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A fundamental problem in quantum computation and quantum information is finding the minimum quantum dimension needed for a task. For tasks involving state preparation and measurements, this problem can be addressed using only the input-output correlations. This has been applied to Bell, prepare-and-measure, and Kochen-Specker contextuality scenarios. Here, we introduce a novel approach to quantum dimension witnessing for scenarios with one preparation and several measurements, which uses the graphs of mutual exclusivity between sets of measurement events. We present the concepts and tools needed for graph-theoretic quantum dimension witnessing and illustrate their use by identifying novel quantum dimension witnesses, including a family that can certify arbitrarily high quantum dimensions with few events.
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We consider a communication network where there exist wiretappers who can access a subset of channels, called a wiretap set, which is chosen from a given collection of wiretap sets. The collection of wiretap sets can be arbitrary. Secure network coding is applied to prevent the source information from being leaked to the wiretappers. In secure network coding, the required alphabet size is an open problem not only of theoretical interest but also of practical importance, because it is closely related to the implementation of such coding schemes in terms of computational complexity and storage requirement. In this paper, we develop a systematic graph-theoretic approach for improving Cai and Yeungs lower bound on the required alphabet size for the existence of secure network codes. The new lower bound thus obtained, which depends only on the network topology and the collection of wiretap sets, can be significantly smaller than Cai and Yeungs lower bound. A polynomial-time algorithm is devised for efficient computation of the new lower bound.
Exploring the graph approach, we restate the extended definition of noncontextuality provided by the contextuality-by-default framework. This extended definition avoids the assumption of nondisturbance, which states that whenever two contexts overlap, the marginal distribution obtained for the intersection must be the same. We show how standard tools for characterizing contextuality can also be used in this extended framework for any set of measurements and, in addition, we also provide several conditions that can be tested directly in any contextuality experiment. Our conditions reduce to traditional ones for noncontextuality if the nondisturbance assumption is satisfied.
We give a family of counter examples showing that the two sequences of polytopes $Phi_{n,n}$ and $Psi_{n,n}$ are different. These polytopes were defined recently by S. Friedland in an attempt at a polynomial time algorithm for graph isomorphism.
Presently, models for the parameterization of cross sections for nodal diffusion nuclear reactor calculations at different conditions using histories and branches are developed from reactor physics expertise and by trial and error. In this paper we describe the development and application of a novel graph theoretic approach (GTA) to develop the expressions for evaluating the cross sections in a nodal diffusion code. The GTA generalizes existing nodal cross section models into a ``non-orthogonal and extensible dimensional parameter space. Furthermore, it utilizes a rigorous calculus on graphs to formulate partial derivatives. The GTA cross section models can be generated in a number of ways. In our current work we explore a step-wise regression and a complete Taylor series expansion of the parameterized cross sections to develop expressions to evaluate them. To establish proof-of-principle of the GTA, we compare numerical results of GTA generated cross section evaluations with traditional models for canonical PWR case matrices and the AP1000 lattice designs.
Instrumental variables allow the estimation of cause and effect relations even in presence of unobserved latent factors, thus providing a powerful tool for any science wherein causal inference plays an important role. More recently, the instrumental scenario has also attracted increasing attention in quantum physics, since it is related to the seminal Bells theorem and in fact allows the detection of even stronger quantum effects, thus enhancing our current capabilities to process information and becoming a valuable tool in quantum cryptography. In this work, we further explore this bridge between causality and quantum theory and apply a technique, originally developed in the field of quantum foundations, to express the constraints implied by causal relations in the language of graph theory. This new approach can be applied to any causal model containing a latent variable. Here, by focusing on the instrumental scenario, it allows us to easily reproduce known results as well as obtain new ones and gain new insights on the connections and differences between the instrumental and the Bell scenarios.
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