All unitaries having operator Schmidt rank 2 are controlled unitaries

All Unitaries Having Operator Schmidt Rank 2 are Controlled Unitaries

Operator-Schmidt decompositions and the Fourier transform, with applications to the operator-Schmidt numbers of unitaries

The operator-Schmidt decomposition is useful in quantum information theory for quantifying the nonlocality of bipartite unitary operations. We construct a family of unitary operators on C ⊗ C whose operatorSchmidt decompositions are computed using the discrete Fourier transform. As a corollary, we produce unitaries on C ⊗ C with operatorSchmidt number S for every S ∈ {1, ..., 9}. This corollary...

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...

Unitaries in a Simple C-algebra of Tracial Rank One

Let A be a unital separable simple infinite dimensional C∗-algebra with tracial rank no more than one and with the tracial state space T (A) and let U(A) be the unitary group of A. Suppose that u ∈ U0(A), the connected component of U(A) containing the identity. We show that, for any ǫ > 0, there exists a selfadjoint element h ∈ As.a such that ‖u− exp(ih)‖ < ǫ. We also study the problem when u c...

C∗-pseudo-multiplicative unitaries

We introduce C∗-pseudo-multiplicative unitaries and (concrete) Hopf C∗-bimodules, which are C∗-algebraic variants of the pseudo-multiplicative unitaries on Hilbert spaces and the Hopf-von Neumann-bimodules studied by Enock, Lesieur, and Vallin [5, 6, 4, 10, 19, 20]. Moreover, we associate to every regular C∗-pseudo-multiplicative unitary two Hopf-C∗bimodules and discuss examples related to loca...