Higher-rank Numerical Ranges and Compression Problems
نویسندگان
چکیده
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 give a complete description in the Hermitian case. We also consider associated projection compression problems.
منابع مشابه
Higher rank numerical ranges of rectangular matrix polynomials
In this paper, the notion of rank-k numerical range of rectangular complex matrix polynomials are introduced. Some algebraic and geometrical properties are investigated. Moreover, for ϵ > 0; the notion of Birkhoff-James approximate orthogonality sets for ϵ-higher rank numerical ranges of rectangular matrix polynomials is also introduced and studied. The proposed denitions yield a natural genera...
متن کاملGENERALIZED JOINT HIGHER-RANK NUMERICAL RANGE
The rank-k numerical range has a close connection to the construction of quantum error correction code for a noisy quantum channel. For noisy quantum channel, a quantum error correcting code of dimension k exists if and only if the associated joint rank-k numerical range is non-empty. In this paper the notion of joint rank-k numerical range is generalized and some statements of [2011, Generaliz...
متن کاملSome Results on the Generalized Higher Rank Numerical Ranges
In this paper, the notion of rank−k numerical range of rectangular complex matrix polynomials are introduced. Some algebraic and geometrical properties are investigated. Moreover, for > 0, the notion of Birkhoff-James approximate orthogonality sets for −higher rank numerical ranges of rectangular matrix polynomials is also introduced and studied. The proposed definitions yield a natural general...
متن کاملThe Ninth Workshop on Numerical Ranges and Numerical Radii
In this talk I will discuss some instances in quantum computing where numerical range techniques arise. I will also try to formulate some open problems. Elliptical range theorems for generalized numerical ranges of quadratic operators Speaker Chi-Kwong Li, William and Mary, [email protected] Co-authors Yiu-Tung Poon, Iowa State University, [email protected]; Nung-Sing Sze, University of Connect...
متن کاملOn higher rank numerical hulls of normal matrices
In this paper, some algebraic and geometrical properties of the rank$-k$ numerical hulls of normal matrices are investigated. A characterization of normal matrices whose rank$-1$ numerical hulls are equal to their numerical range is given. Moreover, using the extreme points of the numerical range, the higher rank numerical hulls of matrices of the form $A_1 oplus i A_2$, where $A_1...
متن کامل