70,002 research outputs found
Complexity of Graph State Preparation
The graph state formalism is a useful abstraction of entanglement. It is used
in some multipartite purification schemes and it adequately represents
universal resources for measurement-only quantum computation. We focus in this
paper on the complexity of graph state preparation. We consider the number of
ancillary qubits, the size of the primitive operators, and the duration of
preparation. For each lexicographic order over these parameters we give upper
and lower bounds for the complexity of graph state preparation. The first part
motivates our work and introduces basic notions and notations for the study of
graph states. Then we study some graph properties of graph states,
characterizing their minimal degree by local unitary transformations, we
propose an algorithm to reduce the degree of a graph state, and show the
relationship with Sutner sigma-game.
These properties are used in the last part, where algorithms and lower bounds
for each lexicographic order over the considered parameters are presented.Comment: 17 page
Exterior convexity and classical calculus of variations
We study the relation between various notions of exterior convexity
introduced in Bandyopadhyay-Dacorogna-Sil \cite{BDS1} with the classical
notions of rank one convexity, quasiconvexity and polyconvexity. To this end,
we introduce a projection map, which generalizes the alternating projection for
two-tensors in a new way and study the algebraic properties of this map. We
conclude with a few simple consequences of this relation which yields new
proofs for some of the results discussed in Bandyopadhyay-Dacorogna-Sil
\cite{BDS1}.Comment: The original publication is available at www.esaim-cocv.org
https://www.esaim-cocv.org/articles/cocv/abs/2016/02/cocv150007/cocv150007.htm
From Logical Calculus to Logical Formality—What Kant Did with Euler’s Circles
John Venn has the “uneasy suspicion” that the stagnation in mathematical logic between J. H. Lambert and George Boole was due to Kant’s “disastrous effect on logical method,” namely the “strictest preservation [of logic] from mathematical encroachment.” Kant’s actual position is more nuanced, however. In this chapter, I tease out the nuances by examining his use of Leonhard Euler’s circles and comparing it with Euler’s own use. I do so in light of the developments in logical calculus from G. W. Leibniz to Lambert and Gottfried Ploucquet. While Kant is evidently open to using mathematical tools in logic, his main concern is to clarify what mathematical tools can be used to achieve. For without such clarification, all efforts at introducing mathematical tools into logic would be blind if not complete waste of time. In the end, Kant would stress, the means provided by formal logic at best help us to express and order what we already know in some sense. No matter how much mathematical notations may enhance the precision of this function of formal logic, it does not change the fact that no truths can, strictly speaking, be revealed or established by means of those notations
On the notion of conditional symmetry of differential equations
Symmetry properties of PDE's are considered within a systematic and unifying
scheme: particular attention is devoted to the notion of conditional symmetry,
leading to the distinction and a precise characterization of the notions of
``true'' and ``weak'' conditional symmetry. Their relationship with exact and
partial symmetries is also discussed. An extensive use of ``symmetry-adapted''
variables is made; several clarifying examples, including the case of
Boussinesq equation, are also provided.Comment: 18 page
Towards a Base UML Profile for Architecture Description
This paper discusses a base UML profile for architecture description as supported by existing Architecture Description Languages (ADLs). The profile may be extended so as to enable architecture modeling both as expressed in conventional ADLs and according to existing runtime infrastructures (e.g., system based on middleware architectures).
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