Approximate vs complete quantum information erasure: constructions and applications

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It is a fundamental, if elementary, observation that to obliterate the quantum information in n qubits by random unitaries, an amount of randomness of at least 2n bits is required. If the randomisation condition is relaxed to perform only approximately, we obtain two answers, depending on the norm used to compare the ideal and the approximation. Using the ''naive'' norm brings down the cost to n bits, while under the more appropriate complete norm it is still essentially 2n.
After reviewing these facts and some constructions, we go on to explore the quantum information theoretical uses of the two notions of erasure. Most prominently, for a given quantum channel and its complementary channel, complete erasure is dual to correctability of the quantum noise; while approximate erasure is dual to the decodability of another task of quantum information, dubbed ''quantum identification''