String Theory Journal Club

We will discuss variants of the singular value decomposition that are invariant under diagonal rescalings (changes of units) in a chosen basis, and show how they lead to new, scale-invariant notions of entanglement. Using these unit-invariant singular values, we define entropy-like measures for reduced density matrices, transition matrices, and other (possibly non-Hermitian) operators. We will illustrate these constructions in biorthogonal quantum mechanics as well as with link complement states in Chern-Simons theory, where in the latter framework, connected sums and framings of knots act naturally as diagonal and unitary transformations, respectively. Finally, we will describe random-matrix aspects, including quarter-circle and stretched quarter-circle laws for the distributions of the new singular values in Ginibre and Wigner Gaussian ensembles.