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.

翻译：测量相对熵与测量Rényi相对熵是量化两个量子态$ρ$与$σ$可区分性的度量。它们被定义为通过对$ρ$和$σ$执行测量所能实现的经典相对熵或Rényi相对熵的最大值，并在渐近量子假设检验中具有解释意义。关键的是，它们可以重写为涉及在正定算子集上优化凹或凸目标函数的变分公式。本文通过分析这些目标函数的矩阵梯度与Hessian超算符，建立了这些目标函数的基础性质；具体而言，我们证明了这些目标函数是$β$-光滑且$γ$-强凸/凹的，其中$β$和$γ$取决于$ρ$与$σ$的最大相对熵。这些性质的一个实际结果是，我们可以执行Nesterov加速投影梯度下降/上升（一种经典的优化技术）来计算测量相对熵与测量Rényi相对熵至任意精度。这些算法通常比我们先前基于半定优化的工作[Huang and Wilde, arXiv:2406.19060]更具内存效率，并且对于良态量子态$ρ$和$σ$，这些算法的速度显著更快。