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We adopt a continuous model to estimate the Grothendieck constants. An analytical formula to compute the lower bounds of Grothendieck constants has been explicitly derived for arbitrary orders, which improves previous bounds. Moreover, our lower bound of the Grothendieck constant of order three gives a refined bound of the threshold value for the nonlocality of the two-qubit Werner states.

We present computable criterion for completely classifying multi-qubit quantum states under local unitary operations. The criterion can be used to detect whether two quantum states in multi-qubit systems are local unitary equivalent or not. Once obtaining the positive answer, we are further able to compute the corresponding unitary operators precisely. Since the scheme is based on the mean values of some quantum mechanical observables, it supplies an experimental way to judge the local equivalence of quantum states.Introduction: Quantum entangled states have become the most important physical resource in the rapidly developing field of quantum information science [1]. In particular, multipartite quantum entanglement plays key roles in many quantum information processing like oneway quantum computing, quantum error correction and quantum secret sharing [2][3][4][5]. However, the nonlocal properties of multipartite mixed states are much more difficult to clarify than bipartite or pure states. Since two quantum states will be of the same power in implementing quantum information processing if they can be transferred to each other by local unitary (LU) transformations, and many crucial characters such as the degree of entanglement [6,7], the maximal violations of Bell inequalities [8][9][10][11], and the teleportation fidelity [12,13] remain invariant under the LU transformations, it has been an important problem to give a complete and operational classification of quantum states under LU transformations.In principle, the LU equivalence problem can be characterized by the complete set of invariants under local unitary transformations. The author in [14] presented a complete set of 18 polynomial invariants for the locally unitary equivalence of two-qubit mixed states. Nice results have been obtained for three qubits states [15,16], some generic mixed states [17][18][19], tripartite pure and mixed states [20]. In [21], Kraus presented a way to determine the local unitary equivalence of n-qubit pure states, which is extended to general multipartite pure states of arbitrary dimensions [22].Mixed states are more general physically. For bipartite mixed quantum systems, Zhou et.al [23] have solved the local unitary equivalence problem by presenting a complete set of invariants such that two density matrices are locally equivalent if and only if all these invari-

We study the relations between quantum coherence and quantum nonlocality, genuine quantum entanglement and genuine quantum nonlocality. We show that the coherence of a qubit state can be converted to the nonlocality of two-qubit states via incoherent operations. The results are also generalized to qudit case. Furthermore, rigorous relations between the quantum coherence of a single-partite state and the genuine multipartite quantum entanglement, as well as the genuine three-qubit quantum nonlocality are established.