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

TL;DR

This paper proves the existence of a universal quantum state approximating any multi-copy state using normalized relative entropy, and applies this to develop a universal quantum lossless data compression scheme.

Contribution

It extends previous results from qubits to general multi-copy states and solves the mini-max approximation error problem, enabling universal quantum data compression.

Findings

Existence of a universal approximating quantum state for multi-copy states.

Solution to the mini-max approximation error problem up to second order.

Construction of a universal quantum lossless data compression scheme.

Abstract

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.