Uncertainty relations with quantum memory for the Wehrl entropy

TL;DR

This paper establishes new fundamental uncertainty relations involving Wehrl entropy with quantum memory, providing bounds that are asymptotically achieved by quantum Gaussian states, with applications in quantum information and quantum optics.

Contribution

It introduces two novel uncertainty relations with quantum memory for Wehrl entropy, applicable to bipartite and tripartite scenarios, with asymptotic optimality demonstrated by Gaussian states.

Findings

Minimum conditional Wehrl entropy characterized

Sum of Wehrl entropies minimized by Gaussian states

Results applicable to quantum key distribution security

Abstract

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.