Applications of Quantum Operations with Indefinite Time Direction

Abstract

This thesis investigates higher-order quantum operations, particularly those exhibiting the property of indefinite time direction (ITD) and their applications to quantum metrology and capacity enhancement. We begin by surveying higher-order quantum theory and examining indefinite causal order (ICO). We focus on the quantum switch, a well-established higher- order operation that implements a causal superposition of two quantum channels. Then we introduce quantum operations with indefinite time directions and the quantum time flip supermap. We go on to introduce classical and quantum metrology and ICO-assisted metrology, which eventually leads to a brief tour of optimal metrology where we illustrate known semidefinite programming algorithms for optimizing metrological protocols. Our work makes two primary contributions: First, we introduce a novel framework for quan- tum metrology using indefinite time direction operations. We develop a universal protocol that achieves Heisenberg-limited precision for parameter estimations of unitaries, demon- strating its e!ectiveness in phase and axis estimation tasks. Notably, we prove that this quantum advantage does not require entanglement, establish the optimality of symmetric pure product states, and show that our approach remains e!ective even with simplified measurement strategies and under realistic noise conditions. Second, we explore the enhancement of classical communication capacity through the com- bined application of ICO and ITD operations. In order to accomplish this, we survey existing literature in quantum capacity and capacity enhancement with various supermaps, including quantum switch, SDPPs, half-switch and quantum time flip. Finally, by composing quantum switch and quantum time flip supermaps, we demonstrate capacity enhancements (Holevo bound) exceeding those achievable with either operation alone. We interpret the obtained results and discuss potential future research directions, including optimizing indefinite time- directed metrology and identifying justifications for the additional capacity enhancement, as well as figuring out optimal protocols when both flip and switch are available. Our results contribute to the growing understanding of higher-order quantum operations and their practical applications in quantum technologies, highlighting how these operations enable novel approaches to fundamental information processing tasks