Test-measured R\'enyi divergences

One possibility of defining a quantum R'enyi $\alpha$-divergence of two quantum states is to optimize the classical R'enyi $\alpha$-divergence of their post-measurement probability distributions over all possible measurements (measured R'enyi divergence), and maybe regularize these quantities over multiple copies of the two states (regularized measured R'enyi $\alpha$-divergence). A key observation behind the theorem for the strong converse exponent of asymptotic binary quantum state discrimination is that the regularized measured R'enyi $\alpha$-divergence coincides with the sandwiched R'enyi $\alpha$-divergence when $\alpha >1$. Moreover, it also follows from the same theorem that to achieve this, it is sufficient to consider $2$-outcome measurements (tests) for any number of copies (this is somewhat surprising, as achieving the measured R'enyi $\alpha$-divergence for $n$ copies might require a number of measurement outcomes that diverges in $n$, in general). In view of this, it seems natural to expect the same when $\alpha <1$; however, we show that this is not the case. In fact, we show that even for commuting states (classical case) the regularized quantity attainable using $2$-outcome measurements is in general strictly smaller than the R'enyi $\alpha$-divergence (which is unique in the classical case). In the general quantum case this shows that the above “regularized test-measured” R'enyi $\alpha$-divergence is not even a quantum extension of the classical R'enyi divergence when $\alpha <1$, in sharp contrast to the $\alpha >1$ case.