Quantum conditional entropies and steerability of states with maximally mixed marginals

Abstract

Quantum steering is an asymmetric correlation which occupies a place between entanglement and Bell nonlocality. In the paradigmatic scenario involving the protagonists Alice and Bob, the entangled state shared between them, is said to be steerable from Alice to Bob if the steering assemblage on Bob's side do not admit a local hidden state (LHS) description. Quantum conditional entropies, on the other hand provide for another characterization of quantum correlations. Contrary to our common intuition conditional entropies for some entangled states can be negative, marking a significant departure from the classical realm. Quantum steering and quantum nonlocality in general share an intricate relation with quantum conditional entropies. In the present contribution, we investigate this relationship. For a significant class, namely the two-qubit Weyl states we show that negativity of conditional R\'enyi 2-entropy and conditional Tsallis 2-entropy is a necessary and sufficient condition for the violation of a suitably chosen three settings steering inequality. With respect to the same inequality we find an upper bound for the conditional R\'enyi 2-entropy, such that the general two-qubit state is steerable. Moving from a particular steering inequality to local hidden state descriptions, we show that some two-qubit Weyl states which admit a LHS model possess non-negative conditional R\'enyi 2-entropy. However, the same does not hold true for some non-Weyl states. Our study further investigates the relation between non-negativity of conditional entropy and LHS models in two-qudits for the isotropic and Werner states. There we find that whenever these states admit a LHS model, they possess a non-negative conditional R\'enyi 2-entropy. We then observe that the same holds true for a noisy variant of the two-qudit Werner state.