Frame Phase-Retrievability and Applications to Quantum Information

Frames provide desirable mathematical representations for signals and information, and consequently play an important role in developing approaches to many challenging questions in science and in engineering applications. Frames have natural connections with the mathematical theory of quantum communications. For example, phase-retrievability of a frame is an important property that allows the recovery of a signal from the magnitudes of its frame coefficient measurements.

This appears in many applications including speech recognition, x-ray crystallography and electron microscopy. In quantum information theory, this is a necessary property for a quantum system to have to distinguish pure states from their quantum measurements. This project addresses several fundamental issues in the area.

The principal investigator (PI) will investigate frame applications to problems involving quantum communication capability and information transmission security. By bringing a new circle of ideas to attack these practical problems, this investigation will advance scientific understanding in quantum information theory as well as in applied functional/harmonic analysis.

Additionally, this project will promote teaching, training, and learning as several graduate and undergraduate students will be directly involved in this research project.

One direction of this investigation is to establish quantified measurements of (mostly, operator-valued) frame phase-retrievability, and then use them to tackle problems of designing (or characterizing) quantum communication channels with prescribed levels of zero-error communication capacity and information transmission security. Special attention will be given to the structured channels, which are usually induced by different kinds of representation frames.

Here, the interplay between operator-valued frames and projective group representations will play a key role. A second direction of this project is in the area of signal/state recovering (or channel detection) from a source of unidentified channels. The PI will establish its theoretical connections with the theory of disjoint frames and quantum channels.

A key ingredient is to use disjoint frames as building blocks to produce a large class of candidates, where any subclass of this set can be used as a source of unidentified channels for a target set of signals or states in the quantum setting. Such an approach requires developing a disjoint frame theory (or multiplexing) for bounded linear maps on von Neumann algebras.

A third direction involves quantum measures. Dilation problems for quantum measures are not only in the scope of aforementioned areas of investigation but also in line with the PI’s long-term goal of establishing a general dilation theory for operator-valued measures in both commutative and non-commutative settings. Such a general dilation theory may lead to a possible classification theory for quantum measures as well as for their associated quantum channels.

The main objective is to establish a dilation theory for quantum measures that can tell us which, when, and how certain important information of a quantum measure can be preserved through dilations.

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.