Application of random matrices and measure concentration to quantum communication theory

2019 Fiscal Year Final Research Report

• Project/Area Number: 16K00005
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Theory of informatics
• Research Institution: Yamagata University
• Principal Investigator: Fukuda Motohisa 山形大学, 理学部, 准教授 (70771161)
• Project Period (FY): 2016-04-01 – 2020-03-31
• Keywords: 量子情報 / ランダム行列 / 自由確率 / 測度集中 / 量子通信 / 量子ガウス状態

Outline of Final Research Achievements

Non-additivity of quantum channels prohibits us from extending Shannon's channel coding theorem to quantum communication. This non-additivite property was proved by M. Hastings in 2009 by using randomly generated quantum channels. In this project, we used random matrices, free probability, and measure concentration to analyse such quantum channels. Our results include finding a new class of non-additive quantum channels. Also, we developed a computer program to calculate abstractly average of tensor networks with random unitary matrices, and then applied it to yield some results on random quantum Gaussian states, aligned with maximal entropy principle, which plays a key role in statistical mechanics. Besides, we showed mathematical connection between our problems of quantum channels to those of meandric systems, and gave partial results on formulas of generating functions of meandric polynomials.

Free Research Field

量子情報理論

Academic Significance and Societal Importance of the Research Achievements

量子通信路の通信路容量を計算するにはエンタングルメントの影響を評価する必要があるが、その影響は限定的であるという主張を支持する数学的な結果を得た。また、ランダムなユニタル行列を含むテンソルネットワークの平均を計算するコンピュータープログラムを作成し、無償で公開した。そのプログラムを用いて量子光学を記述するボソン系ガウス状態の最大エントロピーの原理に沿う結果を証明し、量子統計学への貢献も行った。さらに、ポリマーの折り曲げ問題に関連するメアンダー多項式の母関数の定式化への部分的結果を得た。