Mathematical aspects of quantum entanglement theory

点击翻译按钮获取中文摘要

In quantum information theory, one of the most useful resources is called "entanglement": the ability for particles to be correlated with each other in much stronger ways than those possible in classical mechanics. Entanglement is arguably the main ingredient in most interesting quantum algorithms and protocols, such as quantum teleportation and Shor's factoring algorithm, but many questions remain about its mathematical properties. The proposed program of research aims to answer two of these questions.******The first question that I will tackle is: "How can we construct all possible unextendible product bases in a given quantum system?" An unextendible product basis (UPB) is a set of quantum states that are themselves not entangled, but nonetheless have found numerous uses throughout entanglement theory. For example, UPBs can be used to construct "bound entangled" states: states that are entangled, but their entanglement is so weak that they are useless for many quantum information processing tasks (including quantum teleportation). Despite the many interesting properties of UPBs that have been found, their construction until now has largely been ad hoc. Researchers have found solitary examples in specific cases, but no general-purpose method for characterizing or finding UPBs in general yet exists. I have a computational method in mind that should make substantial progress on this problem, and perhaps solve it completely (up to certain dimensions, depending on how well the algorithm scales).******The second question that I will tackle is: "When is a quantum state absolutely separable?" Absolutely separable states are states that are not entangled, and furthermore it is possible to determine that they are not entangled just by looking at their eigenvalues. Equivalently, these are the states that cannot be made entangled no matter what quantum gate is applied to them, so it is desirable to avoid them in many quantum information processing tasks.******Very little is known about the set of absolutely separable states. Essentially the only results about this set are actually about a set called the "absolutely PPT" states, which was introduced as a coarse approximation of the absolutely separable states that is easier to analyze. However, it seems that these two sets might not be so different after all. I recently proved that these two sets exactly coincide in certain small dimensions, and with some graduate student co-authors we developed a general method for proving "closeness" of these two sets in a certain (somewhat technical) sense. The method that we developed is quite computational in nature, so I believe that it has a natural foothold that an undergraduate student researcher would be able to latch onto, and I believe that there are many new results that can be derived from it.**