Counter-examples for Tensorization Property of Strong Data Processing Inequality for Quantum Divergences
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Abstract
The data processing inequality is a fundamental property that describes the loss of information through noisy channels. A more refined description is characterized by the strong data processing inequality (SDPI). In classical information theory, the tensorization of strong data processing inequality holds for a whole family of $f$-divergences. However, its quantum counterpart is less known. The tensorization of SDPI was shown only for some special cases previously, and the general understanding about the tensorization property of SDPI for quantum divergences remains open. In this work, we report two negative results: the tensorization property fails for certain quantum chi-square divergences, and it also does not hold for the quantum relative entropy.