We first consider a method of centering and a change of variable formula for a quantum integral. We then present three types of quantum integrals. The first considers the expectation of the number of heads in n flips of a "quantum coin". The next computes quantum integrals for destructive pairs examples. The last computes quantum integrals for a (Lebesgue)2 quantum measure. For this last type we prove some quantum counterparts of the fundamental theorem of calculus.
We first consider a method of centering and a change of variable formula for a quantum integral. We then present three types of quantum integrals. The first considers the expectation of the number of heads in n flips of a "quantum coin". The next computes quantum integrals for destructive pairs examples. The last computes quantum integrals for a (Lebesgue)2 quantum measure. For this last type we prove some quantum counterparts of the fundamental theorem of calculus.
We study quantum random variables and generalize several classical limit results to the quantum setting. We prove a quantum analogue of Lebesgue's dominated convergence theorem and use it to prove a quantum martingale convergence theorem. This quantum martingale convergence theorem is of particular interest since it exhibits non-classical behaviour; even though the limit of the martingale exists and is unique, it is not explicitly identifiable. However, we provide a partial classification of the limit through a study of the space of all quantum random variables having quantum expectation zero.
Imagine that a pair of coins are tossed in a black box. The box reports only one of the following three results at random: (1) the outcome of the first coin (heads or tails), (2) the outcome of the second coin (heads or tails), or (3) whether the outcomes of the two coins matched or were different. Our task is to construct a joint probability distribution of the four possible outcomes of the coins (HH, TT, HT, TH) based on many observations of the black box outputs. Now suppose that after many trials, the black box reports that each coin comes up heads two-thirds of the time when measured individually, yet the coins never match when they are compared. (Clearly the results of the coin tosses have been correlated—perhaps a joker in the black box flips the coins and then changes the outcomes appropriately). We seek a distribution that both reflects this correlation and obeys the marginal distributions of each coin as two-thirds heads, one-third tails (see the three tables on page 23). The only way to satisfy both requirements is to force the joint probability P(TT) of getting two tails to be negative! Mathematically, this is because P(HH)+P(TT) is observed to be zero, yet we expect P(HH)to be greater than P(TT), because the individual coins are weighted toward heads.
A common theme in mathematics is the evaluation of Gauss integrals. This, coupled with the fact that they are used in different branches of science, makes the topic always actual and interesting. In these notes we shall analyze a particular class of Gaussian integrals that depends by the quantum parameter ℏ. Starting from classical results, we will present an overview on methods, examples and analogies regarding the practice of solving quantum Gaussian integrals.
In his seminal work, Cleve [ Proceedings of the 18th Annual ACM Symposium on Theory of Computing, 1986, pp. 364--369] has proved that any r-round coin-flipping protocol can be efficiently biased by Θ(1/r). This lower bound was met for the two-party case by Moran, Naor, and Segev [ J. Cryptology, 29 (2016), pp. 491--513] and the three-party case (up to a polylog factor) by Haitner and Tsfadia [ SIAM J. Comput., 46 (2017), pp. 479--542] and was approached for n-party protocols when n<loglogr by Buchbinder et al. [ Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms, 2017, pp. 2580--2600]. For n>loglogr, however, the best bias for n-party coin-flipping protocols remains O(n/r) achieved by the majority protocol of Awerbuch et al. [ How to implement Bracha's O(logn) Byzantine Agreement Algorithm, manuscript, 1985]. Our main result is a tighter lower bound on the bias of coin-flipping protocols, showing that, for every constant ε>0, an rε-party r-round coin-flipping protocol can be efficiently biased by Ω(1/r). As far as we know, this is the first improvement of Cleve's bound and is only n=rε (multiplicative) far from the aforementioned upper bound of Awerbuch et al. We prove the above bound using two new results that we believe are of independent interest. The first result is that a sequence of (``augmented'') weak martingales have large gap: with constant probability there exists two adjacent variables whose gap is at least the ratio between the gap between the first and last variables and the square root of the number of variables. This generalizes over the result of Cleve and Impagliazzo [ Martingales, Collective Coin Flipping and Discrete Control Processes (Extended Abstract), http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.51.1797, 1993], who showed that the above holds for strong martingales, and allows in some setting to exploit this gap by efficient algorithms. We prove the above using a novel argument that does not follow the more complicated approach of R. Cleve and R. Impagliazzo. The second result is a new sampling algorithm that uses a differentially private mechanism to minimize the effect of data divergence.
Quantum stochastic integrals have been constructed in various contexts [2, 3, 4, 5, 9] by adapting the construction of the classical L2-Itô-integral with respect to Brownian motion. Thus, the integral is first defined for simple integrands as a finite sum, then one establishes certain isometry relations or suitable bounds to allow the extension, by continuity, to more general integrands. The integrator is typically operator-valued, the integrand is vector-valued or operator-valued and the quantum stochastic integral is then given as a vector in a Hilbert space, or as an operator on the Hilbert space determined by its action on suitable vectors.