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It is well-known that any quantum channel $\mathcal{E}$ satisfies the dataprocessing inequality (DPI), with respect to various divergences, e.g., quantum$\chi^2_{\kappa}$divergences and quantum relative entropy. More specifically,the data processing inequality states that the divergence between two arbitraryquantum states $\rho$ and $\sigma$ does not increase under the action of anyquantum channel $\mathcal{E}$. For a fixed channel $\mathcal{E}$ and a state$\sigma$, the divergence between output states $\mathcal{E}(\rho)$ and$\mathcal{E}(\sigma)$ might be strictly smaller than the divergence betweeninput states $\rho$ and $\sigma$, which is characterized by the strong dataprocessing inequality (SDPI). Among various input states $\rho$, the largestvalue of the rate of contraction is known as the SDPI constant. An importantand widely studied property for classical channels is that SDPI constantstensorize. In this paper, we extend the tensorization property to the quantumregime: we establish the tensorization of SDPIs for the quantum$\chi^2_{\kappa_{1/2}}$ divergence for arbitrary quantum channels and also fora family of $\chi^2_{\kappa}$ divergences (with $\kappa \ge \kappa_{1/2}$) forarbitrary quantum-classical channels.

Tensorization of the strong data processing inequality for quantum chi-square divergences