Metrics on unitary matrices and their application to quantifying the degree of non-commutativity between unitary matrices

Quantum information processing is the study of methods and efficiency in storage, manipulation and conversion of information represented by quantum states. Many quantum information theoretic concepts are closely related to geometry. For instance, trace distance and fidelity, which come out of the study of distinguishability between quantum states, are closely linked with Bures and Fubini-Study metrics. (See, for example, Ref. [1] for an overview.) A few quantum codes can be constructed by algebraic-geometric means [2]. And finding the optimal quantum circuit can be regarded as the problem of finding the shortest path between two points in a certain curved geometry [3]. Recently, a few metrics on unitary operators with quantum information applications were found. For example, Johnston and Kribs introduced the kth operator norm of an operator acting on a bipartite system by considering the action of the operator on bipartite states with Schmidt rank less than or equal to k. The kth operator norm can be used to study bound entanglement of Werner states as well as to construct several new entanglements witnesses [4, 5]. Rastegin studied the partitioned trace distance which shares similar properties with the standard trace distance [6]. The partitioned trace distance shines new light on exponential indistinguishability and hence can be used to investigate certain quantum cryptographic problems [7]. Interestingly, both the kth operator norm and the partitioned trace distance are related to the Ky Fan norm [4, 6]. By asking the question about the minimum resources needed to perform a unitary transformation (a question of quantum information processing favor), I report families of related