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arXiv:0904.3533 · Journal of Mathematical Physics

Dynamic equations for three different qudits in a magnetic field

E. A. Ivanchenko

Published Apr 1, 2009• 3 citations
arXivHTMLPDF

Abstract

A closed system of equations for the local Bloch vectors and spin correlation functions of three magnetic qudits, which are in an arbitrary, time-dependent, external magnetic field, is obtained using decomplexification of the Liouville–von Neumann equation. The algorithm of the derivation of the dynamic equations is presented. In the basis convenient for the important physical applications, structure constants of algebra su(2S+1) are calculated.

DOI
10.1063/1.3116167

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A closed system of equations for the local Bloch vectors and spin correlation functions of three magnetic qudits, which are in an arbitrary, time-dependent, external magnetic field, is obtained using decomplexification of the Liouville–von Neumann equation. The algorithm of the derivation of the dynamic equations is presented. In the basis convenient for the important physical applications, structure constants of algebra su(2S+1) are calculated.
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A closed system of equations for the local Bloch vectors and spin correlation functions of three magnetic qubits with exchange interaction, which are in an arbitrary, time-dependent, external magnetic field, is obtained using decomplexification of the Liouville–Neuman equations. A numerical comparative analysis of entanglement measures is performed as a function of the initial conditions and modulation of the magnetic field. The investigation could find applications for the analysis of interference experiments and manipulation of quantum qubits.
A closed system of the equations for the local Bloch vectors and spin correlation functions is obtained by decomplexification of the Liouville-von Neumann equation for 4 magnetic particles with the exchange interaction that takes place in an arbitrary time-dependent external magnetic field.The analytical and numerical analysis of the quantum thermodynamic variables is carried out depending on separable mixed initial state and the magnetic field modulation.Under unitary evolution, non-equilibrium reversible dynamics of power production in the finite environment is investigated.
Based on the Liouville–von Neumann equation, we obtain a closed system of equations for the description of a qutrit or coupled qutrits in an arbitrary, time-dependent, external magnetic field. The dependence of the dynamics on the initial states and the magnetic field modulation is studied analytically and numerically. We compare the relative entanglement measure's dynamics in bi-qudits with permutation particle symmetry. We find the magnetic field modulation which retains the entanglement in the system of two coupled qutrits. Analytical formulae for the entanglement measures in finite chains from two to six qutrits or three quartits are presented.
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Abstract In the present article, we consistently develop the main issues of the Bloch vectors formalism for an arbitrary finite-dimensional quantum system. In the frame of this formalism, qudit states and their evolution in time, qudit observables and their expectations, entanglement and nonlocality, etc are expressed in terms of the Bloch vectors—the vectors in the Euclidean space R d 2 − 1 , arising under decompositions of observables and states in different operator bases. Within this formalism, we specify for all d ⩾ 2 the set of Bloch vectors of traceless qudit observables and describe its properties; also, find for the sets of the Bloch vectors of qudit states, pure and mixed, the new compact expressions in terms of the operator norms that explicitly reveal the general properties of these sets and have the unified form for all d ⩾ 2. For the sets of the Bloch vectors of qudit states under the generalized Gell‐Mann representation, these general properties cannot be analytically extracted from the known equivalent specifications of these sets via the system of algebraic equations. We derive the general equations describing the time evolution of the Bloch vector of a qudit state if a qudit system is isolated and if it is open and find for both cases the main properties of the Bloch vector evolution in time. For a pure bipartite state of a dimension d 1 × d 2 , we quantify its entanglement via the characteristics of the Bloch vectors for its reduced states. The introduced general formalism is important both for the theoretical analysis of quantum system properties and for quantum applications, in particular, for optimal quantum control, since, for systems where states are described by vectors in the Euclidean space, the methods of optimal control, analytical and numerical, are well developed.