Uncertainty relations with quantum memory for the Wehrl entropy

We prove two new fundamental uncertainty relations with quantum memory for the Wehrl entropy. These are the first entropic uncertainty relations with quantum memory ever proposed for a single measurement. The first relation applies to the bipartite memory scenario and provides a lower bound to the Wehrl entropy of a quantum state conditioned on the memory quantum system in terms of the von Neumann entropy of the same quantum state conditioned on the same memory quantum system. The second relation applies to the tripartite memory scenario and provides a lower bound to the sum of the Wehrl entropy of a quantum state conditioned on the first memory quantum system with the Wehrl entropy of the same state conditioned on the second memory quantum system. The Wehrl entropy of a quantum state is the Shannon differential entropy of the outcome of a heterodyne measurement performed on the state. The heterodyne measurement is one of the main measurements in quantum optics, and lies at the basis of one of the most promising protocols for quantum key distribution. These fundamental entropic uncertainty relations will be a valuable tool in quantum information, and will e.g. find application in security proofs of quantum key distribution protocols in the asymptotic regime and in entanglement witnessing in quantum optics. ∗I acknowledge financial support from the European Research Council (ERC Grant Agreement no 337603), the Danish Council for Independent Research (Sapere Aude) and VILLUM FONDEN via the QMATH Centre of Excellence (Grant No. 10059). 1