The Exercise

A source emits quantum systems one after another, each prepared in the same state $\rho$ and independently of the others. To store $n$ of them, one encodes the $n$ systems into a register of $nR$ qubits and later decodes the register, asking that what comes out be ever closer to what went in as $n$ grows. The rate $R$ counts qubits per emitted system, and the question is how low it can go. Schumacher’s noiseless channel coding theorem answers with the von Neumann entropy of $\rho$.

A Quantum System

A quantum system $Q$ is a physical system whose state space is a finite dimensional complex Hilbert space.^✦ A state $\rho$ of $Q$ is a positive operator of unit trace on the state space of $Q$, pure or mixed.^✦

The von Neumann entropy of a state $\rho$ of $Q$ is the Shannon entropy, in nats, of the eigenvalues $\lambda_i$ of $\rho$: $$S(\rho) = -\operatorname{tr}(\rho \ln \rho) = -\sum_i \lambda_i \ln \lambda_i .$$(1)^✦

Reliable Compression

A compression scheme of rate $R$ for $Q$ is a sequence of pairs of operations, one for each number $n$ of copies: an encoding $\mathcal{C}^n$ taking states of $n$ copies of $Q$ to states of a space of dimension $2^{nR}$, and a decoding $\mathcal{D}^n$ taking those back to states of $n$ copies of $Q$. A state $\rho$ of $Q$ is reliably compressible at rate $R$ when some compression scheme of rate $R$ restores $n$ copies of $\rho$, the state $\rho^{\otimes n}$, with an entanglement fidelity $F$ that tends to one: $$\lim_{n \to \infty} F\bigl(\rho^{\otimes n}, \mathcal{D}^n \circ \mathcal{C}^n\bigr) = 1 .$$(2)^✦

The Claim

Since the entropy (1) is measured in nats, the threshold of $R$ bits per copy appears in the theorem as $R \ln 2$.

What Is Checked

The claim is stated over the states of a finite dimensional quantum system, their von Neumann entropy (1), and reliable compressibility at a rate (2). What the library proves about these constructions is the equivalence (3), for a state of any finite dimensional system.