Efficient conversion of quantum information resources
This proposal explores the power of quantum information in two respects. The first is the technological power of quantum information in a communication infrastructure, and the second is its descriptive power in many-particle quantum systems. My point of departure is to view quantum information as a resource that can be processed and converted.
In quantum communication, a famous resource conversion is provided by the quantum teleportation protocol, which allows us to send one quantum bit (1 qubit) through the transmission of two classical bits (2 cbits) and the use of one entangled pair of quantum bits (1 ebit):
  1 ebit + 2 cbits > 1 qubit.
Casting quantum protocols in such resource inequalities has proven useful, since the algebraic manipulation of inequalities results in new protocols, but this approach has hitherto largely been limited to point-to-point communication. It is the first goal of this project to overcome this limitation and characterise resource conversion in larger quantum networks. This will result in more efficient communication protocols that will have an impact on the use and design of quantum communication networks, which are currently being built around the globe.
A quantum network involving distant communicating labs is mirrored at the small scale by a set of interacting quantum particles. The quantum state arising from pairwise interactions can be strongly entangled, with an underlying entanglement structure given by a graph with entangled pairs along the edges. There is a surprising and close connection between such entanglement structures and tensor research in the context of algebraic complexity theory. The second goal of the project is to exploit this connection and characterise the resource conversion of entanglement structures. The research will lead to more efficient tensor network representations of many-particle quantum states, and to progress on the computational complexity of matrix multiplication, a long-standing unsolved problem.