Geometry of Bloch vectors in two-qubit system

NASSER METWALLY, Mahmoud Abdel‐Aty, A.‐S. F. Obada

Т.Э. Кренкель

Shuang Cong

The Bloch vector, or vector of coherence, provides a geometric description of the state and density matrix of two-state quantum system. Mathematically, the Bloch vector may be viewed as the adjoint representation of an su(2) object in an so(3) basis. The pure and mixed states details concerning the geometry of the Bloch ball for a qubit are investigated. The system controls with two and three magnetic fields to closed and open system are analyzed in the Bloch ball.

Reinhold A. Bertlmann, Philipp Krammer

We present three different matrix bases that can be used to decompose density matrices of d--dimensional quantum systems, so-called qudits: the generalized Gell-Mann matrix basis, the polarization operator basis, and the Weyl operator basis. Such a decomposition can be identified with a vector --the Bloch vector, i.e. a generalization of the well known qubit case-- and is a convenient expression for comparison with measurable quantities and for explicit calculations avoiding the handling of large matrices. We consider the important case of an isotropic two--qudit state and decompose it according to each basis. Investigating the geometry of entanglement of special parameterized two--qubit and two--qutrit states, in particular we calculate the Hilbert--Schmidt measure of entanglement, we find that the Weyl operator basis is the optimal choice since it is closely connected to the entanglement of the considered states.

Masashi Ban

Simon Morelli, Christopher Eltschka, Marcus Huber +1 more

We present an inequality on the purity of a bipartite state depending solely on the length difference of the local Bloch vectors. For two qubits this inequality is tight for all marginal states and so extends the previously known solution for the two-qubit marginal problem. With this inequality we construct a three-dimensional Bloch model of the two-qubit quantum state space in terms of Bloch lengths, providing a pleasing visualization of this high-dimensional state space. This allows to characterize quantum states by a strongly reduced set of parameters and to investigate the interplay between local properties of the marginal systems and global properties encoded in the correlations.