Higher-Order Quantum Operations with Known States – HOQO-KS

Quantum operations form an important pillar of quantum theory and a key point for many applications to quantum technologies. Traditionally, quantum operations were only viewed as devices to transform quantum states, such as quantum communication channels between distant parties or quantum gate elements in a quantum circuit. However, quantum operations themselves may also be submitted to transformations and play the role of a state by a Higher-Order Quantum Operation (HOQO).

The HOQO approach is useful for manipulating and controlling quantum systems, designing quantum circuits, and tasks involving quantum gates such as channel discrimination. Additionally, higher-Order Quantum Operations have some particularities when compared to its lower-order counterpart, i.e., standard operations on quantum states. Differently from states, operations have a clear notion of input and output, hence, when considering transformations between two or more operations, the concept of causal order emerges naturally. Interestingly, the postulates of quantum mechanics do not explicitly forbid the existence of higher-order transformations which do not respect any definite causal order between the use of the input operations. This leaves room for the existence of quantum process with indefinite causality and for a fruitful analysis on how causality should be understood in quantum theory.