October 6, 2022 — Electrical and computer engineering graduate student Haneul-Kim originally came to the University of Illinois to pursue a master’s degree in applied mathematics. While preparing for this degree, she ended up taking a course in Quantum Information Science taught by Associate Professor ECE Eric Chitambar. This turned into an independent study with him, and she eventually joined his group as a doctoral student in ECE. She used her math background to learn and apply tools and insights not offered by a standard physics or engineering background to problems of quantum information.
This is exactly what Kim did in his recent work with Chitambar published in Physical Review A. They obtained a new result on a well-established theoretical construct called a quantum cloning machine using semi-definite programming, a mathematical methodology that studies how to effectively optimize complex processes.
At first glance, quantum cloning machines pose a threat to communication protocols based on quantum mechanics’ famous no-cloning theorem, which states that no quantum mechanical operation can create an exact copy of a state. quantum. Instead of trying to produce exact copies of quantum states, they try to create approximate replicas that are close enough to fool communicating parties. Such processes are built using the methods of semi-definite programming: the inaccessible cloning operation is approached by an imperfect, but feasible process. However, early research efforts established strong fundamental limitations rendering these processes virtually ineffective.
In their paper “Process-optimized phase-covariant quantum cloning”, Kim and Chitambar note that a detail is missing from the discussion of cloning machines specializing in so-called phase-covariant states (a type of quantum state that is easily characterized and to manipulate) containing several levels. The standard unit of quantum information processing is the two-level qubit, which is widely used for its theoretical simplicity and relative ease of realization. However, multi-level processing units (called “qudits”) are theorized to offer more power and robustness, so it is desirable to know if these features come at the expense of security.
With no result, the researchers went ahead and found one. After using semi-definite programming methods to construct an optimal cloning machine suited to phase covariant states, they demonstrated that process-optimized fidelity, a measure of the quality of replicated states, decreases as the number levels in the processing unit increases. This result is consistent with those of more general cloning machines, confirming that they will not pose a serious threat even if multi-level processing units are adopted.
Kim and Chitambar are attracted to these processes because their functioning depends on symmetry; they hope to apply the mathematical tools they have learned to study newer quantum information problems that also contain symmetry. This symmetry is crucial for devices that approximately copy quantum states may not be obvious. But, as Kim explained, symmetry is present in many physical systems if you know how to look. In this case, a proposed cloning machine must perform equally well on a wide variety of input states. In other words, all input states must be treated equally, so the device must operate symmetrically.
Kim views this article, her first, as an important exposure to the tools she and the Chitambar group will need in the future. She and the group have their eyes on a more current issue with a high degree of symmetry: teleport attacks on quantum localization protocols. They will collaborate with Felix Leditzky, Assistant Professor of Mathematics and expert in the mathematical foundations of quantum information science, to put their foundations to good use and tackle the next step.
Source: Michael O’Boyle, IQIST, University of Illinois