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Density matrix interpretation of solutions of Lie-Nambu equations

Marek Czachor, Marciniak, M

Published Jul 7, 1997• 2 citations
OpenAlex

Abstract

A nonlinear Lie-Nambu generalization of the Liouville-von Neumann equation is considered [M.~Czachor, Phys.~Lett.~A 225, 1 (1997)]. It is shown that spectrum of $\\rho(t)$ is conserved by the Lie-Nambu dynamics if $\\rho(t)$ is a self-adjoint and Hilbert-Schmidt solution of a nonlinear triple-bracket equation. This generalizes to arbitrary separable Hilbert spaces the previous result which was valid for finite-dimensional Hilbert spaces. The result shows that conservation of eigenvalues of density matrices may be a general property which holds independently of linearity of dynamics. The corresponding spectral decomposition plays a role of a nonlinearly generalized convexity principle.

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The Journal of Chemical Physics•Feb 22, 1997•24 citations
A nonlinear Lie-Nambu generalization of the Liouville-von Neumann equation is considered [M.~Czachor, Phys.~Lett.~A 225, 1 (1997)]. It is shown that spectrum of rho(t)\\rho(t)rho(t) is conserved by the Lie-Nambu dynamics if rho(t)\\rho(t)rho(t) is a self-adjoint and Hilbert-Schmidt solution of a nonlinear triple-bracket equation. This generalizes to arbitrary separable Hilbert spaces the previous result which was valid for finite-dimensional Hilbert spaces. The result shows that conservation of eigenvalues of density matrices may be a general property which holds independently of linearity of dynamics. The corresponding spectral decomposition plays a role of a nonlinearly generalized convexity principle.
There are various possibilities in defining Born approximations for the evolution of density matrices. They differ, however, from the ordinary Born approximation for pure states by the fact that they do not, in general, conserve the fundamental properties of physical states. We shall describe these possibilities geometrically in the so-called Liouville space of Hilbert-Schmidt operators and investigate their significance in nonequilibrium dynamics. The results admit of an analogous interpretation in classical statistical mechanics.
Abstract Based oh the properties of Lie algebras, in this work we develop a general framework to linearize the von Neumann equation rendering it in a suitable form for quantum simulations. Departing from the conventional method of expanding the density matrix in the Liouville space formed by matrices unit we express the von Neumann equation in terms of Pauli strings. This provides several advantages related to the quantum tomography of the density matrix and the formulation of the unitary gates that generate the time evolution. The use of Pauli strings facilitates the quantum tomography of the density matrix whose elements are purely real. As for any other basis of Hermitian matrices, this eliminates the need to calculate the phase of the complex entries of the density matrix. This approach also enables to express the evolution operator as a sequence of commuting Hamiltonian gates of Pauli strings that can readily be synthetized using Clifford gates. Additionally, the fact that these gates commute with each other along with the unique properties of the algebra formed by Pauli strings allows to avoid the use of Trotterization hence considerably reducing the circuit depth. The algorithm is demonstrated for three Hamiltonians using the IBM noisy quantum circuit simulator.
We consider solutions of the non-linear von Neumann equation involving Jacobi's elliptic functions sn, cn, and dn, and 3 linearly independent operators. In two cases one can construct a state-dependent Hamiltonian which is such that the corresponding non-linear von Neumann equation is solved by the given density operator. We prove that in a certain context these two cases are the only possibilities to obtain special solutions of this kind. Well-known solutions of the reduced Maxwell-Bloch equations produce examples of each of the two cases. Also known solutions of the non-linear von Neumann equation in dimension 3 are reproduced by the present approach.
A method for the numerical solution of the dissipative Liouville–von Neumann equation is presented. The reduced density operator is expanded in a basis of time-dependent wave functions. This guarantees that the size of the basis required in the representation of the density operator is minimal. Equations of motion for the expansion wave functions and the density matrix elements are obtained from the Dirac–Frenkel variational principle. The numerical effort of the method scales proportional to N, where N is the dimension of the relevant Hilbert space. As a first example, the dynamics of a three-mode system with vibronic coupling (with N≈50 000) coupled to a dissipative bath has been studied, modeling the S1/S2 states of pyrazine. For the cw-absorption spectrum, fast convergence with respect to the number of expansion wave functions has been obtained.