One-dimensional quantum random walks with two entangled coins

We offer theoretical explanations for some recent observations in numerical simulations of quantum random walks (QRW). Specifically, in the case of a QRW on the line with one particle (walker) and two entangled coins, we explain the phenomenon, called "localization", whereby the probability distribution of the walker's position is seen to exhibit a persistent major "spike" (or "peak") at the initial position and two other minor spikes which drift to infinity in either direction. Another interesting finding in connection with QRW's of this sort pertains to the limiting behavior of the position probability distribution. It is seen that the probability of finding the walker at any given location becomes eventually stationary and non-vanishing. We explain these observations in terms of the degeneration of some eigenvalue of the time evolution operator $U(k)$. An explicit general formula is derived for the limiting probability, from which we deduce the limiting value of the height of the observed spike at the origin. We show that the limiting probability decreases {\em quadratically} for large values of the position $x$. We locate the two minor spikes and demonstrate that their positions are determined by the phases of non-degenerated eigenvalues of $U(k)$. Finally, for fixed time $t$ sufficiently large, we examine the dependence on $t$ of the probability of finding a particle at a given location $x$.Comment: 11 page

On limiting distributions of quantum Markov chains

In a quantum Markov chain, the temporal succession of states is modeled by the repeated action of a "bistochastic quantum operation" on the density matrix of a quantum system. Based on this conceptual framework, we derive some new results concerning the evolution of a quantum system, including its long-term behavior. Among our findings is the fact that the Ces$\grave{a}$ro limit of any quantum Markov chain always exists and equals the orthogonal projection of the initial state upon the eigenspace of the unit eigenvalue of the bistochastic quantum operation. Moreover, if the unit eigenvalue is the only eigenvalue on the unit circle, then the quantum Markov chain converges in the conventional sense to the said orthogonal projection. As a corollary, we offer a new derivation of the classic result describing limiting distributions of unitary quantum walks on finite graphs \cite{AAKV01}

Ovarian preservation and prognosis in adnexal torsion surgery — a retrospective analysis

Objectives: This study aims to analyze the conditions of ovarian preservation during adnexal torsion surgery, and safetyof ovarian preservation.Material and methods: A retrospective analysis of 130 patients, who underwent surgery for ovarian benign tumor pedicletorsion in Fujian Provincial Maternal and Child Health Hospital from June 2013 to June 2018, was conducted. This studyanalyses the possible risk factors affecting the operation method using multiple logistic regression and analyses the complicationsand the recovery of ovarian function after the treatment of the ovarian preservation.Results: Among these patients, 58 received ovarian cystectomy, while 72 received ovariectomy. There was no significantdifference in terms of age, preoperative blood, operation time and surgical bleeding volume between the two groups(p > 0.05). However, there was a significant difference in preoperative adnexal blood flow, abdominal pain to the surgicalinterval, and a collection of torsion cycles (p < 0.05). There was an increased risk of ovarian resection in patients whose bloodflow of the annex disappeared, whose time of abdominal pain was long, and whose number of twists were significant. Forthe preservation group, there were no increases in postoperative complications.Conclusions: According to clinical indicators, such as preoperative adnexal blood flow, abdominal pain to the interval ofsurgery and the number of torsion cycles, it was determined whether it was feasible to keep the ovary. Retaining the ovaryis safe, effective and feasible in adnexal torsion