The Church of the Symmetric Subspace

The symmetric subpace has many applications in quantum information theory. This review article begins by explaining key background facts about the symmetric subspace from a quantum information perspective. Then we review, and in some places extend, work of Werner and Chiribella that connects the symmetric subspace to state estimation, optimal cloning, the de Finetti theorem and other topics. In the third and final section, we discuss how the symmetric subspace can yield concentration-of-measure results via the calculation of higher moments of random quantum states. There are no new results in this article, but only some new proofs of existing results, such as a variant of the exponential de Finetti theorem. The purpose of the article is (a) pedagogical, and (b) to collect in one place many, if not all, of the quantum information applications of the symmetric subspace.