Abstract In the present work, we suggest an approach for describing dynamics of finite-dimensional quantum systems in terms of pseudostochastic maps acting on probability distributions, which are obtained via minimal informationally complete quantum measurements. The suggested method for probability representation of quantum dynamics preserves the tensor product structure, which makes it favourable for the analysis of multi-qubit systems. A key advantage of the suggested approach is that minimal informationally complete positive operator-valued measures (MIC-POVMs) are easier to construct in comparison with their symmetric versions (SIC-POVMs). We establish a correspondence between the standard quantum-mechanical formalism and the MIC-POVM-based probability formalism. Within the latter approach, we derive equations for the unitary von-Neumann evolution and the Markovian dissipative evolution, which is governed by the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) generator. We apply the MIC-POVM-based probability representation to the digital quantum computing model. In particular, for the case of spin-1/2 evolution, we demonstrate identifying a transition of a dissipative quantum dynamics to a completely classical-like stochastic dynamics. One of the most important findings is that the MIC-POVM-based probability representation gives more strict requirements for revealing the non-classical character of dissipative quantum dynamics in comparison with the SIC-POVM-based approach. Our results give a physical interpretation of quantum computations and pave a way for exploring the resources of noisy intermediate-scale quantum devices.
In this work, we consider a probability representation of quantum dynamics for finite-dimensional quantum systems with the use of pseudostochastic maps acting on true probability distributions. These probability distributions are obtained via symmetric informationally complete positive operator-valued measure (SIC-POVM) and can be directly accessible in an experiment. We provide SIC-POVM probability representations both for unitary evolution of the density matrix governed by the von Neumann equation and dissipative evolution governed by Markovian master equation. In particular, we discuss whereas the quantum dynamics can simulated via classical random processes in terms of the conditions for the master equation generator in the SIC-POVM probability representation. We construct practical measures of nonclassicality non-Markovianity of quantum processes and apply them for studying experimental realization of quantum circuits realized with the IBM cloud quantum processor.
Quantum machine learning models define probabilistic input--output maps through coherent quantum evolution and measurement. While such models can exhibit computational advantages, their internal functioning and decision making generally resists interpretation in terms of stochastic trajectories through intermediate configurations. In contrast to classical (Markovian) stochastic processes, quantum dynamics generically violates the Chapman--Kolmogorov divisibility condition, preventing a decomposition into probabilistically meaningful intermediate transitions. We develop a probabilistic framework for representing quantum learning models as stochastic processes over configuration spaces where the dynamics are modeled as linear maps on probability distributions. Starting from a fixed POVM, arbitrary quantum channels induce transition kernels on the associated probability representation. For informationally complete POVMs, and in particular SIC-POVMs, these kernels are Markovian but generally quasi-stochastic, with non-classicality appearing as negativity. By contrast, projective spaces admit positive stochastic kernels but generally require non-Markovian dynamics due to the failure of Chapman--Kolmogorov divisibility. This yields a trade-off between negativity and dependence on past configurations, i.e. quantum dynamics can be represented either by Markovian quasi-stochastic maps or by positive stochastic processes with higher Markov order. We discuss how such representations of quantum dynamics can be interpreted as stochastic walks through a memory space in the spirit of Projective Simulation, a model of learning and agency in which decisions arise from random walks over an episodic memory network. We further outline how finite-order stochastic kernels can approximate such quantum deliberation processes and show in what regimes the classical machine learning model is recovered.
Wigner functions provide a way to do quantum physics using quasiprobabilities, that is, ``probability'' distributions that can go negative. Informationally complete positive operator-valued measures, a much younger subject than phase-space formulations of quantum mechanics, are less familiar but provide wholly probabilistic representations of quantum theory. In this paper, we show that the Born rule links these two classes of structure and discuss the art of interconverting between them. In particular, we demonstrate that the operator bases corresponding to minimal discrete Wigner functions (Wigner bases) are orthogonalizations of minimal informationally complete measurements (MICs). By not imposing a particular discrete phase-space structure at the outset, we push Wigner functions to their limits in a suitably quantified sense, revealing an extremality condition satisfied by the symmetric informationally complete measurements (SICs). This contributes another member to the set of optimality criteria that SICs are known to satisfy. Finally, we speculate that astute choices of MICs from the orthogonalization preimages of Wigner bases may in general give quantum measurements that are conceptually illuminating for the quasiprobability representations defined by those Wigner bases.
