Locally-Measured Rényi Divergences

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 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B</i>. 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.