Five Starter Pieces: Quantum Information Science via Semi-definite Programs

Abstract

As the title indicates, this manuscript presents a brief, self-contained introduction to five fundamental problems in Quantum Information Science (QIS) that are especially well-suited to be formulated as Semi-definite Programs (SDP). We have in mind two audiences. The primary audience comprises of Operations Research (and Computer Science) graduate students who have familiarity with SDPs, but have found it daunting to become even minimally conversant with pre-requisites of QIS. The second audience consists of Physicists (and Electrical Engineers) already knowledgeable with modeling of QIS via SDP but interested in computational tools that are applicable more generally. For both audiences, we strive for rapid access to the unfamiliar material. For the first, we provide just enough required background material (from Quantum Mechanics, treated via matrices, and mapping them in Dirac notation) and simultaneously for the second audience we recreate, computationally in Jupyter notebooks, known closedform solutions. We hope you will enjoy this little manuscript and gain understanding of the marvelous connection between SDP and QIS by self-study, or as a short seminar course. Ultimately, we hope this disciplinary outreach will fuel advances in QIS through their fruitful study via SDPs.

Synergies Between Operations Research and Quantum Information Science

“…In fact the defining properties of a state ρ are that it has trace equal to one and is positive semidefinite. For more details, a self-contained account of five accessible applications of semidefinite programming to properties of quantum states and channels appears in [ST21].…”