We investigate the generation of uniform quantum state ensembles via continuous measurement. Using the $SU(d)$ Bloch representation, we derive the associated Langevin and Fokker-Planck equations and identify geometric conditions under which homogeneous monitoring causes global convergence to the uniform pure-state ensemble. We then extend the analysis to mixed states, showing that homogeneous purity-dependent decoherence rates generate uniform Hilbert-Schmidt and Bures ensembles of qubit states through an effective nonlinear stochastic evolution. Additionally, we introduce a post-mixing protocol for qubits: target mixed-state ensembles are assembled by classically sampling trajectories generated with different fixed efficiencies (or decoherence rates). This provides an experimentally feasible route to reconstructing Hilbert-Schmidt and Bures-random mixed-state ensembles, demonstrating that continuous monitoring provides both an exact dynamical generator of Haar-random pure states and a practical route to constructing mixed-state ensembles.We investigate the generation of uniform quantum state ensembles via continuous measurement. Using the $SU(d)$ Bloch representation, we derive the associated Langevin and Fokker-Planck equations and identify geometric conditions under which homogeneous monitoring causes global convergence to the uniform pure-state ensemble. We then extend the analysis to mixed states, showing that homogeneous purity-dependent decoherence rates generate uniform Hilbert-Schmidt and Bures ensembles of qubit states through an effective nonlinear stochastic evolution. Additionally, we introduce a post-mixing protocol for qubits: target mixed-state ensembles are assembled by classically sampling trajectories generated with different fixed efficiencies (or decoherence rates). This provides an experimentally feasible route to reconstructing Hilbert-Schmidt and Bures-random mixed-state ensembles, demonstrating that continuous monitoring provides both an exact dynamical generator of Haar-random pure states and a practical route to constructing mixed-state ensembles.

quant-pharXiv:2606.31848Jul 1, 20260 citations

Time-inhomogeneous and nonlinear quantum evolutions

Alberto Frigerio

Lecture notes in mathematicsJan 1, 19902 citations

Quantum nonlocality and quantum dynamics

S. Gheorghiu-Svirschevski

We argue that usual quantum statics and the dynamical equivalence of mixed quantum states to {\it probabilistic mixtures}suffice to guarantee a linear evolution law, which necessarily complies with the no-signaling condition. Alternatively, there are nonlinear dynamical extensions that treat mixed states as {\it elementary mixtures} and evolve {\it every}pure state linearly and unitarily. But if all {\it entangled} pure states evolve linearly, then elementary mixtures cannot evolve nonlinearly without challenging quantum locality. Conversely, any such extension that is relativistically well behaved demands a nonlinear evolution [decoherence] of pure entangled states. Wherefrom follows that the linear evolution of entangled pure states provides an unequivocal signature of linear quantum dynamics.We argue that usual quantum statics and the dynamical equivalence of mixed quantum states to {\it probabilistic mixtures}suffice to guarantee a linear evolution law, which necessarily complies with the no-signaling condition. Alternatively, there are nonlinear dynamical extensions that treat mixed states as {\it elementary mixtures} and evolve {\it every}pure state linearly and unitarily. But if all {\it entangled} pure states evolve linearly, then elementary mixtures cannot evolve nonlinearly without challenging quantum locality. Conversely, any such extension that is relativistically well behaved demands a nonlinear evolution [decoherence] of pure entangled states. Wherefrom follows that the linear evolution of entangled pure states provides an unequivocal signature of linear quantum dynamics.

arXiv:quant-ph/0203153Mar 29, 20021 citation

Nonlinear Adiabatic Evolution of Quantum Systems

Jie Liu, Sheng-Chang Li, Libin Fu +1 more

Springer eBooksJan 1, 201811 citations

Evolution of Pure States into Mixed States

Jun Liu

In the formulation of Banks, Peskin and Susskind, we show that one can construct evolution equations for the quantum mechanical density matrix $ρ$ with operators which do not commute with hamiltonian which evolve pure states into mixed states, preserve the normalization and positivity of $ρ$ and conserve energy. Furthermore, it seems to be different from a quantum mechanical system with random sources.In the formulation of Banks, Peskin and Susskind, we show that one can construct evolution equations for the quantum mechanical density matrix $ρ$ with operators which do not commute with hamiltonian which evolve pure states into mixed states, preserve the normalization and positivity of $ρ$ and conserve energy. Furthermore, it seems to be different from a quantum mechanical system with random sources.

arXiv:hep-th/9301082Jan 19, 19935 citations

We investigate the generation of uniform quantum state ensembles via continuous measurement. Using the

SU(d)

Bloch representation, we derive the associated Langevin and Fokker-Planck equations and identify geometric conditions under which homogeneous monitoring causes global convergence to the uniform pure-state ensemble. We then extend the analysis to mixed states, showing that homogeneous purity-dependent decoherence rates generate uniform Hilbert-Schmidt and Bures ensembles of qubit states through an effective nonlinear stochastic evolution. Additionally, we introduce a post-mixing protocol for qubits: target mixed-state ensembles are assembled by classically sampling trajectories generated with different fixed efficiencies (or decoherence rates). This provides an experimentally feasible route to reconstructing Hilbert-Schmidt and Bures-random mixed-state ensembles, demonstrating that continuous monitoring provides both an exact dynamical generator of Haar-random pure states and a practical route to constructing mixed-state ensembles.

In the formulation of Banks, Peskin and Susskind, we show that one can construct evolution equations for the quantum mechanical density matrix

ρ

with operators which do not commute with hamiltonian which evolve pure states into mixed states, preserve the normalization and positivity of

ρ

and conserve energy. Furthermore, it seems to be different from a quantum mechanical system with random sources.