Optimal Compression for Identically Prepared Qubit States

We establish the ultimate limits to the compression of sequences of identically prepared qubits. The limits are determined by Holevo’s information quantity and are attained through use of the optimal universal cloning machine, which finds here a novel application to quantum Shannon theory.

Figure 1

Optimal compression for known spectrum and completely unknown basis. The encoder collects information from subspaces with different angular momenta and concentrates it into a system with angular momentum $J_0$. The decoder spreads the information back, modulating the angular momentum by $\sqrt{n}$ units on average.

Figure 2

Optimal full-model compression. The encoder disassembles an arbitrary sequence of $n$ identically prepared qubits into a classical part ($1/2\log n$ bits) and a quantum part ($\log n$ qubits). The decoder recombines these two pieces of information, approximately retrieving the initial state of the sequence.

Figure 3

Spectral distributions of the output states with and without sampling. A comparison of the spectral distributions of the following states: the original state ${\rho }_g^{\otimes n}$ (black, solid line), the output state of the optimal protocol (red, dashed line), and the output state of a protocol with the same encoder of the optimal protocol and a decoder without sampling (blue, dashed line).