Symmetric Informationally Complete Measurements and Quantum Computation

University Of Massachusetts Boston, Dorchester MA

Investigators

Abstract

So far, most of quantum information science has been developed in terms of the textbook presentation of quantum theory, which chiefly amounts to the mathematical language of vectors and matrices of complex numbers. But this language may hide as much as it reveals. This is hinted at by certain quantum foundational approaches such as Quantum Bayesianism where finding alternative ways to represent the theory in terms of probabilities (nonnegative real numbers) does most of the conceptual heavy lifting. This is because these representations live much closer to the goal of analyzing the distinction between classical and quantum in terms of decision theory. This suggests that representations of this nature may also be of critical importance to developing quantum information technologies, for instance by providing benchmarks for how to best simulate quantum systems using conventional computational resources and certifying quantum supremacy. Developing an efficient language for quantum information processing in these terms is the goal of this project, which will have influence across disciplines thanks to the connection it makes between quantum mechanics and number theory, an area of mathematics foreign to the physics curriculum. In recent years much research has focused on the most symmetric possible of such representations—those based on “symmetric informationally complete quantum measurements” or SICs—for their simplifying power and optimality in a surprising number of applications: From optimal quantum-state tomography, to entanglement detectors, novel key distribution schemes, components in device-independent random number generation, dimension witnessing, and more. However, the promise of these representations comes with two catches. First, it is not known whether the conditions for SIC existence can always be satisfied for qudit systems (though SICs are currently known to exist in at least 264 dimensions and strongly believed to exist in all others). Second, when the conditions for existence can be satisfied, except for the global symmetry defining them, the solutions always appear monstrously complex. In this project, the group plans to remedy the latter matter, if not the former, by exploiting recently discovered connections between SICs and algebraic number theory, particularly Hilbert’s 12th problem. Essentially what is called for is the development of a tool pack of special (transcendental) functions by which to make the representation more manageable. With the tool pack in hand, the group will reexamine a number of phenomena from this more efficient perspective and build an open-source code base to facilitate applied research more broadly. 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.