We present an analysis of Wiesner's quantum money scheme, as well as some natural generalizations of it, based on semidefinite programming. For Wiesner's original scheme, it is determined that the optimal probability for a counterfeiter to create two copies of a bank note from one, where both copies pass the bank's test for validity, is (3/4) n for n being the number of qubits used for each note. Generalizations in which other ensembles of states are substituted for the one considered by Wiesner are also discussed, including a scheme recently proposed by Pastawski, Yao, Jiang, Lukin, and Cirac, as well as schemes based on higher dimensional quantum systems. In addition, we introduce a variant of Wiesner's quantum money in which the verification protocol for bank notes involves only classical communication with the bank. We show that the optimal probability with which a counterfeiter can succeed in two independent verification attempts, given access to a single valid n-qubit bank note, is (3/4 + √ 2/8) n . We also analyze extensions of this variant to higher-dimensional schemes.

Yao's 1995 publication ‘Quantum circuit complexity’ in Proceedings of the 34th Annual IEEE Symposium on Foundations of Computer Science , pp. 352–361, proved that quantum Turing machines and quantum circuits are polynomially equivalent computational models: t ≥ n steps of a quantum Turing machine running on an input of length n can be simulated by a uniformly generated family of quantum circuits with size quadratic in t , and a polynomial-time uniformly generated family of quantum circuits can be simulated by a quantum Turing machine running in polynomial time. We revisit the simulation of quantum Turing machines with uniformly generated quantum circuits, which is the more challenging of the two simulation tasks, and present a variation on the simulation method employed by Yao together with an analysis of it. This analysis reveals that the simulation of quantum Turing machines can be performed by quantum circuits having depth linear in t , rather than quadratic depth, and can be extended to variants of quantum Turing machines, such as ones having multi-dimensional tapes. Our analysis is based on an extension of method described by Arright, Nesme and Werner in 2011 in Journal of Computer and System Sciences 77 , 372–378. ( doi:10.1016/j.jcss.2010.05.004 ), that allows for the localization of causal unitary evolutions.

This paper studies the potential advantages of correlated answers in hypothetical tests of a simple interactive type. In these tests, an individual is presented with a question, to which he or she must respond with an answer, leading to an outcome of pass or fail. In quantum information theoretic variants of these tests, we show it is possible for an individual to correlate their answers across multiple tests in a way that produces a striking non-classical behaviour. Specifically, there exist tests for which no strategy can lead to the outcome pass with certainty, but where answers to multiple, independently administered tests can be correlated so that at least one test is passed with certainty. This phenomenon, which gives rise to a perfect form of hedging when considered in a game-theoretic setting, is quantum in nature: it is not possible in classical variants of the tests we consider.

The application of mathematical tools can be necessary to provide an integrated analysis and interpretation of the abundant information that can be collected in air quality monitoring networks. This article develops a methodology based on the use of SelfOrganizing Map (SOM) artificial neural networks for integrating data about multiple measured pollutants to group monitoring stations according to their similar air quality. The proposed method considers the subsequent geographical mapping of the clusters of stations observed with the SOM, which can make it possible to detect geographically different areas but that share similar air pollution problems. This methodology is illustrated with its application to a case study in which 517 stations of the Spanish air quality monitoring network were classified considering simultaneously their levels of regulated pollutants in 2005, highlighting some implications of data normalization in the process. In particular, the use of legal limit values to normalize the concentrations of pollutants proved to be especially advisable. Results obtained with the SOM-based methodology, when compared to classifications based directly on legislation, provided more useful classifications for further air quality management actions, and revealed that these types of tools can facilitate the design of air pollution reduction programs by discovering different areas with similar problems.

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