We study the stochastic local operation and classical communication (SLOCC) equivalence for arbitrary dimensional multipartite quantum states. For multipartite pure states, we present a necessary and sufficient criterion in terms of their coefficient matrices. This condition can be used to classify some SLOCC equivalent quantum states with coefficient matrices having the same rank. For multipartite mixed state, we provide a necessary and sufficient condition by means of the realignment of matrix. Some detailed examples are given to identify the SLOCC equivalence of multipartite quantum states.
We study the uncertainty relation in the product form of variances and obtain some new uncertainty relations with weight, which are shown to be tighter than those derived from the Cauchy-Schwarz inequality.
We study the fully entangled fraction of a quantum state. An upper bound is obtained for arbitrary bipartite system. This upper bound only depends on the Frobenius norm of the state. [6-9] etc. For instance, in the process of teleportation, the fidelity of optimal teleportation is given by fully entangled fraction (FEF) [10]. Thus an analytic formula for FEF is of great importance. In [11] an elegant formula for a two-qubit system is derived analytically by using the method of Lagrange multipliers. Concerning the estimation of entanglement of formation and concurrence, exact results have been obtained not only for two-qubit case, but also for some higher dimensional states, isotropic and Werner states [12]. Analytical lower bounds have also been obtained for general cases [13,14]. In [15] an estimate of the upper bound of FEF was given. Some relations between FEF with eigenvalues of the density matrix were studied in [16]. Nevertheless, analytical computation of FEF remains formidable and few results have been known for higher dimensional quantum states.
1The aim of this work is to give an upper bound of the FEF for arbitrary higher dimensional state. Our main techniques come from a careful analysis of the Frobenious norm.
We study the trade-off relations given by the l1-norm coherence of general multipartite states. Explicit trade-off inequalities are derived with lower bounds given by the coherence of either bipartite or multipartite reduced density matrices. In particular, for pure three-qubit states, it is explicitly shown that the trade-off inequality is lower bounded by the three tangle of quantum entanglement.
We study the equivalence of mixed states under local unitary transformations in bipartite system and three partite system. First we express quantum states in Bloch representation. Then based on the coefficient matrices, some invariants are constructed in terms of the products, trace and determinant of matrices. This method and results can be extended to multipartite high dimensional system.
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