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A Quantum Key Distribution (QKD) protocol describes how two remote parties can establish a secret key by communicating over a quantum and a public classical channel that both can be accessed by an eavesdropper. QKD protocols using energy-time entangled photon pairs are of growing practical interest because of their potential to provide a higher secure key rate over long distances by carrying multiple bits per entangled photon pair. We consider a system where information can be extracted by measuring random times of a sequence of entangled photon arrivals. Our goal is to maximize the utility of each such pair. We propose a discrete-time model for the photon arrival process, and establish a theoretical bound on the number of raw bits that can be generated under this model. We first analyze a well-known simple binning encoding scheme, and show that it generates a significantly lower information rate than what is theoretically possible. We then propose three adaptive schemes that increase the number of raw bits generated per photon, and compute and compare the information rates they offer. Moreover, the effect of public channel communication on the secret key rates of the proposed schemes is investigated. 

This paper considers a generalized version of the coin weighing problem with a spring scale that lies at the intersection of group testing and compressed sensing problems. Given a collection of n ≥ 2 coins of total weight d (for a known integer d), where the weight of each coin is an unknown integer in the range of {0, 1, ..., k} (for a known integer k ≥ 1), the goal is to determine the weight of each coin by weighing subsets of coins in a spring scale. The problem is to devise a weighing strategy that minimizes the average number of weighings over all possible weight configurations. For d = k = 1, an adaptive bisecting weighing strategy is known to be optimal. However, even the simplest non-trivial case of the problem, i.e., d = k = 2, is still open. For this case, we propose and analyze a simple and effective adaptive weighing strategy. Our analysis shows that the proposed strategy requires about 1.365log2n-0.5 weighings on average. As n grows unbounded, the proposed strategy, when compared to an optimal strategy within the commonly-used class of nested strategies, requires about 31.75% less number of weighings on average; and in comparison with the information-theoretic lower bound, it requires at most about 8.16% extra number of weighings on average.