On the distance from a matrix to nilpotents

We prove that the distance from an $n\times n$ complex matrix $M$ to the set of nilpotents is at least $\frac{1}{2}\sec\frac{\pi}{n+2}$ if there is a nonzero projection $P$ such that $PMP=M$ and $M^*M\geq P$. In the particular case where $M$ equals $P$, this verifies a conjecture by G.W. MacDonald in 1995. We also confirm a related conjecture in D.A. Herrero's book.Comment: 4 page

On the shape of correlation matrices for unitaries

For a positive integer $n$, we study the collection $\mathcal{F}_{\mathrm{fin}}(n)$ formed of all $n\times n$ matrices whose entries $a_{ij}$, $1\leq i,j\leq n$, can be written as $a_{ij}=\tau(U_j^*U_i)$ for some $n$-tuple $U_1, U_2, \ldots, U_n$ of unitaries in a finite-dimensional von Neumann algebra $\mathcal{M}$ with tracial state $\tau$. We show that $\mathcal{F}_{\mathrm{fin}}(n)$ is not closed for every $n\geq 8$. This improves a result by Musat and R{\o}rdam which states the same for $n\geq 11$.Comment: 4 page

The structure of maps on the space of all quantum pure states that preserve a fixed quantum angle

Abstract Let $H$ be a Hilbert space and $P(H)$ be the projective space of all quantum pure states. Wigner’s theorem states that every bijection $\phi \colon P(H)\to P(H)$ that preserves the quantum angle between pure states is automatically induced by either a unitary or an antiunitary operator $U\colon H\to H$. Uhlhorn’s theorem generalizes this result for bijective maps $\phi$ that are only assumed to preserve the quantum angle $\frac{\pi }{2}$ (orthogonality) in both directions. Recently, two papers, written by Li–Plevnik–Šemrl and Gehér, solved the corresponding structural problem for bijections that preserve only one fixed quantum angle $\alpha$ in both directions, provided that 0 < \alpha \leq \frac{\pi }{4} holds. In this paper we solve the remaining structural problem for quantum angles $\alpha$ that satisfy \frac{\pi }{4} < \alpha < \frac{\pi }{2}, hence complete a programme started by Uhlhorn. In particular, it turns out that these maps are always induced by unitary or antiunitary operators, however, our assumption is much weaker than Wigner’s