This chapter describes the principle of compression in quantum communication channels. The underlying concept is that it is possible to convey “faithfully” a quantum message with a large number of qubits, while transmitting a compressed version of this message with a reduced number of qubits through the channel. Beyond the mere notion of fidelity, which characterizes the quality of quantum message transmission, the description brings the new concept of typicality in the space defined by all possible “quantum codewords.” The theorem of Schumacher's quantum compression states that for a qubit source with von Neumann entropy S, the message compression factor R has S − ε for the lower bound, where ε is any nonnegative parameter that can be made arbitrarily small for sufficiently long messages (hence, R ≈ S is the best possible compression factor). An original graphical and numerical illustration of the effect of Schumacher's quantum compression and the evolution of the typical quantum-codeword subspace with increasing message length is provided.

Quantum data compression and fidelity

In this chapter, we have reached the stage where it is possible to start addressing the issues that are central to information theory, namely, “How efficiently can we code information in a quantum communication channel?” both in terms of economy of means – the concept of data compression – and accuracy of transmission – the concept of message integrity or minimal data error, referred to here as fidelity.