Linear and nonlinear evolution of reduced density matrices

A.G. Hall

K. M. Fonseca-Romero, Peter Talkner A, Peter Hänggi A

We study the influence of the preparation of an open quantum system on its reduced time evolution. In contrast to the frequently considered case of an initial preparation where the total density matrix factorizes into a product of a system density matrix and a bath density matrix the time evolution generally is no longer governed by a linear map nor is this map affine. Put differently, the evolution is truly nonlinear and cannot be cast into the form of a linear map plus a term that is independent of the initial density matrix of the open quantum system. As a consequence, the inhomogeneity that emerges in formally exact generalized master equations is in fact a nonlinear term that vanishes for a factorizing initial state. The general results are elucidated with the example of two interacting spins prepared at thermal equilibrium with one spin subjected to an external field. The second spin represents the environment. The field allows the preparation of mixed density matrices of the first spin that can be represented as a convex combination of two limiting pure states, i.e., the preparable reduced density matrices make up a convex set. Moreover, the map from these reduced density matrices onto the corresponding density matrices of the total system is affine only for vanishing coupling between the spins. In general, the set of the accessible total density matrices is nonconvex.

Eugene S. Kryachko

Gustavo E. Massaccesi, P. Capuzzi, Ofelia B. Oña +4 more

The N-representability problem of reduced density matrices represents a fundamental challenge in electronic structure theory. In this work, we focus on the N-representability of transition reduced density matrices and propose a practical approach to embed a p-body transition reduced density matrix (RDM) of an N-particle system into a (p + 1)-body RDM of an (N + 1)-particle system. This embedding allows us to apply a unitary evolution algorithm based on a recently developed adaptive derivative-assembled pseudo-Trotter variational quantum algorithm to determine the N-representability of reduced density matrices. The algorithm works by effectively applying a sequence of unitary transformations to a given (N + 1)-particle initial state in such a way that the distance of a projection of its corresponding (p + 1)-body RDM (embedded p-body transition RDM) to a target is minimized. Our methodology enables the purification of transition reduced density matrices and the reconstruction of approximate wave functions involved in the studied transitions, providing an effective strategy to correct and refine transition density matrices. We validate our approach with numerical simulations on N-particle systems, including a three-electron model system and the H3 molecule, demonstrating its robustness and accuracy.

Andrei T. Patrascu

Abstract We formulate and analyze an operational test of a tacit assumption frequently made in quantum device modeling: that the reduced density matrix ρS of a subsystem S is a sufficient state variable for predicting its future reduced evolution under fixed declared controls. We formalize this assumption as Local Evolution Completeness (LEC): if two global preparations share the same reduced state ρS (0), then they must yield identical ρS ( t ) under identical device dynamics. We first establish a strict-locality null result: when the applied Hamiltonian factorizes as HS ⊗ IE , LEC holds exactly, independent of correlations with ancillary degrees of freedom. We then prove, via an explicit process-tensor counterexample, that LEC can fail even when all intermediate maps are completely positive and trace-preserving and the global dynamics is strictly unitary. In such cases, no reduced map Φ exists with ρS ( t ) = Φ( ρS (0)) for all embeddings realizing the same ρS (0). To operationalize this distinction, we construct a fully solvable qubit–ancilla model and derive closed-form reduced trajectories, along with a statistical inference framework for detecting preparation-dependent deviations under fixed controls. We provide microscopic realizations of the embedding variable responsible for LEC failure, including correlation-invariant and control-reference mechanisms. The proposed protocol isolates a distinct experimental axis from traditional nonlinear quantum mechanics tests: it probes the predictive sufficiency of reduced states rather than convex linearity of state evolution. A violation of LEC would not imply a modification of global unitary dynamics, but rather demonstrate that single-time reduced density matrices are not sufficient statistics for multi-time prediction in the tested setting. This clarifies the structural role of reduced states within the operational foundations of quantum theory.