arXiv subjects
Publications and source records attributed to Edward Poon.
We determine the structure of linear maps on complex (real) square matrices sending unitary (orthogonal) matrices to multiples of unitary (orthogonal) matrices. The result is used to determine the linear preservers of matrix pairs satisfying the extremal norm properties $\|AB\| = \|A\| \|B\|$, $\|A^*B\| = \|A\| \|B\|$, or $\|AB^*\| = \|A\| \|B\|$, for the spectral norm $\|\cdot\|$.
Research on the projective measurement of gravitons increasingly supports Dysons conclusions that the detection of single gravitons is not physically possible. It is therefore prudent to consider alternative signatures of non-classicality in gravitational wave detections to determine if gravity is quantized. Coincident multiple detector operations make it possible to consider the bipartite measurement-induced entanglement, in the detection process, as a signature of non-classicality. By developing a model of measurement-induced entanglement, based on a fixed number of gravitons for the bipartite system, we demonstrate that the entanglement entropy is on the order of a few percent of the mean number of gravitons interacting with the detectors. The bipartite measurement-induced entanglement is part of the detection process, which avoids the challenges associated with developing signatures of production-induced entanglement, due to the extremely low gravitational wave detector efficiencies. The calculation of normalized measurement-induced entanglement entropy demonstrates the potential of developing physically meaningful signatures of non-classicality based on bipartite detections of gravitational radiation. This result is in stark contrast to the discouraging calculations based on single-point detections.
We study the problem of calculating noncommutative distances on graphs, using techniques from linear algebra, specifically, Birkhoff-James orthogonality. A complete characterization of the solutions is obtained in the case when the underlying graph is a path.
Let $1 \leq k < n$ be integers. Two $n \times n$ matrices $A$ and $B$ form a parallel pair with respect to the $k$-numerical radius $w_k$ if $w_k(A + \mu B) = w_k(A) + w_k(B)$ for some scalar $\mu$ with $|\mu| = 1$; they form a TEA (triangle equality attaining) pair if the preceding equation holds for $\mu = 1$. We classify linear bijections on $\mathbb M_n$ and on $\mathbb H_n$ which preserve parallel pairs or TEA pairs. Such preservers are scalar multiples of $w_k$-isometries, except for some exceptional maps on $\mathbb H_n$ when $n=2k$.
Two bounded linear operators $A$ and $B$ are parallel with respect to a norm $\|\cdot\|$ if $\|A+\mu B\| = \|A\| + \|B\|$ for some scalar $\mu$ with $|\mu| = 1$. Characterization is obtained for bijective linear maps sending parallel bounded linear operators to parallel bounded linear operators with respect to the Ky-Fan $k$-norms.
We characterize under what conditions $n\times n$ Hermitian matrices $A_1$ and $A_2$ have the property that the spectrum of $\cos t A_1 + \sin t A_2$ is independent of $t$ (thus, the trigonometric pencil $\cos t A_1 + \sin t A_2$ is isospectral). One of the characterizations requires the first $\lceil \frac{n}{2} \rceil$ higher rank numerical ranges of the matrix $A_1+iA_2$ to be circular disks with center 0. Finding the unitary similarity between $\cos t A_1 + \sin t A_2$ and, say, $A_1$ involves finding a solution to Lax's equation.
Which matrix norms induce proper measures for quantifying quantum coherence? We study this problem for two important classes of norms and show that (i) coherence measures cannot be induced by any unitary similarity invariant norm, and (ii) the $\ell_{q,p}$-norm induces a coherence measure if and only if $q=1$ and $1 \leq p \leq 2$, thus giving a new class of coherence measures with simple closed forms that are easy to compute. These results extend and unify previously known facts about norm-induced coherence measures, and lead to a broader framework for understanding what functionals can be coherence measures.
This paper is the result of a research project completed in the context of the first author's Undergraduate Student Research Award from the Natural Sciences and Engineering Research Council of Canada (NSERC). We prove a nesting phenomenon for twisted Frobenius extensions. Namely, suppose $R \subseteq B \subseteq A$ are graded superrings such that $A$ and $B$ are both twisted Frobenius extensions of $R$, $R$ is contained in the center of $A$, and $A$ is projective over $B$. Our main result is that, under these assumptions, $A$ is a twisted Frobenius extension of $B$. This generalizes a result of Pike and the second author, which considered the case where $R$ is a field.
Alice and Bob are given an unknown initial state chosen from a set of pure quantum states. Their task is to transform the initial state to a corresponding final pure state using local operations only. We prove necessary and sufficient conditions on the existence of such a transformation. We also provide efficient algorithms that can quickly rule out the possibility of transforming a set of initial states to a set of final states.
Given two sets of quantum states {A_1, ..., A_k} and {B_1, ..., B_k}, represented as sets of density matrices, necessary and sufficient conditions are obtained for the existence of a physical transformation T, represented as a trace-preserving completely positive map, such that T(A_i) = B_i for i = 1, ..., k. General completely positive maps without the trace-preserving requirement, and unital completely positive maps transforming the states are also considered.