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Given a sequence of numbers ${p_n}$ in $[0,1]$, consider the following experiment. First, we flip a fair coin and then, at step $n$, we turn the coin over to the other side with probability $p_n$, $nge 2$. What can we say about the distribution of the empirical frequency of heads as $ntoinfty$? We show that a number of phase transitions take place as the turning gets slower (i.e. $p_n$ is getting smaller), leading first to the breakdown of the Central Limit Theorem and then to that of the Law of Large Numbers. It turns out that the critical regime is $p_n=text{const}/n$. Among the scaling limits, we obtain Uniform, Gaussian, Semicircle and Arcsine laws.
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In this letter we present the first implementation of a quantum coin tossing protocol. This protocol belongs to a class of ``two-party cryptographic problems, where the communication partners distrust each other. As with a number of such two-party protocols, the best implementation of the quantum coin tossing requires qutrits. In this way, we have also performed the first complete quantum communication protocol with qutrits. In our experiment the two partners succeeded to remotely toss a row of coins using photons entangled in the orbital angular momentum. We also show the experimental bounds of a possible cheater and the ways of detecting him.
Let $S$ be the random walk obtained from coin turning with some sequence ${p_n}_{nge 1}$, as introduced in [6]. In this paper we investigate the scaling limits of $S$ in the spirit of the classical Donsker invariance principle, both for the heating and for the cooling dynamics. We prove that an invariance principle, albeit with a non-classical scaling, holds for not too small sequences, the order const$cdot n^{-1}$ (critical cooling regime) being the threshold. At and below this critical order, the scaling behavior is dramatically different from the one above it. The same order is also the critical one for the Weak Law of Large Numbers to hold. In the critical cooling regime, an interesting process emerges: it is a continuous, piecewise linear, recurrent process, for which the one-dimensional marginals are Beta-distributed. We also investigate the recurrence of the walk and its scaling limit, as well as the ergodicity and mixing of the $n$th step of the walk.
We give a game-theoretic proof of the celebrated Erdos-Feller-Kolmogorov-Petrowsky law of the iterated logarithm for fair coin tossing. Our proof, based on Bayesian strategy, is explicit as many other game-theoretic proofs of the laws in probability theory.
We study capital process behavior in the fair-coin game and biased-coin games in the framework of the game-theoretic probability of Shafer and Vovk (2001). We show that if Skeptic uses a Bayesian strategy with a beta prior, the capital process is lucidly expressed in terms of the past average of Realitys moves. From this it is proved that the Skeptics Bayesian strategy weakly forces the strong law of large numbers (SLLN) with the convergence rate of O(sqrt{log n/n})$ and if Reality violates SLLN then the exponential growth rate of the capital process is very accurately described in terms of the Kullback divergence between the average of Realitys moves when she violates SLLN and the average when she observes SLLN. We also investigate optimality properties associated with Bayesian strategy.
The security of quantum communication using a weak coherent source requires an accurate knowledge of the sources mean photon number. Finite calibration precision or an active manipulation by an attacker may cause the actual emitted photon number to deviate from the known value. We model effects of this deviation on the security of three quantum communication protocols: the Bennett-Brassard 1984 (BB84) quantum key distribution (QKD) protocol without decoy states, Scarani-Acin-Ribordy-Gisin 2004 (SARG04) QKD protocol, and a coin-tossing protocol. For QKD, we model both a strong attack using technology possible in principle, and a realistic attack bounded by todays technology. To maintain the mean photon number in two-way systems, such as plug-and-play and relativistic quantum cryptography schemes, bright pulse energy incoming from the communication channel must be monitored. Implementation of a monitoring detector has largely been ignored so far, except for ID Quantiques commercial QKD system Clavis2. We scrutinize this implementation for security problems, and show that designing a hack-proof pulse-energy-measuring detector is far from trivial. Indeed the first implementation has three serious flaws confirmed experimentally, each of which may be exploited in a cleverly constructed Trojan-horse attack. We discuss requirements for a loophole-free implementation of the monitoring detector.
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