Quantum conditional entropies and fully entangled fraction of states with maximally mixed marginals

The fully entangled fraction (FEF) measures the proximity of a quantum state to maximally entangled states. FEF $>\frac{1}{d}$, in $d \otimes d$ systems is a significant benchmark for various quantum information processing protocols including teleportation. Quantum conditional entropy (QCE) on the other hand is a measure of correlation in quantum systems. Conditional entropies for quantum systems can be negative, marking a departure from conventional classical systems. The negativity of quantum conditional entropies plays a decisive role in tasks like state merging and dense coding. In the present work, we investigate the relation of these two important yardsticks. Our probe is mainly done in the ambit of states with maximally mixed marginals, with a few illustrations from other classes of quantum states. We start our study in two qubit systems, where for the Werner states, we obtain lower bounds to its FEF when the conditional R\'enyi $\alpha-$entropy is negative. We then obtain relations between FEF and QCE for two qubit Weyl states. Moving on to two qudit states we find a necessary and sufficient condition based on FEF, for the isotropic state to have negative conditional entropy. In two qudit systems the relation between FEF and QCE is probed for the rank deficient and generalized Bell diagonal states. FEF is intricately linked with $k$- copy nonlocality and $k$- copy steerability. The relations between FEF and QCE facilitates to find conditions for $k$- copy nonlocality and $k$- copy steerability based on QCE. We obtain such conditions for certain classes of states in two qubits and two qudits. As a corollary to the relations obtained between QCE and FEF we obtain lower bounds to minimal deterministic work cost for two qubit Werner states and two qudit generalized Bell states.