相关论文: Higher-Rank Numerical Ranges of Unitary and Normal…
We consider higher-rank versions of the standard numerical range for matrices. A central motivation for this investigation comes from quantum error correction. We develop the basic structure theory for the higher-rank numerical ranges, and…
Some problems of the quantum error-correcting codes theory can be reduced to the investigation of the higher-rank numerical ranges of the operators related to the error operators. We constructively verify a conjecture on the structure of…
The higher rank numerical range is closely connected to the construction of quantum error correction code for a noisy quantum channel. It is known that if a normal matrix $A \in M_n$ has eigenvalues $a_1, \..., a_n$, then its higher rank…
The higher-rank numerical range is a convex compact set generalizing the classical numerical range of a square complex matrix, first appearing in the study of quantum error correction. We will discuss some of the real algebraic and convex…
Results on matrix canonical forms are used to give a complete description of the higher rank numerical range of matrices arising from the study of quantum error correction. It is shown that the set can be obtained as the intersection of…
We introduce a notion of nuclear numerical range defined as the set of expectation values of a given operator $A$ among normalized pure states, which belong to the nucleus of an auxiliary operator $Z$. This notion proves to be applicable to…
We introduce and initiate the study of a family of higher rank matricial ranges, taking motivation from hybrid classical and quantum error correction coding theory and its operator algebra framework. In particular, for a noisy quantum…
The recently developed theory of higher--rank numerical ranges originated in problems of error correction in quantum information theory but its mathematical implications now include a quite satisfactory understanding of \emph{scalar}…
Sum-rank metric codes are a natural extension of both linear block codes and rank-metric codes. They have several applications in information theory, including multishot network coding and distributed storage systems. The aim of this…
Matrices with low-rank structure are ubiquitous in scientific computing. Choosing an appropriate rank is a key step in many computational algorithms that exploit low-rank structure. However, estimating the rank has been done largely in an…
A presentation of numerical range for rectangular matrices is undertaken in this paper, introducing two different definitions and elaborating basic properties. Then we are extended to the treatment of rank-k numerical range.
For a noisy quantum channel, a quantum error correcting code exists if and only if the joint higher rank numerical ranges associated with the error operators of the channel is non-empty. In this paper, geometric properties of the joint…
We consider the problem of exact low-rank matrix completion from a geometric viewpoint: given a partially filled matrix M, we keep the positions of specified and unspecified entries fixed, and study how the minimal completion rank depends…
In this paper we study properties and invariants of matrix codes endowed with the rank metric, and relate them to the covering radius. We introduce new tools for the analysis of rank-metric codes, such as puncturing and shortening…
We study low-rank matrix estimation for a generic inhomogeneous output channel through which the matrix is observed. This generalizes the commonly considered spiked matrix model with homogeneous noise to include for instance the dense…
The paper explores further the computation of the quaternionic numerical range of a complex matrix. We prove a modified version of a conjecture by So and Tompson. Specifically, we show that the shape of the quaternionic numerical range for…
The ranks and kernels of generalized Hadamard matrices are studied. It is proven that any generalized Hadamard matrix $H(q,\lambda)$ over $F_q$, $q>3$, or $q=3$ and $\gcd(3,\lambda)\not =1$, generates a self-orthogonal code. This result…
In this work we show how to approach the problem of manimulating the numerical range of a unitary matrix. This task has far-reaching impact on the study of discrimination of quantum measurements. We achieve the aforementioned manipulation…
The notion of the higher rank numerical range $\Lambda_{k}(L(\lambda))$ for matrix polynomials $L(\lambda)=A_{m}\lambda^{m}+...+A_{1}\lambda+A_{0}$ is introduced here and some fundamental geometrical properties are investigated. Further,…
In this article, we are going to introduce the weighted numerical range which is a further generalization both the c-numerical range and the rank k numerical range. If the boundaries of weighted numerical ranges of two matrices (possibly of…