state $\sum_i p_i \rho_i$ of $Q$.^✦

The Source

For $0 \le p \le 1$, the qubit source $\rho(p)$ is the mixture of the basis states $|0\rangle$ and $|1\rangle$ with weights $p$ and $1-p$: $$\rho(p) = p\,|0\rangle\langle 0| + (1-p)\,|1\rangle\langle 1| = \begin{pmatrix} p & 0 \\ 0 & 1-p \end{pmatrix}.$$(1)^✦

Entropy and Compression

The Shannon entropy of a probability distribution $(q_1, \dots, q_d)$ is $-\sum_i q_i \ln q_i$, measured in nats; for the distribution $(p, 1-p)$ it is $H(p) = -p \ln p - (1-p) \ln (1-p)$, and $H(p)/\ln 2$ is the same entropy in bits.^✦ The von Neumann entropy of a state $\rho$ of $Q$ is the Shannon entropy of the eigenvalues of $\rho$: $$S(\rho) = -\operatorname{tr}(\rho \ln \rho).$$(2)^✦

A state $\rho$ of $Q$, taken as a source that emits $\rho$ independently at each use, is reliably compressible at rate $R$ when for each block length $n$ there is a compression scheme of rate $R$, an encoding $\mathcal{C}^n$ followed by a decoding $\mathcal{D}^n$ that stores each block of $n$ qubits in $nR$ qubits, whose entanglement fidelity $F$, the fidelity with which the scheme preserves the entanglement of a purification of $\rho^{\otimes n}$ with a reference system, tends to one on $n$ copies of $\rho$: $$\lim_{n \to \infty} F\bigl(\rho^{\otimes n}, \mathcal{D}^n \circ \mathcal{C}^n\bigr) = 1.$$(3)^✦

The Claim

What Is Checked

The claim is stated over the qubit and its basis states $|0\rangle$ and $|1\rangle$, the source $\rho(p)$ built from their projectors by the mixture (1), the von Neumann entropy (2), and reliable compressibility (3).