Test-Measured Renyi Divergences

One possibility of defining a quantum Renyi a-divergence of two quantum states is to optimize the classical Renyi a-divergence of their post-measurement probability distributions over all possible measurements (measured Renyi divergence), and maybe regularize these quantities over multiple copies of the two states (regularized measured Renyi a-divergence). A key observation behind the theorem for the strong converse exponent of asymptotic binary quantum state discrimination is that the regularized measured Renyi a-divergence coincides with the sandwiched Renyi a-divergence when a > 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 Renyi a-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 a < 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 Renyi a-divergence (which is unique in the classical case). In the general quantum case this shows that the above "regularized test-measured" Renyi a-divergence is not even a quantum extension of the classical Renyi divergence when a < 1, in sharp contrast to the a > 1 case.