Quantum Correlations in Information Theory, Their Interplay, and Their Limitations

This project will advance mathematical knowledge of quantum information theory. This work is in line with Quantum Leap, one of the NSF's 10 Big Ideas. One of the goals of the Quantum Leap Big Idea is to develop next-generation quantum technologies.

But any technological advancement has usually been long-preceded through theoretical mathematical progress. This project deals with questions in quantum resource theory, in particular a theory of entanglement. Quantum devices use quantum systems to encode and process information.

These systems make it possible to harness the effects of purely quantum phenomena such as superposition (the ability of being in "two places" at once) and entanglement (strong connections between two parts of the system) to drastically increase the security and the speed of computational devices. Because of entanglement two spatially separated particles can affect each other instantaneously, a phenomenon which was initially considered to be paradoxical, even by Einstein.

One of the best-known protocols - quantum teleportation - relies on entanglement to destroy a quantum state in one place and perfectly recreate it in another distant location. While entanglement became one of the most valuable resources in quantum theory, many attributes of its behavior remain unknown. The broader impact of this award is to promote progress, diversity, and access to science on all levels of society and to broaden participation from underrepresented communities in STEM.

The major broader impact goals are: 1) to enhance training of graduate students in quantum information at the University of Houston; 2) to improve educational opportunities in STEM for students in local middle-schools in the economically challenged neighborhoods; 3) to increase public education in STEM and quantum information science.

Entanglement is only one quantifier of "quantumness" in a system. Other measures include coherence, discord, and mutual information. Entanglement theory has gained its strong mathematical footing in the 1990's, but a closely related coherence theory started only recently.

This project will establish proper mathematical limits on and relations between these measures, unify them, and classify them. It is well-known that under certain parameters, entanglement and coherence are dual to each other, in a sense that coherence can be converted to entanglement via incoherent operations, and vice versa in the asymptotic limit.

This project will investigate this relation for different entanglement and coherence measures. Moreover, the Principal Investigator will develop and investigate properties of newly proposed correlation measures. A closely related question is the distinguishability between two quantum states.

How well we can distinguish different physical states determines how much information we can encode into a certain system and how quickly we can manipulate it. One of the main problems that this project will focus on is the continuity of these quantifiers, which is a natural property of any quantum system. In other words, if the states are close when measured by one of these distances, how close are they when measured by another?

Or how well can these states be distinguished? Or what is the structure of states that are close to each other?

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.