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arXiv:1302.1330 · Physica Scripta

Stochastic evolution of finite level systems: classical versus quantum

Dariusz Chruściński, Vladimir I. Man’ko, Giuseppe Marmo, F. Ventriglia

Published Mar 26, 2013• 8 citations
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Abstract

Quantum dynamics of the density operator in the framework of a single probability vector is analyzed. In this framework quantum states define a proper convex quantum subset in an appropriate simplex. It is showed that the corresponding dynamical map preserving a quantum subset needs not be stochastic contrary to the classical evolution which preserves the entire simplex. Therefore, violation of stochasticity witnesses quantumness of evolution.

DOI
10.1088/0031-8949/87/04/045015

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Quantum dynamics of the density operator in the framework of a single probability vector is analyzed. In this framework quantum states define a proper convex quantum subset in an appropriate simplex. It is showed that the corresponding dynamical map preserving a quantum subset needs not be stochastic contrary to the classical evolution which preserves the entire simplex. Therefore, violation of stochasticity witnesses quantumness of evolution.
We develop a dynamical framework for quantum measurement based on stochastic but unitary evolution in projective state space. Random Hamiltonians drawn from the Gaussian Unitary Ensemble generate stochastic unitary dynamics of the quantum state, while equivalence classes reflecting finite detector resolution define classical observables as well as classical configuration-space and phase-space submanifolds. When the evolution is constrained to the phase-space submanifold, free Schrödinger dynamics reduces to Newtonian motion, while stochastic motion constrained to the classical configuration-space submanifold yields ordinary Brownian motion in classical space. Transition probabilities under the stochastic dynamics satisfy the Born rule, whereas the constrained classical evolution produces the normal probability distributions characteristic of classical measurements. We show that, in this setting, measurement, state reduction, and the quantum-classical transition emerge from unitary dynamics alone, without invoking nonunitary collapse or additional postulates. Entanglement and EPR correlations arise geometrically from the evolution of joint states in composite state space, preserving locality in spacetime. The framework provides a unified dynamical account of measurement and classicality compatible with the structure of quantum mechanics.
The probabilistic description of the time evolution of a physical system can take two conceptually distinct forms: a trajectory of probabilities, which specifies how probabilities evolve over time, and a probability on trajectories, which assigns probabilities to possible histories. A lack of a clear distinction between these two probabilistic descriptions has given rise to a number of conceptual difficulties, particularly in recent analyses of stochastic-quantum correspondence. This paper provides a systematic account of their relationship. We define probability dynamics and stochastic process families together with a precise notion of implementation that connects the two descriptions. We show that implementations are generically non-unique, that every probability dynamics admits a Markovian implementation, and characterize when non-Markovian implementations are possible. We expose fallacies in common arguments for the linearity of probability dynamics based on the law of total probability and clarify the proper interpretation of ``transition matrices'' by distinguishing dynamics-level maps from the conditional probability matrices of implementing processes. We further introduce decomposability as the appropriate general notion of stepwise evolution for (possibly nonlinear) probability dynamics, relate it to divisibility in the linear case -- showing that the two can come apart -- and disentangle both notions from Markovianity and time-homogeneity. Finally, we connect these results to what we call statistical dynamics, in which linearity is indeed physically motivated, and contrast the framework with quantum mechanics.
We start with the simplest quantum system (a two-level system, i.e., a qubit) and discuss a one-to-one mapping of the quantum state in a two-dimensional Hilbert space to a vector in an eight dimensional probability space (probability simplex). We then show how the usual transformations of the quantum state, specifically the Hadamard gate and the single-qubit phase gate, can be accomplished with appropriate transformations of the mapped vector in the probability simplex. One key defining feature of both the mapping to the simplex and the transformations in the simplex is that they are not linear. These results show that both the initial state and the time evolution of a qubit can be fully captured in an eight dimensional probability simplex (or equivalently using three classical probabilistic bits). We then discuss multi-partite quantum systems and their mapping to the probability simplex. Here, the key tool is the identical tensor product structure of combining multiple quantum systems as well as multiple probability spaces. Specifically, we explicitly show how to implement an analog of the two-qubit controlled-not (CNOT) gate in the simplex. We leave it an open problem how much the quantum dynamics of NNN qubits can be captured in a probability simplex with 3N3N3N classical probabilistic bits. Finally, we also discuss the equivalent of the Schrodinger's equation for the wavefunction (in a Hilbert space of arbitrary dimension), which dictates the time evolution of the vectors in the simplex.
The description of the evolution of open quantum systems is a topic of great interest both for its wide range of technological applications and for the understanding of fundamental aspects of quantum theory. In this context, the quantum dynamical mappings formalism is a widely used tool for generally characterizing open systems dynamics. Nonetheless, for this formalism there is no agreement on how to define Markovianity, which is an extremely important concept for most informational analysis of quantum systems. In this undergraduate thesis we compare the quantum dynamical mappings framework with the stochastic processes theory, which has a precise definition of Markovianity, and show that there is a fundamental difference between them that prevents us from properly defining Markovianity in the quantum case. In the end, we briefly present an alternative approach, the process tensor framework, in which is possible to consistently define quantum Markovianity. We then argue that, for this reason, the process tensor framework might be more adequate than quantum dynamical mappings for generally describing the evolution of open quantum systems
Abstract We consider a closed quantum system subject to a stochastic resetting process. The generic expression for the resulting density operator is formulated for arbitrary resetting dynamics, fully characterised by the distribution of times between consecutive reset events. We analyse the behaviour of the state in the long-time regime, as well as the evolution of relevant quantities in the study of quantum coherence and closed- versus open-system dynamics. Our general results are complemented with examples involving particular resetting distributions, and special attention is paid to the two-level (qubit) system, in which we elucidate the effects of the renewal process on the speed of evolution toward an orthogonal state, and gain insight into the resetting applied to open systems.