模糊函数的加权范数与Wigner分布
Weighted Norms of Ambiguity Functions and Wigner Distributions
摘要：
本文给出了模糊函数和Wigner函数的加权p-范数的新的界。这种规范经常出现在物理学和工程学的几个领域。例如，在广义平稳不相关散射信道中的Weyl-Heisenberg信令的脉冲优化中，对于给定的散射统计量找到最佳波形是关键步骤，这在雷达和声纳波形优化中也是众所周知的问题。在量子信息处理和光通信中，当优化纯量子态以用于玻色子量子信道中的通信时，即，找到最佳信道输入状态以使纯态信道保真度最大化时，出现相同的情况。由于这个问题的非凸性质的最佳和最大化本身一般很难找到，数值和分析。因此，可实现的性能上的上限是重要的，这将提供此贡献。基于E. Lieb，主要定理陈述了一个新的上界，该上界与波形无关，并且仅对于高斯权重和波形变得紧密。讨论这个特别重要的情况下，收紧最近的结果高斯量子保真度和相干态，将给出。另一个边界的情况下，散射是由相空间中的一些任意区域。
英文摘要：
In this article new bounds on weighted p-norms of ambiguity functions and Wigner functions are derived. Such norms occur frequently in several areas of physics and engineering. In pulse optimization for Weyl--Heisenberg signaling in wide-sense stationary uncorrelated scattering channels for example it is a key step to find the optimal waveforms for a given scattering statistics which is a problem also well known in radar and sonar waveform optimizations. The same situation arises in quantum information processing and optical communication when optimizing pure quantum states for communicating in bosonic quantum channels, i.e. find optimal channel input states maximizing the pure state channel fidelity. Due to the non-convex nature of this problem the optimum and the maximizers itself are in general difficult find, numerically and analytically. Therefore upper bounds on the achievable performance are important which will be provided by this contribution. Based on a result due to E. Lieb, the main theorem states a new upper bound which is independent of the waveforms and becomes tight only for Gaussian weights and waveforms. A discussion of this particular important case, which tighten recent results on Gaussian quantum fidelity and coherent states, will be given. Another bound is presented for the case where scattering is determined only by some arbitrary region in phase space.
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