Reversible arithmetic coding for quantum data compression

We study the problem of compressing a block of symbols (a block quantum state) emitted by a memoryless quantum Bernoulli source. We present a simple-to-implement quantum algorithm for projecting, with high probability, the block quantum state onto the typical subspace spanned by the leading eigenstates of its density matrix. We propose a fixed-rate quantum Shannon–Fano code to compress the projected block quantum state using a per-symbol code rate that is slightly higher than the von Neumann entropy limit. Finally, we propose quantum arithmetic codes to efficiently implement quantum Shannon–Fano codes. Our arithmetic encoder and decoder have a cubic circuit and a cubic computational complexity in the block size. Both the encoder and decoder are quantum-mechanical inverses of each other, and constitute an elegant example of reversible quantum computation.