Entanglement-assisted entropic uncertainty principle

Abstract

In quantum mechanics, the ability to simultaneously predict the precise outcomes of two conjugate observables, such as the position and momentum, for a particle is restricted by the uncertainty principle [1]. For example, the more precisely the location of the particle is determined, the less accurate the momentum determination will be. Originally given by Heisenberg, the uncertainty principle is best known as the Heisenberg–Robertson [2] commutation ΔRΔS 12 |〈[R,S]〉|, with ΔR(ΔS) representing the standard deviation of the corresponding variable R (S). Obviously, the bound on the right-hand side of above equation is state-dependent and can vanish even when R and S are non-commuting. To avoid this defect, the uncertainty relation is further extended to the entropic form to precisely reflect its physical meanings. The entropic uncertainty relation for any two general observables was first given by Deutsch [3]. Soon afterwards, an improved version was proposed by Kraus [4] and then proved by Maassen and Uffink [5]. The improved relation reads as H(R)+H(S) log2 1c , where H is the Shannon entropy and c represents the overlap between observables R and S. The uncertainty principle is the essential characteristic of quantum mechanics. However, the possibility of violating Heisenberg’s uncertainty relation has been considered early. In 1935, Einstein, Podolsky, and Rosen published the famous paper in which they considered using two particles entangled in position and momentum freedoms to violate Heisenberg’s uncertainty relation and to challenge the correctness of quantum mechanics (EPR paradox) [6]. Popper also proposed an experimental scheme using entangled systems to demonstrate the violation of the Heisenberg’s uncertainty relation [7]. After a long debate and many experimental work, it is known that these violations do not contradict the quantum theory and they are now implemented as a signature of entanglement which is the fundamental feature of quantum mechanics and important resource of quantum information processing. One way to think about uncertainty relations is through the uncertainty game [8] between two players, Alice and Bob. Before the game commences, Alice and Bob agree on two measurements, R and S. The game proceeds as follows (Fig. 1). Bob prepares a particle in a chosen quantum state and sends it to Alice. Alice then carries out one of the two measurements and announces her choice to Bob. Bob’s task is to minimize his uncertainty about Alice’s measurement outcome.