The exact statistics of an arbitrary quantum observable is analytically obtained. Due to the probabilistic nature of a sequence of intermediate measurements and stochastic fluctuations induced by the interaction with the environment, the measurement outcomes at the end of the system's evolution are random variables. Here, we provide the exact large-deviation form of their probability distribution, which is given by an exponentially decaying profile in the number of measurements. The most probable distribution of the measurement outcomes in a single realization of the system transformation is then derived, thus achieving predictions beyond the expectation value. The theoretical results are confirmed by numerical simulations of an experimentally reproducible two-level system with stochastic Hamiltonian.
The exact statistics of an arbitrary quantum observable is analytically obtained. Due to the probabilistic nature of a sequence of intermediate measurements and stochastic fluctuations induced by the interaction with the environment, the measurement outcomes at the end of the system's evolution are random variables. Here, we provide the exact large-deviation form of their probability distribution, which is given by an exponentially decaying profile in the number of measurements. The most probable distribution of the measurement outcomes in a single realization of the system transformation is then derived, thus achieving predictions beyond the expectation value. The theoretical results are confirmed by numerical simulations of an experimentally reproducible two-level system with stochastic Hamiltonian.
The exact statistics of an arbitrary measuring observable, describing specific features of an open quantum system, is analytically obtained. Due to the probabilistic nature of a sequence of intermediate measurements and stochastic fluctuations induced by the interaction with an external environment, the measurement outcomes at the end of the system evolution are random variables. Here, we provide the exact large deviation form of their probability distribution, given by an exponentially decaying profile in the number of measurements. The most probable distribution of the measurement outcomes in a single realization of the system transformation is then derived, thus achieving predictions beyond the expectation value.
Quantum measurements are crucial for observing the properties of a quantum system, which, however, unavoidably perturb its state and dynamics in an irreversible way. Here we study the dynamics of a quantum system being subjected to a sequence of projective measurements applied at random times. In the case of independent and identically distributed intervals of time between consecutive measurements, we analytically demonstrate that the survival probability of the system to remain in the projected state assumes a large deviation (exponentially decaying) form in the limit of an infinite number of measurements. This allows us to estimate the typical value of the survival probability, which can therefore be tuned by controlling the probability distribution of the random time intervals. Our analytical results are numerically tested for Zeno-protected entangled states, which also demonstrate that the presence of disorder in the measurement sequence further enhances the survival probability when the Zeno limit is not reached (as it happens in experiments). Our studies provide a new tool for protecting and controlling the amount of quantum coherence in open complex quantum systems by means of tunable stochastic measurements.
We develop a statistical model of microscopic stochastic deviation from classical mechanics based on a stochastic process with a transition probability that is assumed to be given by an exponential distribution of infinitesimal stationary action. We apply the statistical model to stochastically modify a classical mechanical model for the measurement of physical quantities reproducing the prediction of quantum mechanics. The system+apparatus always has a definite configuration at all times, as in classical mechanics, fluctuating randomly following a continuous trajectory. On the other hand, the wavefunction and quantum mechanical Hermitian operator corresponding to the physical quantity arise formally as artificial mathematical constructs. During a single measurement, the wavefunction of the whole system+apparatus evolves according to a Schrödinger equation and the configuration of the apparatus acts as the pointer of the measurement so that there is no wavefunction collapse. We will also show that while the outcome of each single measurement event does not reveal the actual value of the physical quantity prior to measurement, its average in an ensemble of identical measurements is equal to the average of the actual value of the physical quantity prior to measurement over the distribution of the configuration of the system.
We study three definitions of large deviation in many-body quantum dynamics: (i) via the full distribution of an extensive observable, (ii) via the distribution of measurement outcomes (from a continuous monitoring of the observable) over a time interval t≤tmax, and (iii) via the distribution of expectation values over t≤tmax. In generic systems without conservation laws, the large deviation function (i) reaches its longtime limit at t∼O(1), independently of system size N. (ii) and (iii) reach their longtime limit at t∼eN and t∼exp(eN), respectively. Before that, there is a {\it sharp} frontier between the explored and unexplored outcomes/expectation values; their distribution equals the longtime limit truncated at values that drift with tmax. We propose that the evolution of these values with tmax provides a measure of quantum complexity.
The state of an open quantum system undergoing an adiabatic process evolves by following the instantaneous stationary state of its time-dependent generator. This observation allows one to characterize, for a generic adiabatic evolution, the average dynamics of the open system. However, information about fluctuations of dynamical observables, such as the number of photons emitted or the time-integrated stochastic entropy production in single experimental runs, requires controlling the whole spectrum of the generator and not only the stationary state. Here, we show how such information can be obtained in adiabatic open quantum dynamics by exploiting tools from large deviation theory. We prove an adiabatic theorem for deformed generators, which allows us to encode, in a biased quantum state, the full counting statistics of generic time-integrated dynamical observables. We further compute the probability associated with an arbitrary "rare" time history of the observable and derive a dynamics which realizes it in its typical behavior. Our results provide a way to characterize and engineer adiabatic open quantum dynamics and to control their fluctuations.
The state of an open quantum system undergoing an adiabatic process evolves by following the instantaneous stationary state of its time-dependent generator. This observation allows one to characterize, for a generic adiabatic evolution, the average dynamics of the open system. However, information about fluctuations of dynamical observables, such as the number of photons emitted or the time-integrated stochastic entropy production in single experimental runs, requires controlling the whole spectrum of the generator and not only the stationary state. Here, we show how such information can be obtained in adiabatic open quantum dynamics by exploiting tools from large deviation theory. We prove an adiabatic theorem for deformed generators, which allows us to encode, in a biased quantum state, the full counting statistics of generic time-integrated dynamical observables. We further compute the probability associated with an arbitrary "rare" time-history of the observable and derive a dynamics which realizes it in its typical behavior. Our results provide a way to characterize and engineer adiabatic open quantum dynamics and to control their fluctuations.