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A Complete Set of Local Invariants for a Family of Multipartite Mixed States.pdf

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A Complete Set of Local Invariants for a Family of Multipartite Mixed States

a r X i v : 0 8 0 1 .1 7 0 6 v 1 [ q u a n t - p h ] 1 1 J a n 2 0 0 8 A Complete Set of Local Invariants for a Family of Multipartite Mixed States Xiao-Hong Wang1, Shao-Ming Fei1, 2 and Ke Wu1 1 Department of Mathematics, Capital Normal University, Beijing, China 2 Max-Planck-Institute for Mathematics in the Sciences, 04103 Leipzig, Germany Key words: Invariants, Local equivalence, Multipartite states PACS number(s): 03.67.-a, 02.20.Hj, 03.65.-w Abstract We study the equivalence of quantum states under local unitary trans- formations by using the singular value decomposition. A complete set of invariants under local unitary transformations is presented for several classes of tripartite mixed states in CK ? CM ? CN composite systems. Two density matrices in the same class are equivalent under local unitary transformations if and only if all these invariants have equal values for these density matrices. 1 Introduction Quantum entanglement is playing very important roles in quantum information processing. Quantum entangled states are the key resource in quantum information processing [1] such as teleportation, super-dense coding, key distribution, error cor- rection and quantum repeater. Therefore it is of great importance to classify and characterize the quantum states. The nature of the entanglement among the parts of a composite system does not depend on the labeling of the basis states of the individual subsystems. It is there- fore invariant under unitary transformations of the individual state spaces. Such transformations are referred to as local unitary transformations. The polynomial invariants of local unitary transformations have been discussed in [2, 3, 4]. General methods, which allow in principle to compute all such invariants, but are in fact not really operational, were introduced in [5, 6, 7, 8]. More explicit complete and partial solutions have been found for some special cases: two qubits [9] and three qubits [10, 11] systems, three qutrits [