Measured f-Divergences

Milán Mosonyi, Fumio Hiai

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$; however, we show that this is not the case. In fact, we show that even for commuting states (classical case) the regularized quantity attainable using 2-outcome measurements is in general strictly smaller than the Rényi$\alpha $-divergence (which is unique in the classical case). In the general quantum case this shows that the above “regularized test-measured” Rényi$\alpha $-divergence is not even a quantum extension of the classical Rényi divergence when$\alpha < 1$, in sharp contrast to the$\alpha >1$case.

Takahiro Sagawa

Tobias Rippchen, Sreejith Sreekumar, Mario Berta

We propose an extension of the classical Rényi divergences to quantum states through an optimization over probability distributions induced by restricted sets of measurements. In particular, we define the notion of locally-measured Rényi divergences, where the set of allowed measurements originates from variants of locality constraints between (distant) partiesAandB. We then derive variational bounds on the locally-measured Rényi divergences and systematically discuss when these bounds become exact characterizations. As an application, we evaluate the locally-measured Rényi divergences on variants of highly symmetric data-hiding states, showcasing the reduced distinguishing power of locality-constrained measurements. For n-fold tensor powers, we further employ our variational formulae to derive corresponding additivity results, which gives the locally-measured Rényi divergences operational meaning as optimal rate exponents in asymptotic locally-measured hypothesis testing.

Yves Pomeau A, Minh-Binh Tran

Tobias Rippchen, Sreejith Sreekumar, Mario Berta

We propose an extension of the classical Rényi divergences to quantum states through an optimization over probability distributions induced by restricted sets of measurements. In particular, we define the notion of locally-measured Rényi divergences, where the set of allowed measurements originates from variants of locality constraints between (distant) parties $A$ and $B$. We then derive variational bounds on the locally-measured Rényi divergences and systematically discuss when these bounds become exact characterizations. As an application, we evaluate the locally-measured Rényi divergences on variants of highly symmetric data-hiding states, showcasing the reduced distinguishing power of locality-constrained measurements. For $n$-fold tensor powers, we further employ our variational formulae to derive corresponding additivity results, which gives the locally-measured Rényi divergences operational meaning as optimal rate exponents in asymptotic locally-measured hypothesis testing.

F. Beck, Benno Fuchssteiner