Well-posedness and blowup of the geophysical boundary layer problem

Under the assumption that the initial velocity and outflow velocity are analytic in the horizontal variable, the local well-posedness of the geophysical boundary layer problem is obtained by using energy method in the weighted Chemin-Lerner spaces. Moreover, when the initial velocity and outflow velocity satisfy certain condition on a transversal plane, it is proved that the $W^{1,\infty}-$norm of any smooth solution decaying exponentially in the normal variable to the geophysical boundary layer problem blows up in a finite time

An efficient implementation of the decoy-state measurement-device-independent quantum key distribution with heralded single-photon sources

We study the decoy-state measurement-device-independent quantum key distribution using heralded single-photon sources. This has the advantage that the observed error rate in X basis is in higher order and not so large. We calculate the key rate and transmission distance for two cases: one using only triggered events, and the other using both triggered and non-triggered events. We compare the key rates of various protocols and find that our new scheme using triggered and non-triggered events can give higher key rate and longer secure distance. Moreover, we also show the different behavior of our scheme when using different heralded single-photon sources, i.e., in poisson or thermal distribution. We demonstrate that the former can generate a relatively higher secure key rate than the latter, and can thus work more efficiently in practical quantum key distributions.Comment: 15 pages, 4 figure

A review on the decoy-state method for practical quantum key distribution

We present a review on the historic development of the decoy state method, including the background, principles, methods, results and development. We also clarify some delicate concepts. Given an imperfect source and a very lossy channel, the photon-number-splitting (PNS) attack can make the quantum key distribution (QKD) in practice totally insecure. Given the result of ILM-GLLP, one knows how to distill the secure final key if he knows the fraction of tagged bits. The purpose of decoy state method is to do a tight verification of the the fraction of tagged bits. The main idea of decoy-state method is changing the intensities of source light and one can verify the fraction of tagged bits of certain intensity by watching the the counting rates of pulses of different intensities. Since the counting rates are small quantities, the effect of statistical fluctuation is very important. It has been shown that 3-state decoy-state method in practice can work even with the fluctuations and other errors.Comment: 17 page

Graphs induced by iterated function systems

For an iterated function system (IFS) of simillitidues, we define two graphs on the representing symbolic space. We show that if the self-similar set $K$ has positive Lebesgue measure or the IFS satisfies the weak separation condition, then the graphs are hyperbolic, moreover the hyperbolic boundaries are homeomorphic to the self-similar sets

On the role of coherent attacks in a type of strategic problem related to quantum key distribution

We consider a strategic problem of the Evesdropping to quantum key distribution. Evesdropper hopes to obtain the maxium information given the disturbance to the qubits is often For this strategy, the optimized individual attack have been extensively constructed under various conditions. However, it seems a difficult task in the case of coherent attack, i.e., Eve may treat a number of intercepted qubits collectively, including the collective unitary transformations and the measurements. It was conjectured by Cirac and Gisin that no coherent attack can be more powerful for this strategy for BB84 protocol. In this paper we give a general conclusion on the role of coherent attacks for the strategy of maxmizing the information given the disturbance. Suppose in a quantum key distribution(QKD) protocol, all the transmitted bits from Alice are independent and only the individual disturbances to each qubits are examined by Alice and Bob. For this type of protocols(so far almost all QKD protocols belong to this type), in principle no coherent attack is more powerful than the product of optimized individual attack to each individual qubits. All coherent attacks to the above QKD protocols can be disregarded for the strategy above.Comment: This is identical to quant-ph/0110073. The the title of the old version is inappropriate. What I have proven is the Cirac-Gisin conjecture in Phys. Lett. A229, 1(1997). The actual conclusion is that coherent attack can be ignored for the strategy of maximizing Eve's total information given disturbance. But this does not mean coherent attack can be ignored in QK

Criteria for unconditional entanglement purification

We show that the fidelity result of advantage distillation (Bennett et al, Phys. Rev. Lett., 76, 722(1996)) is not only for the product state of raw pairs, it is actually correct with whatever form of state of raw pairs. We then give a general theorem for unconditional entanglement purification. This theorem lists the conditions on which the fidelity result of a purification protocol keeps unchanged from product form of raw-pair state and arbitrary form of raw-pair state. Using this theorem, we find that all existing purification puricfication can work for arbitrary initial state of raw pairs.Comment: The motivation and the title are change

Perfect random number generator is unnecessary for secure quantum key distribution

Game G: Clare passes a string s which is either from perfect random number generator R0 or from good imperfect number generator R1, with equal probability. Alice's information about whether it is from R0 or R1 is bounded by small value h. Alice use s as the input random numbers for QKD protocol with Bob. Suppose Eve may have very small information about the final key if s is from R0 and Eve has large information if s is from R1, then after the protocol, Alice announce the final key, Eve's information about whether s is from R0 or R1 is unreasonablly large, i.e., breaks the known bound, h. Explicit formulas are given in the article.Comment: The proof is VERY simpl

Revisiting differentially private linear regression: optimal and adaptive prediction & estimation in unbounded domain

We revisit the problem of linear regression under a differential privacy constraint. By consolidating existing pieces in the literature, we clarify the correct dependence of the feature, label and coefficient domains in the optimization error and estimation error, hence revealing the delicate price of differential privacy in statistical estimation and statistical learning. Moreover, we propose simple modifications of two existing DP algorithms: (a) posterior sampling, (b) sufficient statistics perturbation, and show that they can be upgraded into **adaptive** algorithms that are able to exploit data-dependent quantities and behave nearly optimally **for every instance**. Extensive experiments are conducted on both simulated data and real data, which conclude that both AdaOPS and AdaSSP outperform the existing techniques on nearly all 36 data sets that we test on.Comment: Uncertainty in Artificial Intelligence (UAI-2018), Monterey, C

A practically feasible entanglement assisted quantum key distribution protocol

We give an entanglement assisted scheme for quantum key distribution. The scheme requires the maximally entangled 2-qubit state but does not require any quantum storage. The scheme is unconditionally secure under whatever Eve's attack. Given the symmetric noisy channel with uncorrelated noise, our scheme can tolerate the bit error rate up to 26% in the 4-state case and 30% in the 6-state respectively, respectively. These values are higher than those of all currently known two-level-state schemes without using a quantum storage.Comment: 13 pages, 3 figure

Calculating the day of the week: null-days algorithm

In this paper, we propose a new algorithm of calculating the day of the week for any given century, year, month and day in Gregorian calendar. We provide two simple formulas to convert the century and the year into two integers. Then we introduce a list of null-days to transform the month and the day into another integer. Adding these three integers together and calculating the sum's residue modulo $7$ gives a number between $0$ and $6$, which corresponds to Sunday until Saturday. Slight modification is needed if we have a leap year and the given month is either January or February. Our null-days algorithm is simple enough to be done by mental calculation, and the list of null-days has memorable patterns