Canonical Coset Parameterization and the Bures Metric of the Three-level Quantum Systems (original) (raw)
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This article gives a brief overview of some recent progress in the characterization and parametrization of density matrices of finite dimensional systems. We discuss in some detail the Bloch-vector and Jarlskog parametrizations and mention briefly the coset parametrization. As applications of the Bloch parametrization we discuss the trace invariants for the case of time dependent Hamiltonians and in some detail the dynamics of three-level systems. Furthermore, the Bloch vector of two-qubit systems as well as the use of the polarization operator basis is indicated. As the main application of the Jarlskog parametrization we construct density matrices for composite systems. In addition, some recent related articles are mentioned without further discussion.
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We give an algorithm allowing to construct bases of local unitary invariants of pure k-qubit states from the knowledge of polynomial covariants of the group of invertible local filtering operations. The simplest invariants obtained in this way are explicited and compared to various known entanglement measures. Complete sets of generators are obtained for up to four qubits, and the structure of the invariant algebras is discussed in detail. 1.
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In quantum mechanics, sets of density matrices are important for numerous reasons. For example, their compact notation make them useful for describing decoherence and entanglement properties of multi-particle quantum systems. In particular, two two-state density matrices, otherwise known as two qubit density matrices, are important for their role in explaining quantum teleportation, dense coding, and computation theorems. The aim of this paper is to show an explicit parameterization for the Hilbert space of all two qubit density matrices. Such a parameterization would be extremely useful for numerical calculations concerning entanglement and other quantum information parameters. We would also like to know the properties of such parameterized two qubit density matrices; in particular, the representation of their convex sets, their subsets , and their set boundaries in terms of our parameterization. Here we present a generalized Euler angle parameterization for SU(4) and all possible two qubit density matrices as well as the corrected Haar Measures for SU(3) and SU(4) from this parameterization. The role of the parameterization in the Peres-Horodecki criteria will also be introduced as well as its usefulness in calculating entangled two qubit states.
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