Test-Measured Rényi Divergences
Milán Mosonyi, Fumio Hiai · IEEE Transactions on Information Theory · 2022
One possibility of defining a quantum Rényi$\alpha $-divergence of two quantum states is to optimize the classical Rényi$\alpha $-divergence of their post-measurement probability distributions over all possible measurements (measured Rényi divergence), and maybe regularize these quantities over multiple copies of the two states (regularized measured Rényi$\alpha $-divergence). A key observation behind the theorem for the strong converse exponent of asymptotic binary quantum state discrimination is that the regularized measured Rényi$\alpha $-divergence coincides with the sandwiched Rényi$\alpha $-divergence when$\alpha >1$. Moreover, it also follows from the same theorem that to achieve this, it is sufficient to consider 2-outcome measurements (tests) for any number of copies (this is somewhat surprising, as achieving the measured Rényi$\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$case.