Quantum Nonlocality For Three-qubit State

Quantum entanglement,quantum steering,and the Bell nonlocality are a typical nonlocality,which are an important resource in quantum information.Their properties are a basic problems in quantum mechanics and information.Quantum steering is a form of quantum nonlocality between quantum entanglement and the Bell nonlocality.The discussion of quantum entanglement and the Bell nonlocality has been deeply studied in literature.However,quantum steering has only been studied in recent years and compared with the other two types of quantum nonlocality.Therefore,in this paper,we will focus on the quantum steering property for a three-qubit state,and discuss the possible relationship among quantum coherence,steering parameters,and entanglement negativity.Firstly,we give the relevant theoretical basis,and introduce the concepts of entangled states,density matrix,reduced density matrix,and quantum nonlocality,which focuses on the relevant knowledge of quantum steering.In addition,we also introduce the measures of quantum coherence,quantum steering parameters,and entanglement negativity.Secondly,we consider two general states for three qubits:the generalized W state and the generalized GHZ state.We calculate,modulate,and analyze quantum steering,quantum coherence,and negativity for the generalized W state and GHZ state,respectively.The results show that,for the generalized W state,whether one qubit is in the excited state or two qubits are in the excited state,there are both steerable and un-steerable state.The maximum value of quantum coherence decreases gradually with the increase of entanglement negativity for the steerable W state.The maximum value of quantum coherence of the un-steerable state also decreases gradually with the increase of entanglement negativity,but its minimum value is firstly unchanged and then decreases.For the given negativity and quantum coherence,it may be either a steerable state or an un-steerable W state.For the steerable W state,the maximum value of its steering parameter is basically stable firstly and then decreases sharply with the increase of negativity.It is worthwhile to mention that for the generalized W state with one qubit in the excited state,the steering is asymmetric,while for the generalized W state with two qubits in the excited state,it is symmetric.For the generalized GHZ state,there are only un-steerable state.For a typical GHZ state,the corresponding negativity reaches the maximum value while quantum coherence degree reaches the minimum value.In short,we have studied the steering,coherence,and negativity for general three-qubit states.We hope that those could provide useful help in the selection of quantum states for quantum information processing in future work,and stimulate interest in the research of quantum steering and coherence for multi-partite states.