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arXiv:quant-ph/0005030

Darboux-integration of id\rho/dt=[H,f(\rho)]

N. V. Ustinov, Sergey Leble, Marek Czachor, Maciej Kuna

Published May 6, 2000• 13 citations
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Abstract

A Darboux-type method of solving the nonlinear von Neumann equation $i\\dot \\rho=[H,f(\\rho)]$, with functions $f(\\rho)$ commuting with $\\rho$, is developed. The technique is based on a representation of the nonlinear equation by a compatibility condition for an overdetermined linear system. von Neumann equations with various nonlinearities $f(\\rho)$ are found to possess the so-called self-scattering solutions. To illustrate the result we consider the Hamiltonian $H$ of a one-dimensional harmonic oscillator and $f(\\rho)=\\rho^q-2\\rho^{q-1}$ with arbitary real $q$. It is shown that self-scattering solutions possess the same asymptotics for all $q$ and that different nonlinearities may lead to effectively indistinguishable evolutions. The result may have implications for nonextensive statistics and experimental tests of linearity of quantum mechanics.

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A Darboux-type method of solving the nonlinear von Neumann equation idotrho=[H,f(rho)]i\\dot \\rho=[H,f(\\rho)]idotrho=[H,f(rho)], with functions f(rho)f(\\rho)f(rho) commuting with rho\\rhorho, is developed. The technique is based on a representation of the nonlinear equation by a compatibility condition for an overdetermined linear system. von Neumann equations with various nonlinearities f(rho)f(\\rho)f(rho) are found to possess the so-called self-scattering solutions. To illustrate the result we consider the Hamiltonian HHH of a one-dimensional harmonic oscillator and f(rho)=rhoq−2rhoq−1f(\\rho)=\\rho^q-2\\rho^{q-1}f(rho)=rhoq−2rhoq−1 with arbitary real qqq. It is shown that self-scattering solutions possess the same asymptotics for all qqq and that different nonlinearities may lead to effectively indistinguishable evolutions. The result may have implications for nonextensive statistics and experimental tests of linearity of quantum mechanics.
Starting with a time-independent Hamiltonian h and an appropriately chosen solution of the von Neumann equation $i\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\rho}}(t)=[h,\ensuremath{\rho}(t)]$ we construct its binary-Darboux partner h1(t){h}_{1}(t)h1​(t) and an exact scattering solution of $i{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\rho}}}_{1}(t)=[{h}_{1}(t),{\ensuremath{\rho}}_{1}(t)],$ where h1(t){h}_{1}(t)h1​(t) is time dependent and not isospectral to h. The method is analogous to supersymmetric quantum mechanics but is based on a different version of a Darboux transformation. We illustrate the technique by the example where h corresponds to a one-dimensional harmonic oscillator. The resulting h1(t){h}_{1}(t)h1​(t) represents a scattering of a solitonlike pulse on a three-level system.
A combination of the Lie-Poisson equation with the qqq-averaged energy Uq=〈H〉q{U}_{q}=〈H{〉}_{q}Uq​=〈H〉q​ leads to a microscopic framework for nonextensive qqq thermodynamics. The resulting von Neumann equation is nonlinear: $i\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\rho}}=[H,{\ensuremath{\rho}}^{q}].$ In spite of its nonlinearity the dynamics is consistent with linear quantum mechanics of pure states. The free energy ${F}_{q}{=U}_{q}\ensuremath{-}{\mathrm{TS}}_{q}$ is a stability function for the dynamics. This implies that qqq-equilibrium states are dynamically stable. The (microscopic) evolution of \ensuremath{\rho} is reversible for any q,q,q, but for $q\ensuremath{\ne}1$ the corresponding macroscopic dynamics is irreversible.
Solving linear and nonlinear differential equations with large degrees of freedom is an important task for scientific and industrial applications. In order to solve such differential equations on a quantum computer, it is necessary to embed classical variables into a quantum state. While the Carleman and Koopman-von Neumann embeddings have been investigated so far, the class of problems that can be mapped to the Schrödinger equation is not well understood even for linear differential equations. In this work, we investigate the conditions for linear differential equations to be mapped to the Schrödinger equation and solved on a quantum computer. Interestingly, we find that these conditions are identical for both Carleman and Koopman-von Neumann embeddings. We also compute the computational complexity associated with estimating the expected values of an observable. This is done by assuming a state preparation oracle, block encoding of the mapped Hamiltonian via either Carleman or Koopman-von Neumann embedding, and block encoding of the observable using O(log⁡M)O(\log M)O(logM) qubits with MMM is the mapped system size. Furthermore, we consider a general classical quadratic Hamiltonian dynamics and find a sufficient condition to map it into the Schrödinger equation. As a special case, this includes the coupled harmonic oscillator model [Babbush et al., \cite{babbush_exponential_2023}]. We also find a concrete example that cannot be described as the coupled harmonic oscillator but can be mapped to the Schrödinger equation in our framework. These results are important in the construction of quantum algorithms for solving differential equations of large-degree-of-freedom.
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.