Research problems on numerical ranges in quantum computing

Abstract

We describe some instances in quantum information processing where numerical range techniques arise. We focus on two basic settings: higher-rank numerical ranges and their relevance in theoretical quantum error correction, and the classical numerical range and its use for comparing quantum information processing operations. We present the basic theory, discuss examples and formulate open problems.Keywords: numerical range; higher-rank numerical range; quantum information processing; quantum error correction; gate fidelity AMS Subject Classifications: 15A60; 15A90; 47N50; 81P68 IntroductionThe tools of matrix analysis and operator theory arise in a growing number of diverse scientific settings. Wherever matrices and operators are in use, it is also natural to expect that numerical range techniques will find application. The emerging disciplines of quantum information science [17] are no different. As two examples in quantum computing, for instance, higher-rank numerical ranges have been recently introduced in the context of quantum error correction [3,4], and numerical range techniques have recently been applied in quantum information processing and quantum optimal control [6,20,21]. There are certainly other instances of note, but for the sake of this article we shall focus on these two. We thus begin this article with a brief introduction to the basic mathematical setting for quantum computing. Then we describe in some detail these two scenarios, including examples and open problems.

Higher rank numerical ranges of Jordan-like matrices

“…2 > m and |β − α| ≤ cos ψ k,m , even though k > m we have C k,m = ∅; then the exact reasoning from previous paragraph applies. (5)…”

“…The (closure of, in the infinite-dimensional case) the numerical range of T always contains the spectrum σ(T ). The numerical range has applications in and is related to many areas, like matrix analysis, inequalities, operator theory, numerical analysis, perturbation theory, quantum computing, and others, see [1][2][3][4][5][6][7][8][9] for a few examples. We refer a reader who is not familiar with the numerical range to [10,Chapter 1].…”

A generalization of Gudder–Nagy's theorem with numerical ranges of operators

“…If A is normal, we have a 12 = 0 immediately, and a 11 = 0 by (5). So, A is a scalar multiple of the identity.…”

“…[1][2][3][4][5] Particularly, some new conclusions are found in the field of physics combined with mathematics. The research of the numerical range of the linear operator on Hilbert space H has an extensive history, and further research on these concepts and their generalizations attracted a number of scholars.…”

On the C-determinantal range for special classes of matrices

“…The numerical range and its generalizations are related to and have applications in several branches of both pure and applied science. For some incidences of this versatile algebraic tool in quantum theory and related problems, see [6,12,15,18,25].…”