Accelerated optimization of measured relative entropies
Zixin Huang, Mark M. Wilde
TL;DR
This work derives explicit matrix gradients and Hessian superoperators for the measured relative entropy and measured Rényi relative entropy, proving that the associated objective functions are $β$-smooth and $γ$-strongly convex/concave on operator intervals tied to max-relative entropies. These smoothness/convexity properties enable the use of Nesterov accelerated projected gradient methods to compute $D^{M}(ρ∥σ)$ and $D^{M}_{α}(ρ∥σ)$ to arbitrary precision, with iterative costs scaling as $O(\sqrt{κ}\,d^{3}\log(1/ε))$ (and $κ$ depending on $ρ$ and $σ$). Compared to previous SDP-based approaches, the new algorithms are more memory efficient and, for well-conditioned states, significantly faster. The results provide practical, scalable tools for quantum distinguishability measures across all Rényi parameters and contribute foundational understanding of the variational objectives underlying these quantities.
Abstract
The measured relative entropy and measured Rényi relative entropy are quantifiers of the distinguishability of two quantum states $ρ$ and $σ$. They are defined as the maximum classical relative entropy or Rényi relative entropy realizable by performing a measurement on $ρ$ and $σ$, and they have interpretations in terms of asymptotic quantum hypothesis testing. Crucially, they can be rewritten in terms of variational formulas involving the optimization of a concave or convex objective function over the set of positive definite operators. In this paper, we establish foundational properties of these objective functions by analyzing their matrix gradients and Hessian superoperators; namely, we prove that these objective functions are $β$-smooth and $γ$-strongly convex / concave, where $β$ and $γ$ depend on the max-relative entropies of $ρ$ and $σ$. A practical consequence of these properties is that we can conduct Nesterov accelerated projected gradient descent / ascent, a well known classical optimization technique, to calculate the measured relative entropy and measured Rényi relative entropy to arbitrary precision. These algorithms are generally more memory efficient than our previous algorithms based on semi-definite optimization [Huang and Wilde, arXiv:2406.19060], and for well conditioned states $ρ$ and $σ$, these algorithms are notably faster.