Higher-Order Quantum Operations: Foundations and Applications
The understanding of the dynamics of quantum systems is an important pillar of quantum theory and a key point for many applications to quantum technologies. Even if the laws of quantum mechanics allow us, in principle, to analyse the dynamics of quantum states for any particular time, several important applications in quantum information, such as quantum computing and quantum communications, do not require a general dynamics approach. Instead, one may focus on quantum operations, devices which can transform a quantum input state into a quantum output state. Quantum operations provide a description of, for instance, quantum communication channels between distant parties or quantum gate elements in a quantum circuit.
Although not stated explicitly in the laws of quantum mechanics, quantum operations themselves can also be transformed.
Transformations between quantum operations are referred to as higher-order quantum operations and find applications in manipulating and controlling quantum systems and transforming quantum circuits. Higher-order operations have some particularities when compared to 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 predict higher-order transformations with an indefinite causal order between the use of the input operations. This research project seeks better understanding of higher-order operations with and without a definite causal order. From a foundational perspective, the study of indefinite causal order has thus far brought us relevant insights on quantum theory itself, motivating further investigation and promising impactful applications to channel discrimination tasks, quantum metrology, and quantum circuit design.