We define non-Markovian quantum dynamics as evolution in which the current state depends on all past states, and completely characterize its structure under the assumptions of complete positivity and non-signalling. The resulting continuous-time dynamics is an integro-differential equation that augments the Gorini-Kossakowski-Sudarshan-Lindblad equation with a memory integral, and is capable of describing the quantum state of systems exposed to noise with any integrable power spectral density with no further approximations. We then establish a formalism to evaluate multi-time correlations of measurement outcomes in this general setting, obviating the need for a regression theorem. As an application, we derive the emission spectrum of a driven two-level system coupled to a non-Markovian bath: the familiar Mollow triplet acquires a frequency-dependent linewidth that encodes the memory of the bath. Our work provides a rigorous yet transparent description of the quantum state of non-Markovian systems, opening the door for state estimation and state-based quantum control beyond the Markovian regime.
This thesis presents a comprehensive study of phase-space measurements, decoherence, and the emergence of classicality in quantum systems. Motivated by limitations of position-based localisation in standard decoherence models, we develop a phase-space decoherence formalism using coherent-state positive operator-valued measures (POVMs). These generalised measurements, constructed from overcomplete sets of coherent states, provide a natural and physically motivated framework for modelling coarse-grained quantum observations. We develop an iterative measurement scheme for phase-space decoherence in systems with Heisenberg-Weyl,
SU(2),
SU(1,1), and
SU(N) symmetries. In each case, repeated weak measurements gradually suppress quantum interference, driving the system toward a maximally mixed state. A universal structure is shown to emerge: a single iteration maps the Wigner function to the Husimi
Q-function, removing negativity, while full decoherence maps the singular Glauber-Sudarshan
P-function to the
Q-function, eliminating all singularities. This establishes a group-independent mechanism by which phase-space measurements induce classicality. We compare this model to Lindblad-type dynamics, which describe open quantum systems via completely positive, trace-preserving semigroups. For Heisenberg-Weyl and tomographic
SU(N) systems, the measurement channel coincides exactly with isotropic Lindblad evolution. In
SU(2) and
SU(1,1) cases, the correspondence is only partial, with exact agreement in asymptotic regimes. A universal decoherence timescale,
t∗, emerges beyond which quasi-probability distributions become strictly positive. We then construct a stochastic unravelling based on Brownian motion on phase space, offering a geometric alternative to standard stochastic Schr\"odinger equations. This reveals that decoherence occurs without GKSL-type dynamics in noncompact systems. Such unravellings preserve the intrinsic phase-space symmetry and provide a natural setting for studying semiclassical limits and quantum-to-classical transitions. Finally, we extend the framework to relativistic quantum systems by defining a covariant coherent-state POVM on the future tube and constructing a Bargmann transform to the forward light cone. Preliminary results on relativistic phase-space decoherence are presented, along with possible directions for future research.
This paper studies a minimal time-symmetric operational framework in which information from initial preparation and post-selection is used to define an operational description of quantum evolution. Quantum randomness is interpreted as epistemic incompleteness rather than ontological indeterminacy. The central dynamical object is the informational coupling map Λκ(ρ) = (1 − κ)ρ + κσZ, which mixes the actual state ρ with a fixed reference state σZ. The channel itself is mathematically standard; the contribution of the manuscript is the operational interpretation of the dimensionless coupling parameter κ as a measure of boundary alignment, its entropy-based estimation under explicitly stated assumptions, and its embedding in a delayed-choice quantum random number generator (QRNG) protocol. A Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) representation of the corresponding continuous-time dynamics is given using a separate relaxation rate γ, complete positivity is explicit, and no-signaling is preserved under the proposed local action. In the limit γ → 0, standard unitary quantum mechanics is recovered, whereas nonzero γ produces small deviations that can be interpreted as effective informational bias associated with post-selection. In the present framework, σZ is not treated as a universal microscopic final state, but as a context-dependent operational reference state determined by the post-selection structure used in the model.