Failure of the mean-field Hartree approximation for a bosonic many-body system with non-Hermitian Hamiltonian

Matias Ginzburg; Simone Rademacher; Giacomo De Palma

doi:10.22331/q-2026-10-22-2244

Failure of the mean-field Hartree approximation for a bosonic many-body system with non-Hermitian Hamiltonian

Matias Ginzburg¹, Simone Rademacher², and Giacomo De Palma³

¹Scuola Internazionale Superiore di Studi Avanzati, Via Bonomea, 265, 34136 Trieste TS, Italy
²Department of Mathematics, LMU Munich, Theresienstrasse 39, 80333 Munich, Germany
³University of Bologna, Department of Mathematics, Piazza di Porta San Donato 5, 40126 Bologna BO, Italy

Published:	2026-10-22, volume 10, page 2244
Editor:	Tommaso Macrì
Eprint:	arXiv:2601.13038v3
Doi:	https://doi.org/10.22331/q-2026-10-22-2244
Citation:	Quantum 10, 2244 (2026).

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Abstract

Mean-field Hartree theory is a central tool for reducing many-body dynamics to an effective one-particle evolution. This approximation has been employed when the Hamiltonian governing the many-body dynamics is not Hermitian. Indeed, non-Hermitian Hamiltonians model particle gain/loss or the evolution of open quantum systems between consecutive quantum jumps. Furthermore, the validity of the Hartree approximation for generic non-Hermitian Hamiltonians lies at the basis of a quantum algorithm for nonlinear differential equations. In this work, we show that this approximation can fail.
First, we consider the most general bosonic mean-field Hamiltonian with two-body interactions and we prove that, for non-Hermitian Hamiltonian, the Hartree equation does not capture the correct second time derivative of the one-particle marginal state at $t=0$ in the thermodynamic limit. Then, we analytically solve a model of $N$ bosonic qubits with a purely anti-Hermitian Hamiltonian, compute the limit for $N\to\infty$ of the one-particle marginal state and show that it does not agree with the solution of the non-Hermitian Hartree equation. We further show that there exists an initial condition such that the exact one-particle marginal state undergoes a finite-time transition to a mixed state, a phenomenon that is completely absent in the case of Hermitian Hamiltonians. Finally, we simulate a non-Hermitian Hamiltonian with real spectrum and show that, also in this case, there are initial conditions for which the Hartree equation does not reproduce the correct one-particle marginal state in the limit $N\to\infty$.
Our findings challenge the validity of the mean-field Hartree approximation for non-Hermitian Hamiltonians, and call for additional conditions for the validity of the mean-field regime to model the dynamics of particle gain and loss and the open-system dynamics in bosonic many-body systems.

► BibTeX data

@article{Ginzburg2026failureofmeanfield,
  doi = {10.22331/q-2026-10-22-2244},
  url = {https://doi.org/10.22331/q-2026-10-22-2244},
  title = {Failure of the mean-field {H}artree approximation for a bosonic many-body system with non-{H}ermitian {H}amiltonian},
  author = {Ginzburg, Matias and Rademacher, Simone and De Palma, Giacomo},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {10},
  pages = {2244},
  month = oct,
  year = {2026}
}

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