  We prove two new fundamental uncertainty relations with quantum memory for
the Wehrl entropy. The first relation applies to the bipartite memory scenario.
It determines the minimum conditional Wehrl entropy among all the quantum
states with a given conditional von Neumann entropy and proves that this
minimum is asymptotically achieved by a suitable sequence of quantum Gaussian
states. The second relation applies to the tripartite memory scenario. It
determines the minimum of 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 and proves that also
this minimum is asymptotically achieved by a suitable sequence of quantum
Gaussian states. 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, for example, find
application in security proofs of quantum key distribution protocols in the
asymptotic regime and in entanglement witnessing in quantum optics.
