Spontaneous unitarity violations were recently proposed as a cause of objective quantum state reduction. This complements proposals based on stochastic modifications of Schr\"odinger's equation, but also differs from them in several aspects. Here, we formalize the description of models for spontaneous unitarity violation, and show that they generically imply models of dynamical quantum state reduction driven by colored noise. We present a formalism for exploring such models as well as a prescription for enforcing explicit norm-preservation, and we show that the resulting pure state dynamics is described by a modified von Neumann-Liouville equation which in a particular limit reduces to the Gorini-Kossakowski-Sudarshan-Lindblad master equations. We additionally show adherence to Born's rule emerging in the same limit from a physical constraint relating fluctuating and dissipating components of the model.
Spontaneous unitarity violations were recently proposed as a cause of objective quantum state reduction. This complements proposals based on stochastic modifications of Schr"odinger's equation, but also differs from them in several aspects. Here, we formalize the description of models for spontaneous unitarity violation, and show that they generically imply models of dynamical quantum state reduction driven by colored noise. We present a formalism for exploring such models as well as a prescription for enforcing explicit norm-preservation, and we show that the resulting pure state dynamics is described by a modified von Neumann-Liouville equation which in a particular limit reduces to the Gorini-Kossakowski-Sudarshan-Lindblad master equations. We additionally show adherence to Born's rule emerging in the same limit from a physical constraint relating fluctuating and dissipating components of the model.
The inability of Schrödinger's unitary time evolution to describe the measurement of a quantum state remains a central foundational problem. It was recently suggested that the unitarity of Schrödinger dynamics can be spontaneously broken, resulting in measurement as an emergent phenomenon in the thermodynamic limit. Here, we introduce a family of models for spontaneous unitarity violation that apply to generic initial superpositions over arbitrarily many states, using either single or multiple state-independent stochastic components. Crucially, we show that Born's probability rule emerges spontaneously in all cases.
Objective collapse theories propose modifications to Schrödinger’s equation that solve the quantum measurement problem by interpolating between microscopic quantum dynamics and projective evolution of macroscopic objects. Coloured-noise driven collapse theories extending the equilibrium description of spontaneous symmetry breaking (SSB) to spontaneous unitary violation (SUV) in quantum dynamics were recently shown to possess a Markovian white-noise limit when applied to initial two-state superpositions. Here, we show that this limit coincides with a subclass of continuous spontaneous localization (CSL) models when collapsing in a basis of spatially localized energy eigenstates. We show that the energy expectation value remains conserved throughout this process, and we also extend the model to a form that can be applied to any initial state. We furthermore show that, as for the SUV models, the emergence of Born rule statistics in the Markovian limit is enforced by a fluctuation–dissipation relation (FDR), which results in ensemble averaged probability densities following a linear quantum semi-group guaranteeing the absence of superluminal signalling.
Objective collapse theories propose a solution to the quantum measurement problem by predicting deviations from Schrödinger’s equation that can be tested experimentally. A class of objective theories based on spontaneous unitarity violation was recently introduced, in which the stochastic field required for obtaining Born’s rule does not depend on the state of the system being measured. Here, we classify possible models for the stochastic field dynamics in theories of spontaneous unitarity violation. We show that for correlated stochastic dynamics, the field must be defined on a closed manifold. In two or more dimensions, it is then always possible to find stochastic dynamics yielding Born’s rule, independent of the state being measured or the correlation time of the stochastic field. We show that the models defined this way are all isomorphic to the definition on a two-sphere, which we propose to be a minimal physical model for the stochastic field in models of spontaneous unitarity violation.
The stochastic Schrödinger equation (SSE) provides a trajectory-level route to simulate the dynamics of open quantum systems with applications ranging from molecular processes to quantum technologies. We study a colored-noise extension of the SSE based on an Ornstein-Uhlenbeck (OU) noise drive and benchmark its ensemble-averaged dynamics against the standard white-noise SSE and against a fluctuating OU random Hamiltonian. When the environment exhibits a finite correlation time, averaging over pure-state trajectories yields master equations that are generally open-form and not of Lindblad type, yet remain positive by construction. By considering the differential of the OU process, we define an effective correlated noise, whose properties we analyze and use to formulate an SSE unraveling of its associated open-form quantum master equation. We show that the averaged dissipator separates into a Lindblad contribution stemming from the white-noise component and additional correlation terms arising from the fluctuations of the OU Hamiltonian. To obtain a practical closed description and physical intuition, we introduce a Redfield-inspired perturbative closure for these correlation terms, providing an effective master equation for the colored SSE. For a two-level system, the resulting dynamics exhibit long-lived coherences, nontrivial stationary (including oscillatory) states, and multi-timescale relaxation, rationalized through the components of a time-dependent Redfield tensor.
Within the framework of the quantum-statistical approach, utilizing both non-Hermitian Hamiltonian and Lindblad’s jump operators, one can derive various generalizations of the von Neumann equation for reduced density operators, also known as hybrid master equations. If one considers the evolution of pure states only, i.e., disregarding the coherence between states and spontaneous transitions from pure to mixed states, then one can resort to quantum-mechanical equations of the Schrödinger type. We derive them from the hybrid master equations and study their main properties, which indicate that our equations have a larger range of applicability compared to other generalized Schrödinger equations proposed hitherto. Among other features, they can describe not only systems which remain in the stationary eigenstates of the Hamiltonian as time passes, but also those which evolve from those eigenstates. As an example, we consider a simple but important model, a quantum harmonic oscillator driven by both Hamiltonian and non-Hamiltonian terms, and derive its classical limit, which turns out to be the damped harmonic oscillator. Using this model, we demonstrate that the effects of dissipative environments of different types can cancel each other, thus resulting in an effectively dissipation-free classical system. Another discussed phenomenon is whether a non-trivial quantum system can reduce to a classical system in free motion, i.e., without experiencing any classical Newtonian forces. This uncovers a large class of quantum-mechanical non-Hamiltonian systems whose dynamics are not determined by conventional mechanics’ potentials and forces, but rather come about through quantum statistical effects caused by the system’s environment.
Relativity and quantum mechanics are two cornerstones of modern physics, yet their unification within a single-particle path integral and a dynamical explanation of quantum measurement remain unresolved. Historically, these two problems have been treated as separate, but here we show they are intimately linked. We construct a self-consistent relativistic path integral that yields the Dirac and other standard wave equations under differetialable potentials. More importantly, we find that this propagator contains a latent, nonlocal correlation that is activated by realistic electromagnetic noise. This correlation unifies unitary evolution and wave-function collapse into a single dynamical mechanism: while differentiable potentials preserve unitary driving, nondifferentiable noise activates a bounded-martingale stochastic process that induces collapse. We show that the characteristics of quantum measurement are naturally derived from this stochastic dynamical process, thereby turning the axioms of quantum measurement from postulates into dynamical consequences. Furthermore, averaging this stochastic evolution over the noise record recovers the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation, providing a first-principles derivation of decoherence free from the method of Born-Markov approximation. Extending this approach to composite systems establishes a stochastic foundation for effective non-Hermitian descriptions while preserving relativistic causality. Finally, because the noise spectrum governs the collapse process, engineering ``colored'' noise can actively accelerate or steer state reduction, suggesting new routes toward fast qubit reset and enhanced quantum control.