YQI Colloquium - Ivan Deutsch, Center for Quantum Information and Control, “The Power of Being Positive”

Diagnosing the performance of quantum information processors is a grand challenge. This is because quantum information is stored in a manner quite different from classical information. We can’t just look under the hood and measure everything we want about the device in one fell swoop. Quantum measurements disturb the system and we can’t learn everything about noncommuting observables at the same time. This becomes particularly challenging for systems that encode information in a large dimensional Hilbert space, e.g. d=2^N, for N qubits. Thus, we must devise special methods for characterizing, verifying, and validating the device performance. One tool we can borrow from classical estimation theory is “compressed sensing,” which employs prior information about the structure of a signal to greatly reduce the resources we need to reconstruct it. In this talk, I will show how compressed sensing follows almost automatically in estimation of quantum states and processes due to fact that the physics imposes a constraint — they are described by positive matrices. I will describe the fundamental theory of compressed-sensing tomography and its implementation in an experimental testbed consisting of hyperfine spins in ultracold cesium atoms, defined by a 16-dimensional Hilbert space. Experimental work is has been carried out by my collaborator, Prof. Poul Jessen, University of Arizona.