  We have proven that there exists a quantum state approximating any multi-copy
state universally when we measure the error by means of the normalized relative
entropy. While the qubit case was proven by Krattenthaler and Slater (IEEE
Trans. IT, 46, 801-819 (2000); quant-ph/9612043), the general case has been
open for more than ten years. For a deeper analysis, we have solved the
mini-max problem concerning `approximation error' up to the second order.
Furthermore, we have applied this result to quantum lossless data compression,
and have constructed a universal quantum lossless data compression.
