Locally-Measured R\'enyi Divergences

TL;DR

This paper extends classical Re9nyi divergences to quantum states by incorporating measurement locality constraints, deriving bounds, and exploring their implications in quantum hypothesis testing and data hiding.

Contribution

It introduces locally-measured Re9nyi divergences for quantum states, providing variational bounds and analyzing their properties under locality constraints.

Findings

Derived variational bounds for locally-measured Re9nyi divergences

Showcased reduced distinguishability in data-hiding states due to locality constraints

Established additivity results for tensor powers in hypothesis testing

Abstract

We propose an extension of the classical R\'enyi 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 R\'enyi 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 R\'enyi divergences and systematically discuss when these bounds become exact characterizations. As an application, we evaluate the locally-measured R\'enyi 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 R\'enyi divergences operational meaning as optimal rate exponents in asymptotic locally-measured hypothesis testing.