Oct. 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 working towards that degree, she wound up taking a class on quantum information science taught by ECE associate professor Eric Chitambar. This turned into an independent study with him, and she eventually joined his group as a PhD candidate in ECE. She has been using her mathematics background to learn and apply tools and perspectives not offered by a standard physics or engineering training to problems in quantum information.

Kim did just this in her 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 semidefinite programming, a mathematical methodology that studies how to efficiently optimize complicated processes.

On the surface, quantum cloning machines pose a threat to communications protocols based on the famous no-cloning theorem of quantum mechanics, which states that no quantum mechanical operation can create an exact duplicate of a quantum state. Instead of trying to produce exact copies of quantum states, they try to create approximate replicas close enough to fool the communicating parties. Such processes are constructed using the methods of semidefinite programming: the unattainable cloning operation is approximated by an imperfect, but realizable process. However, early research efforts established strong, fundamental limits rendering these processes practically ineffective.

In their article “Process-optimized phase-covariant quantum cloning,” Kim and Chitambar note that there is a missing detail in the discussion of cloning machines specialized to so-called phase covariant states (a type of quantum state easy to characterize and manipulate) that contain multiple levels. The standard quantum information processing unit is the two-level qubit, which is widely used for its theoretical simplicity and comparative 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 cost of security.

In the absence of a result, the researchers went ahead and found one. After using methods from semidefinite programming to construct an optimal cloning machine tuned to phase covariant states, they demonstrated that the process-optimized fidelity, a measure of the quality of replicated states, decreases as the number of levels in the processing unit increases. This result is consistent with those for more general cloning machines, confirming they will not pose a serious threat even if multi-level processing units are adopted.

Kim and Chitambar are drawn to these processes because their operation depends on symmetry; they hope to apply the mathematical tools they learned to study more recent problems in quantum information that also contain symmetry. That symmetry is crucial to devices approximately copying 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 work equally well on a large variety of input states. Put differently, all input states must be treated the same way, so the device must operate in a symmetric manner.

Kim sees this paper, her first, as an important exposure to the tools she and Chitambar’s group will need in the future. She and the group have their eyes on a more current problem with a high degree of symmetry: teleportation attacks on quantum location protocols. They will collaborate with Felix Leditzky, an assistant professor of mathematics and an expert in the mathematical underpinnings of quantum information science, to put their groundwork to use and tackle the next thing.

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Source: Michael O’Boyle, IQUIST, University of Illinois