Strongly trapped two-dimensional quantum walks

Discrete time quantum walks (DTQWs) are nontrivial generalizations of random walks with a broad scope of applications. In particular, they can be used as computational primitives, and they are suitable tools for simulating other quantum systems. DTQWs usually spread ballistically due to their quantumness. In some cases, however, they can remain localized at their initial state (trapping). The trapping and other fundamental properties of DTQWs are determined by the choice of the coin operator. We introduce and analyze an up to now uncharted type of walks driven by a coin class leading to strong trapping, complementing the known list of walks. This class of walks exhibit a number of exciting properties with the possible applications ranging from light pulse trapping in a medium to topological effects and quantum search.Comment: 5 pages, 4 figures, Accepted for publication in Physical Review

Discrete time quantum walks on percolation graphs

Randomly breaking connections in a graph alters its transport properties, a model used to describe percolation. In the case of quantum walks, dynamic percolation graphs represent a special type of imperfections, where the connections appear and disappear randomly in each step during the time evolution. The resulting open system dynamics is hard to treat numerically in general. We shortly review the literature on this problem. We then present our method to solve the evolution on finite percolation graphs in the long time limit, applying the asymptotic methods concerning random unitary maps. We work out the case of one dimensional chains in detail and provide a concrete, step by step numerical example in order to give more insight into the possible asymptotic behavior. The results about the case of the two-dimensional integer lattice are summarized, focusing on the Grover type coin operator.Comment: 22 pages, 3 figure

Impact of Cancer and 12 Weeks of Chemotherapy on the Balance of the Autonomic Nervous System in Cancer Patients

Cancer and its treatment itself (especially chemotherapy) is associated with number of negative effects on the human body. These include mainly cardiac toxicity, peripheral neuropathy, bone loss, depression, anxiety, nausea, pain, cognitive changes, fatigue, fitness reduction and more. Fatigue is one of the most common negative effects, often persists long after treatment and is described as insurmountable and is associated with lower parasympathetic activity. The autonomic nervous system (ANS) is the main homeostatic regulatory system of the body, it regulates involuntary physiological processes. We believe that this part of the peripheral nervous system can be negatively affected by cancer and chemotherapy, which can have a negative impact on all the processes that control this system. Most drugs used in oncology lead to chemotherapy-induced peripheral neuropathy and are expected to have an influence on the autonomic nervous system. Activity and balance of the autonomic nervous system depend on a range of dynamically changing and quantitatively different conditions such as age, stress, physical activity, sleep, illness, fatigue and more. Methods: To evaluate ANS activity, spectral analysis of heart rate variability (HRV) was assessed. During treatment, 19 oncological patients with prescribed adjuvant chemotherapy measured HRV 3 times a week using a chest strap with a HRV monitor mySASY and mySASY software. Parasympathetic activity (PA), sympathetic activity (S), total score (TS) and total power (TP) were selected as indicators of ANS activity. The patients were women aged 50.38 ± 10,29 with BMI 25,72 ± 4,16. The mean values for the first 14 days of treatment and then for 14 days after 12 weeks of treatment were compared. Data normality was verified by Kolmogorov-Smirnov test (K-S) and static significance was calculated by t-test. All statistical tests were performed at a significance level of 5%. Results: There was a significant decrease in values for three parameters. The PA decreased from 3,80 ± 1,56 to 3,14 ± 1,67 (p = .03), TP decreased from 3,65 ± 1,87 to 2,82 ± 2,08 (p = .04), TS decreased from 3,40 ± 1,67 to 2,67 ± 1,73 (p = .01). Sympathetic activity was somewhat but not significantly higher, increased from 6,74 ± 1,22 to 6,80 ± 1,44 (p = .83). Conclusion: During the 12 weeks of treatment with adjuvant chemotherapy, there was a significant reduction in parasympathetic activity, total score, and total power. A decrease in PA is usually associated with lower regenerative abilities of the organism, a decrease in TP is associated with a decrease in the activity of the entire ANS. TS evaluates the total power and balance of both ANS branches. The higher the values, the more the body is regenerated and ready for further stress and response to stress

