Fidelity of holonomic quantum computations in the case of random errors in the values of control parameters

We investigate the influence of random errors in external control parameters on the stability of holonomic quantum computation in the case of arbitrary loops and adiabatic connections. A simple expression is obtained for the case of small random uncorrelated errors. Due to universality of mathematical description our results are valid for any physical system which can be described in terms of holonomies. Theoretical results are confirmed by numerical simulations.Comment: 7 pages, 3 figure

Waiting time distribution for electron transport in a molecular junction with electron-vibration interaction

On the elementary level, electronic current consists of individual electron tunnelling events that are separated by random time intervals. The waiting time distribution is a probability to observe the electron transfer in the detector electrode at time $t+\tau$ given that an electron was detected in the same electrode at earlier time $t$. We study waiting time distribution for quantum transport in a vibrating molecular junction. By treating the electron-vibration interaction exactly and molecule-electrode coupling perturbatively, we obtain master equation and compute the distribution of waiting times for electron transport. The details of waiting time distributions are used to elucidate microscopic mechanism of electron transport and the role of electron-vibration interactions. We find that as nonequilibrium develops in molecular junction, the skewness and dispersion of the waiting time distribution experience stepwise drops with the increase of the electric current. These steps are associated with the excitations of vibrational states by tunnelling electrons. In the strong electron-vibration coupling regime, the dispersion decrease dominates over all other changes in the waiting time distribution as the molecular junction departs far away from the equilibrium

Classical no-cloning theorem under Liouville dynamics by non-Csisz\'ar f-divergence

The Csisz\'ar f-divergence, which is a class of information distances, is known to offer a useful tool for analysing the classical counterpart of the cloning operations that are quantum mechanically impossible for the factorized and marginality classical probability distributions under Liouville dynamics. We show that a class of information distances that does not belong to this divergence class also allows for the formulation of a classical analogue of the quantum no-cloning theorem. We address a family of nonlinear Liouville-like equations, and generic distances, to obtain constraints on the corresponding functional forms, associated with the formulation of classical analogue of the no-cloning principle.Comment: 6 pages, revised, published versio

On the Validity of the 0-1 Test for Chaos

In this paper, we present a theoretical justification of the 0-1 test for chaos. In particular, we show that with probability one, the test yields 0 for periodic and quasiperiodic dynamics, and 1 for sufficiently chaotic dynamics

Fractional generalization of Fick's law: a microscopic approach

In the study of transport in inhomogeneous systems it is common to construct transport equations invoking the inhomogeneous Fick law. The validity of this approach requires that at least two ingredients be present in the system. First, finite characteristic length and time scales associated to the dominant transport process must exist. Secondly, the transport mechanism must satisfy a microscopic symmetry: global reversibility. Global reversibility is often satisfied in nature. However, many complex systems exhibit a lack of finite characteristic scales. In this Letter we show how to construct a generalization of the inhomogeneous Fick law that does not require the existence of characteristic scales while still satisfying global reversibility.Comment: 4 pages. Published versio

Poisson-noise induced escape from a metastable state

We provide a complete solution of the problems of the probability distribution and the escape rate in Poisson-noise driven systems. It includes both the exponents and the prefactors. The analysis refers to an overdamped particle in a potential well. The results apply for an arbitrary average rate of noise pulses, from slow pulse rates, where the noise acts on the system as strongly non-Gaussian, to high pulse rates, where the noise acts as effectively Gaussian

Full counting statistics for transport through a molecular quantum dot magnet

Full counting statistics (FCS) for the transport through a molecular quantum dot magnet is studied theoretically in the incoherent tunneling regime. We consider a model describing a single-level quantum dot, magnetically coupled to an additional local spin, the latter representing the total molecular spin s. We also assume that the system is in the strong Coulomb blockade regime, i.e., double occupancy on the dot is forbidden. The master equation approach to FCS introduced in Ref. [12] is applied to derive a generating function yielding the FCS of charge and current. In the master equation approach, Clebsch-Gordan coefficients appear in the transition probabilities, whereas the derivation of generating function reduces to solving the eigenvalue problem of a modified master equation with counting fields. To be more specific, one needs only the eigenstate which collapses smoothly to the zero-eigenvalue stationary state in the limit of vanishing counting fields. We discovered that in our problem with arbitrary spin s, some quartic relations among Clebsch-Gordan coefficients allow us to identify the desired eigenspace without solving the whole problem. Thus we find analytically the FCS generating function in the following two cases: i) both spin sectors lying in the bias window, ii) only one of such spin sectors lying in the bias window. Based on the obtained analytic expressions, we also developed a numerical analysis in order to perform a similar contour-plot of the joint charge-current distribution function, which have recently been introduced in Ref. [13], here in the case of molecular quantum dot magnet problem.Comment: 17 pages, 5 figure

Theory of Second and Higher Order Stochastic Processes

This paper presents a general approach to linear stochastic processes driven by various random noises. Mathematically, such processes are described by linear stochastic differential equations of arbitrary order (the simplest non-trivial example is $\ddot x = R(t)$, where $R(t)$ is not a Gaussian white noise). The stochastic process is discretized into $n$ time-steps, all possible realizations are summed up and the continuum limit is taken. This procedure often yields closed form formulas for the joint probability distributions. Completely worked out examples include all Gaussian random forces and a large class of Markovian (non-Gaussian) forces. This approach is also useful for deriving Fokker-Planck equations for the probability distribution functions. This is worked out for Gaussian noises and for the Markovian dichotomous noise.Comment: 35 pages, PlainTex, accepted for publication in Phys Rev. E

Coarse graining of master equations with fast and slow states

We propose a general method for simplifying master equations by eliminating from the description rapidly evolving states. The physical recipe we impose is the suppression of these states and a renormalization of the rates of all the surviving states. In some cases, this decimation procedure can be analytically carried out and is consistent with other analytical approaches, like in the problem of the random walk in a double-well potential. We discuss the application of our method to nontrivial examples: diffusion in a lattice with defects and a model of an enzymatic reaction outside the steady state regime.Comment: 9 pages, 9 figures, final version (new subsection and many minor improvements

Connecting protein and mRNA burst distributions for stochastic models of gene expression

The intrinsic stochasticity of gene expression can lead to large variability in protein levels for genetically identical cells. Such variability in protein levels can arise from infrequent synthesis of mRNAs which in turn give rise to bursts of protein expression. Protein expression occurring in bursts has indeed been observed experimentally and recent studies have also found evidence for transcriptional bursting, i.e. production of mRNAs in bursts. Given that there are distinct experimental techniques for quantifying the noise at different stages of gene expression, it is of interest to derive analytical results connecting experimental observations at different levels. In this work, we consider stochastic models of gene expression for which mRNA and protein production occurs in independent bursts. For such models, we derive analytical expressions connecting protein and mRNA burst distributions which show how the functional form of the mRNA burst distribution can be inferred from the protein burst distribution. Additionally, if gene expression is repressed such that observed protein bursts arise only from single mRNAs, we show how observations of protein burst distributions (repressed and unrepressed) can be used to completely determine the mRNA burst distribution. Assuming independent contributions from individual bursts, we derive analytical expressions connecting means and variances for burst and steady-state protein distributions. Finally, we validate our general analytical results by considering a specific reaction scheme involving regulation of protein bursts by small RNAs. For a range of parameters, we derive analytical expressions for regulated protein distributions that are validated using stochastic simulations. The analytical results obtained in this work can thus serve as useful inputs for a broad range of studies focusing on stochasticity in gene expression