Applications of algebraic methods in quantum physics

这个建议沿着两个平行的轨道进行。一个是关于具有饱和海森堡测不准关系性质的量子力学状态的结构和性质。这样的状态被称为智能，并且有许多应用，特别是在光学干涉计量学中。此外，它们在探测纠缠方面也有重要的理论作用，这是量子信息论的一个基本特征。第二个轨道涉及有限维量子系统中的互补性。这里，目的是研究描述量子系统最大不相容准备的状态集的构造。这样的状态对于量子信息的各种协议的最佳实施是有用的。其目的是考虑这样的情况，例如在6维的量子系统中，维度不是质数或素数的幂。在后一种情况下，互补状态的集合是已知的并被构造。

This proposal runs along two parallel tracks.  One deals with the construction and properties of quantum mechanical states that have the property of saturating the Heisenberg uncertainty relation.  Such states are called intelligent, and have numerous applications, particularly in optical interferometry.  They also have an important theoretical role in the detection of entanglement, a feature essential to quantum information theory.A second track deals with complementarity in finite dimensional quantum systems.  Here, the objective is to study the construction of sets of states that describe maximally incompatible preparations of a quantum system.  Such states are useful for the optimal implementation of various protocols of quantum information.  The intention is to consider cases, as in a quantum system of dimenion 6, where the dimension is not a prime number or a power of a prime number.  In the latter cases, sets of complementary states are already known and constructed.