Abstract. We give a one-to-one correspondence between classes of density matrices under local unitary invariance and the double cosets of unitary groups. We show that the interrelationship among classes of local unitary equivalent multi-partite mixed states is independent from the actual values

The equivalence of tripartite pure states under local unitary transformations is investigated. The nonlocal properties for a class of tripartite quantum states in C K ⊗ C M ⊗ C N composite systems are investigated and a complete set of invariants under local unitary transformations for these states

The equivalence of arbitrary dimensional bipartite states under local unitary transformations (LUT) is studied. A set of invariants and ancillary invariants under LUT is presented. We show that two states are equivalent under LUT if and only if they have the same values for all of these invariants.

The equivalence of arbitrary dimensional bipartite states under local unitary transformations (LUT) is studied. A set of invariants and an-cillary invariants under LUT is presented. We show that two states are equivalent under LUT if and only if they have the same values for all of these invariants.

We give necessary conditions for the simulation of both bipartite and multipartite Hamiltonians, which are independent of the eigenvalues of Hamiltonians and based on the algebraic-geometric invariants recently introduced in [1] [2]. The results show that the problem of simulation of Hamiltonians in arbitrary bipartite or multipartite quantum systems cannot be described by only using eigenvalues, which is quite different to the two-qubit case. Historically, the idea of simulating Hamiltonian time evolutions was the first motivation for quantum computation [3]. Recently the ability of nonlocal Hamiltonians to simulate one another is a popular topic, which has applications in quantum control theory [4], quantum compuation [5],[6],[7],[8] and 1 the task of generating enatnglement [9] [10]. The problem to parametrize the nonlocal properties of interaction Hamiltonians, so as to characterize the efficiency with which they can be used to simulate one another is theoretically

In terms of the analysis of fixed point subgroup and tensor decomposability of certain matrices, we study the equivalence of of quantum bipartite mixed states under local unitary transformations. For non-degenerate case an operational criterion for the equivalence of two such mixed bipartite states

1 Quaternionic conformal map 2 In this paper the geometry of two-qubit systems under local unitary group SO(2) ⊗ SU(2) is discussed. It is shown that the quaternionic conformal map intertwines between this local unitary subgroup of Sp(2) and the quaternionic Möbius transformation which is rather a

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to characterize the equivalence classes under local unitary transformations for the set of tripartite states whose partial trace with respect to one of the subsystems belongs to the class of CHG mixed states.

The equivalence of multipartite quantum mixed states under local unitary transformations is studied. A criterion for the equivalence of non-degenerate mixed multipartite quantum states under local unitary transformations is presented. PACS numbers: 03.67.-a, 02.20.Hj, 03.65.-w Quantum entangled