OMC for scalar multiples of unitaries

Quadratic Form Expansions for Unitaries

We introduce techniques to analyze unitary operations in terms of quadratic form expansions, a form similar to a sum over paths in the computational basis when the phase contributed by each path is described by a quadratic form over R. We show how to relate such a form to an entangled resource akin to that of the one-way measurement model of quantum computing. Using this, we describe various co...

All Unitaries Having Operator Schmidt Rank 2 are Controlled Unitaries

Almost Commuting Unitaries with Spectral Gap Are near Commuting Unitaries

Let Mn be the collection of n×n complex matrices equipped with operator norm. Suppose U, V ∈ Mn are two unitary matrices, each possessing a gap larger than ∆ in their spectrum, which satisfy ‖UV −V U‖ ≤ ǫ. Then it is shown that there are two unitary operators X and Y satisfying XY −Y X = 0 and ‖U−X‖+‖V −Y ‖ ≤ E(∆/ǫ) “ ǫ ∆2 ” 1 6 , where E(x) is a function growing slower than x 1 k for any posit...

Morphisms of Multiplicative Unitaries

In this paper, we will give a natural definition for morphisms between multiplicative unitaries. We will then discuss some equivalences of this definition and some interesting properties of them. Moreover, we will define normal sub-multiplicative unitaries for multiplicative unitaries of discrete type and prove an imprimitivity type theorem for discrete multiplicative unitaries.

Proving Admissibility of Irreducible Unitaries

• Higher commutators • Godement’s principle: bounding dimensions of irreducibles • The von Neumann (strong) density theorem • Lie’s Theorem on irreducible finite-dimensional representations of connected solvable groups • Separating families of irreducible finite-dimensional representations • Subrepresentation Theorem for finite-dimensional representations of reductive linear real Lie groups • T...