Abstract
Some time ago, Steven Weinberg wrote an article for the New York Review of Books with the title, “Symmetry: A ’Key to Nature’s Secrets’.” So too, I would like to say of quantum information: Only by identifying Hilbert space's most stringent and hard-to-attain symmetries will we be able to unlock quantum information's deepest secrets and greatest potential. In this talk, I introduce the “symmetric informationally complete” (SIC) sets of quantum states as a candidate for that structure. By their aid, one can rewrite quantum states so that they become simply probability distributions, unitary transformations so that they become doubly stochastic matrices, and the Born rule so that it becomes a rather simple variant of the classical law of total probability. These representations hold the potential for entirely new ways of analyzing quantum communication channels and algorithms. Surprisingly however, despite the way they can be used to make quantum theory look formally close to classical information theory, there is also a sense in which the SIC states are as far from classical as possible: For instance, by some measures these states are as sensitive to quantum eavesdropping as any alphabet of quantum states can be. Time permitting, I will show off some of the latest things known about the SICs.
© 2013 Optical Society of America
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