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We have proven that there exists a quantum state approximating any multi-copystate universally when we measure the error by means of the normalized relativeentropy. While the qubit case was proven by Krattenthaler and Slater (IEEETrans. IT, 46, 801-819 (2000); quant-ph/9612043), the general case has beenopen for more than ten years. For a deeper analysis, we have solved themini-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.

Universal approximation of multi-copy states and universal quantum lossless data compression