Studies of Symmetric Informationally-Complete Positive Operator Valued Measures for Quantum Information and Quantum Foundations

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There has never been any theory more successful than quantum mechanics: for over a hundred years in its history no experiment has ever contradicted the prediction of the theory up to the best experimental precision. Applications of quantum mechanics can be seen everywhere in the modern world, from tiny lasers in DVD players and sensors in digital cameras, to MRI scanners and nuclear power plants. Ironically, till today the mathematical structure lying at the heart of the theory is not well understood, and there are numerous conceptual issues in physical interpretations of quantum mechanics. We would like to propose the study of a special hypothetical mathematical object called a symmetric informationally-complete set of quantum states, or a SIC in short, whose symmetric properties could potentially provide important hints about the mathematical structure of quantum mechanics. SICs play a crucial role in the Quantum Bayesianism program, a newly developed interpretation aiming to understand quantum mechanics from the full machinery of probability theory. SICs have also been found useful in various areas in quantum information, for example they help provide more robust measurements and more secure communication. Being able to construct SICs in general, or even just to prove their existence, would bring us important new understandings about quantum mechanics as well as new applications in quantum information, and no one can yet know what more quantum mechanics can bring to our lives once it is fully understood.