  It is well-known that any quantum channel $\mathcal{E}$ satisfies the data
processing 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 arbitrary
quantum states $\rho$ and $\sigma$ does not increase under the action of any
quantum 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 between
input states $\rho$ and $\sigma$, which is characterized by the strong data
processing inequality (SDPI). Among various input states $\rho$, the largest
value of the rate of contraction is known as the SDPI constant. An important
and widely studied property for classical channels is that SDPI constants
tensorize. In this paper, we extend the tensorization property to the quantum
regime: we establish the tensorization of SDPIs for the quantum
$\chi^2_{\kappa_{1/2}}$ divergence for arbitrary quantum channels and also for
a family of $\chi^2_{\kappa}$ divergences (with $\kappa \ge \kappa_{1/2}$) for
arbitrary quantum-classical channels.
