On Variational Expressions for Quantum Relative Entropies

Distance measures between quantum states like the trace distance and the fidelity can naturally be defined by optimizing a classical distance measure over all measurement statistics that can be obtained from the respective quantum states. In contrast, Petz showed that the measured relative entropy, defined as a maximization of the Kullback-Leibler divergence over projective measurement statistics, is strictly smaller than Umegaki’s quantum relative entropy whenever the states do not commute. We extend this result in two ways. First, we show that Petz’ conclusion remains true if we allow general positive valued measures. To do this, we prove that maximizing the Kullback-Leibler divergence over general positive valued measures results in the measured relative entropy. Second, we extend the result to Rényi relative entropies and show that for non-commuting states the sandwiched Rényi relative entropy is strictly larger than the measured Rényi relative entropy for α ∈ ( 2 ,∞), and strictly smaller for α ∈ [0, 1 2 ). The latter statement provides counterexamples for the data-processing inequality of the sandwiched Rényi relative entropy for α < 1 2 . Our main tool is a new variational expression for the measured Rényi relative entropy, which we further exploit to show that certain lower bounds on quantum conditional mutual information are superadditive.