Detection and Quantification of Quantum Resources

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In quantum information theory, there are several different notions of "usefulness" of quantum states, depending on what is to be done with them. For example, sometimes the states that are "useful" are those that are entangled (i.e., have non-local properties not present in classical mechanics), and sometimes the states that are "useful" are those that are coherent (i.e., in a superposition). These different notions of "usefulness" give rise to what is called a quantum resource theory: a collection of distinguished quantum states that we think of as "useful", and some way of quantifying how useful a quantum state is. My research program aims to develop a better understanding of the various resources that are of interest in quantum information theory, and also explore how they are related to each other. I have numerous specific projects within this overall research program that I plan to explore over the next few years: 1. An entanglement witness is a mathematical device that can be used to detect entanglement in a quantum system. We often do not have complete information about a quantum state or an entanglement witness, so we ask questions like "what can we say about a state if we only know its eigenvalues?" I intend to explore this question of what we can say about entanglement if we only know a witness's eigenvalues or a state's eigenvalues. 2. Similarly, can we determine whether or not a state is useful in the resource theory of coherence if we only know its eigenvalues? I (with a few other researchers) have a conjecture that would completely solve this problem. We also have a few ideas for how we could prove the conjecture, but each of them is quite involved and will require a significant amount of work. 3. Numerous connections have recently been made between quantum resource theories and the set of "completely positive" matrices, which have been studied for decades by researchers in the fields of matrix analysis and convex optimization. However, there has not been any thorough investigation into how the various known properties of completely positive matrices can be used to tell us, for example, if a quantum state is separable or entangled. I will tackle this question, and use completely positive matrices to develop some new tests for separability. 4. If we have a collection of infinitely many quantum states, how can we be sure that all of them are entangled? In general, this problem is very difficult, but I (with a few other researchers) have developed a method that seems to work quite well in practice. We will explore what can be done with this method, and pin down exactly when it works well (and not so well).