Information geometrical study of quantum channel estimation problems

Vishal Katariya, Mark M. Wilde

Kimikazu Kato

In quantum information theory, a geometric approach, known as "quantum information geometry," has been considered as a powerful method. In this thesis, we give a computational geometric interpretation to the geometric structure of a quantum system. Especially we introduce the concept of the Voronoi diagram and the smallest enclosing ball problem to the space of quantum states. With those tools in computational geometry, we analyze the adjacency structure of a point set in the quantum state space. Additionally, as an application, we show an effective method to compute the capacity of a quantum channel.

Vishal Katariya, Mark M. Wilde

Quantum channel estimation and discrimination are fundamentally related information processing tasks of interest in quantum information science. In this paper, we analyze these tasks by employing the right logarithmic derivative Fisher information and the geometric Rényi relative entropy, respectively, and we also identify connections between these distinguishability measures. A key result of our paper is that a chain-rule property holds for the right logarithmic derivative Fisher information and the geometric Rényi relative entropy for the interval α∈ (0 , 1) of the Rényi parameter α. In channel estimation, these results imply a condition for the unattainability of Heisenberg scaling, while in channel discrimination, they lead to improved bounds on error rates in the Chernoff and Hoeffding error exponent settings. More generally, we introduce the amortized quantum Fisher information as a conceptual framework for analyzing general sequential protocols that estimate a parameter encoded in a quantum channel. We then use this framework, beyond the aforementioned application, to show that Heisenberg scaling is not possible when a parameter is encoded in a classical–quantum channel. We then identify a number of other conceptual and technical connections between the tasks of estimation and discrimination and the distinguishability measures involved in analyzing each. As part of this work, we present a detailed overview of the geometric Rényi relative entropy of quantum states and channels, as well as its properties, which may be of independent interest.

Demetris P. K. Ghikas

Masahito Hayashi