Fourier multiport devices in which the creation and annihilation operators at the output are related to those at the input through a finite Fourier transform are studied. A general method for the calculation of the output for arbitrary input states is presented. The case of a squeezed state at one of the inputs and vacua at all other inputs is discussed. In the case of thermal states at the input, the device can be used as a thermometer. A factorization technique inspired by the fast Fourier transform leads to a substantial reduction in the number of beam splitters that are required for the experimental implementation of these devices.
The rapid progress of the Internet of Things (IoT) has continued to offer humanity numerous benefits, including many security and safety-critical applications. However, unlocking the full potential of IoT applications, especially in high-consequence domains, requires the assurance that IoT devices will not constitute risk hazards to the users or the environment. To design safe, secure, and reliable IoT systems, numerous frameworks have been proposed to analyse the safety and security, among other properties. This paper reviews some of the prominent classical and model-based system engineering (MBSE) approaches for IoT systems’ safety and security analysis. The review established that most analysis frameworks are based on classical manual approaches, which independently evaluate the two properties. The manual frameworks tend to inherit the natural limitations of informal system modelling, such as human error, a cumbersome processes, time consumption, and a lack of support for reusability. Model-based approaches have been incorporated into the safety and security analysis process to simplify the analysis process and improve the system design’s efficiency and manageability. Conversely, the existing MBSE safety and security analysis approaches in the IoT environment are still in their infancy. The limited number of proposed MBSE approaches have only considered limited and simple scenarios, which are yet to adequately evaluate the complex interactions between the two properties in the IoT domain. The findings of this survey are that the existing methods have not adequately addressed the analysis of safety/security interdependencies, detailed cyber security quantification analysis, and the unified treatment of safety and security properties. The existing classical and MBSE frameworks’ limitations obviously create gaps for a meaningful assessment of IoT dependability. To address some of the gaps, we proposed a possible research direction for developing a novel MBSE approach for the IoT domain’s safety and security coanalysis framework.
A quantum system Σ(n) with variables in Z(n), where n = Q pi (with pi prime numbers), is considered. The non-near-linear geometry G(n) of the phase space Z(n) × Z(n), is studied. The lines through the origin are factorized in terms of 'prime factor lines' in Z(pi)×Z(pi). Weak mutually unbiased bases (WMUB) which are products of the mutually unbiased bases in the 'prime factor Hilbert spaces' H(pi), are also considered. The factorization of both lines and WMUB is analogous to the factorization of integers in terms of prime numbers. The duality between lines and WMUB is discussed. It is shown that there is a partial order in the set of subgeometries of G(n), isomorphic to the partial order in the set of subsystems of Σ(n).
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