Finite Operator-Valued Frames

TL;DR

This paper develops the theory of finite operator-valued frames, including dilation, duals, existence, optimality under data loss, and construction with specified properties, relevant for quantum computing and data encoding.

Contribution

It introduces new theoretical results on operator-valued frames in finite Hilbert spaces, including optimality conditions and explicit construction methods.

Findings

Characterization of operator-valued frames under data loss

Existence results for frames with given properties

Construction of frames with specified frame operator and scaling

Abstract

Operator-valued frames are natural generalization of frames that have been used in quantum computing, packets encoding, etc. In this paper, we focus on developing the theory about operator-valued frames for finite Hilbert spaces. Some results concerning dilation, alternate dual, and existence of operator-valued frames are given. Then we characterize the optimal operator-valued frames under the case which one packet of data is lost in transmission. At last we construct the operator-valued frames $\{V_j\}_{j=1}^m$ with given frame operator $S$ and satisfying $V_jV_j^*=\alpha_jI$, where $\alpha_j's$ are positive numbers.