This thesis investigates quantum information processing within the framework of higher-order operations, focusing on the transformation and discrimination of quantum channels, as well as on the role of indefinite causal order in enhancing quantum tasks. Following a preliminary chapter on the fundamentals of higher-order quantum operations, we present two chapters of original research.
The first research chapter addresses the transformation of unitary quantum operations, examining the problem of converting multiple calls of an arbitrary unitary operation into its inverse, transpose, and complex conjugate. We analyse the performance of parallel, sequential, and indefinite-causal-order strategies, revealing insights into the conditions under which sequential strategies can be parallelised, and conditions where indefinite-causal-order strategies are a useful resource. There we establish one-to-one relationships with parallel unitary inversion problems, to unitary estimation problems, and the seemingly unrelated task of port-based teleportation.
The second research chapter explores quantum channel discrimination when multiple calls are available. As with the unitary transformation analysis, we evaluate parallel, sequential, and indefinite-causal-order strategies. Traditionally, the field of quantum channel discrimination has focused on sets of unitary channels that form a group and are uniformly distributed, or discrimination tasks involving only two channels. Here, we combine a computer-assisted proof method with semidefinite programming techniques, arising from the framework of higher-order quantum operations, to broaden the scope of problems that are tractable. Using our methods, we identify and present various examples of ensembles of channels where sequential strategies offer an advantage when compared to parallel ones.
Finally, we provide an outlook on these different research directions and list recent contributions to other fields.