RMTQIT

Quantum Information Theory(QIT) is one of the most dynamic and exciting areas of research in science and technology. It overlaps many different fields of physics and mathematics such as Quantum Mechanics, Operator Theory, and Probability Theory. One of the most interesting branches of Probability Theory is Random Matrix Theory (RMT) and lately it has been proven that many ideas originating from RMT can be exploited for given reliable answers to the open question of QIT. Our project focuses on a systematic exploration of theoretical questions in QIT using tools developed by the mathematicians and physicists working in Random Matrix Theory. The principal goal of our three-year project is to systematically explore and provide answers for questions in QIT about random quantum states and random quantum channels. Such problems have attracted the attention lately in a very naturally connection to fundamental issues of QIT theory, such as entanglement theory and classical (or quantum) capacities for channels. Since explicit answers for non-trivial dimensions are hard (if not impossible) to give, physicists and computer scientists turned to typical states and channels, that is random states or channels. These examples proved to be very rich and they were the key ingredient to some very spectacular results, such as the non-additivity of the Holevo quantity for quantum channels or thresholds for the entanglement of typical mixed quantum states. Therefore, techniques coming from Random Matrix Theory can be considered natural tools to tackle the above problems. Another interesting example of this is the question of the additivity of the minimum output entropy (MOE) for a quantum channel. Quantum information theorists have shown that this question was equivalent to understanding the set of singular values of linear maps sampled from vector spaces of rectangular matrices. Hastings’ answer to this question, in the negative, does not shed light on how big the non-additivity can be or on what are the minimal dimensions in which this extraordinary phenomenon can occur. We believe that this question, natural from the perspective of RMT, can have a more precise answer and we are working towards a better understanding of the respective sets of singular values. Another problem of interest is to find the image of a finite set of states through random quantum channel. There are known attempts to solve this kind of problem using techniques coming from RMT. The image of classes of bipartite quantum states under a tensor product of random quantum channels have been derived lately. Interesting properties are found such as that the Bell state gives asymptotically, among some large classes of input states, the output with the least entropy. Such questions are relevant also in the framework of additive problems. Indeed, it does not prove that the Bell state gives the largest violation of additivity, but it stands as a solid mathematical evidence towards the fact that the physically intuitive choice of the Bell state is indeed close to being optimal. Our approach to these open question is to use techniques coming from RMT; it is an ambitious task, demanding for various mathematical skills that we believe that our team possesses in its ensemble.