Fertility of Roma Minorities in Central and Eastern Europe

We analyse Roma fertility in four neighbouring countries in Central and Eastern Europe with a large Roma minority: in Hungary, Slovakia, Romania and Serbia. The sources of data are the respective national population censuses from 2011. Fertility is measured at the birth cohort level as the average number of children ever born. We make an international comparison of the fertility of Roma and non-Roma majority population women on the basis of completed education. In the case of Hungary, we also explore how the correlation between fertility and ethnic identity is modified when completed education and ethnic residential segregation are controlled. The fertility of Roma women is far above the majority population average in all birth cohorts and in each country. Educational attainment modifies this relationship. The fertility of highly educated Roma and majority population women is converging. The exposure to majority behaviour also has an effect. The lower the level of ethnic residential segregation, the smaller the difference between the fertility of Roma and majority population women. Completed education and residential segregation may exert different forces at the two ends of the educational hierarchy when their joint effect is explored. At the upper end of the social hierarchy, neither segregation nor ethnicity matters; at the lower end, however, both exposure to ethnic majority behaviour and ethnicity matter

Complete classification of trapping coins for quantum walks on the 2D square lattice

One of the unique features of discrete-time quantum walks is called trapping, meaning the inability of the quantum walker to completely escape from its initial position, albeit the system is translationally invariant. The effect is dependent on the dimension and the explicit form of the local coin. A four state discrete-time quantum walk on a square lattice is defined by its unitary coin operator, acting on the four dimensional coin Hilbert space. The well known example of the Grover coin leads to a partial trapping, i.e., there exists some escaping initial state for which the probability of staying at the initial position vanishes. On the other hand, some other coins are known to exhibit strong trapping, where such escaping state does not exist. We present a systematic study of coins leading to trapping, explicitly construct all such coins for discrete-time quantum walks on the 2D square lattice, and classify them according to the structure of the operator and the manifestation of the trapping effect. We distinguish three types of trapping coins exhibiting distinct dynamical properties, as exemplified by the existence or non-existence of the escaping state and the area covered by the spreading wave-packet.Comment: 15 pages, 6 figures; updated to the published versio

Sensitivity to initial noise in measurement-induced nonlinear quantum dynamics

We consider a special iterated quantum protocol with measurement-induced nonlinearity for qubits, where all pure initial states on the Bloch sphere can be considered chaotic. The dynamics is ergodic with no attractive �fixed cycles. We show that initial noise radically changes this behavior. The completely mixed state is an attractive �xed point of the dynamics induced by the protocol. Our numerical simulations strongly indicate that initially mixed states all converge to the completely mixed state. The presented protocol is an example, where gaining information from measurements and employing it to control an ensemble of quantum systems enables us to create ergodicity, which in turn is destroyed by any initial noise

Phase Transition in Iterated Quantum Protocols for Noisy Inputs

Quantum information processing exploits all the features quantum mechanics offers. Among them there is the possibility to induce nonlinear maps on a quantum system by involving two or more identical copies of the given system in the same state. Such maps play a central role in distillation protocols used for quantum key distribution. We determine that such protocols may exhibit sensitive, quasi-chaotic evolution not only for pure initial states but also for mixed states, i.e. the complex dynamical behavior is not destroyed by small initial uncertainty. We show that the appearance of sensitive, complex dynamics associated with a fractal structure in the parameter space of the system has the character of a first order phase transition. The purity of the initial state plays the role of the control parameter and the dimension of the fractal structure is independent of the purity value after passing the phase transition point. The critical purity coincides with the purity of a repelling fixed point of the dynamics and we show that all the pre-images of states from the close neighborhood of pure chaotic initial states have purity larger than this. Initial states from this set can be considered as quasi-chaotic