2201.05477Jan 14, 2022v3: 30 pages, minor improvements. Thanks to a comment by an anonymous reviewer, we can now show that the two different ways to regularize the test-measured Rényi $α$-divergence lead to different quantities
One possibility of defining a quantum Rényi α-divergence of two quantum states is to optimize the classical Rényi α-divergence of their post-measurement probability distributions over all possible measurements (measured Rényi divergence), and maybe regularize these quantities over multiple copies of the two states (regularized measured Rényi α-divergence). A key observation behind the theorem for the strong converse exponent of asymptotic binary quantum state discrimination is that the regularized measured Rényi α-divergence coincides with the sandwiched Rényi α
Nearby in the stack
-divergence when
α>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 Rényi
α
-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
α<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 Rényi
α
-divergence (which is unique in the classical case). In the general quantum case this shows that the above "regularized test-measured" Rényi
α
-divergence is not even a quantum extension of the classical Rényi divergence when