Nonlocal advantage of quantum coherence

Abstract

A bipartite state is said to be steerable if and only if it does not have a single system description, i.e., the bipartite state cannot be explained by a local hidden state model. Several steering inequalities have been derived using different local uncertainty relations to verify the ability to control the state of one subsystem by the other party. Here, we derive complementarity relations between coherences measured on mutually unbiased bases using various coherence measures such as the l1-norm, relative entropy and skew information. Using these relations, we derive conditions under which non-local advantage of quantum coherence can be achieved and the state is steerable. We show that not all steerable states can achieve such advantage. Steering is a kind of non-local correlation introduced by Schrödinger [1] to reinterpret the EPR-paradox [2]. According to Schrödinger, the presence of entanglement between two subsystems in a bipartite state enables one to control the state of one subsystem by its entangled counter part. Wiseman et al. [3] formulated the operational and mathematical definition of quantum steering and showed that steering lies between quantum entanglement and Bell non-locality on the basis of their strength [4]. The notion of the steerability of quantum states is also intimately connected [5] to the idea of remote state preparation [6,7].As introduced in Ref.[3], let us consider a hypothetical game to explain the steerability of quantum states. Suppose, Alice prepares two quantum systems, say, A and B in an entangled state ρ AB and sends the system B to Bob. Bob does not trust Alice but agrees with the fact that the system B is quantum. Therefore, Alice's task is to convince Bob that the prepared state is indeed entangled and they share non-local correlation. On the other hand, Bob thinks that Alice may cheat by preparing the system B in a single quantum system, on the basis of possible strategies [8,9]. Bob agrees with Alice that the prepared state is entangled and they share non-local correlation if and only if the state of Bob cannot be written by local hidden state model (LHS) [3]where {P(λ), ρ Q B } is an ensemble of LHS prepared by Alice and P(a|A, λ) is Alice's stochastic map to convince Bob. Here, we consider λ to be a hidden variable with the constraint λ P(λ) = 1 and ρ Q B (λ) is a quantum state received by Bob. The joint probability distribution on such states, P (a Ai , b Bi ) of obtaining outcome a for the measurement of observables chosen from the set {A i } by Alice and outcome b for the measurement of observables chosen from the set {B i } by Bob can be written aswhere P Q (b i |λ) is the quantum probability of the measurement outcome b i due to the measurement of B i . Several steering conditions have been derived on the basis of Eq. (2) and the existence of single system description of a part of the bi-partite systems [8][9][10]. It has also been quantified for two-qubit systems [11]. In the last few years, several experiments have been performed to demonstrate the steering e...

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