Nuclear Numerical Range and Quantum Error Correction Codes for non-unitary noise models

TL;DR

This paper introduces the nuclear numerical range, a new geometric tool for analyzing quantum noise models with block-diagonal Kraus operators, aiding in the construction of quantum error correction codes.

Contribution

It defines the nuclear numerical range and demonstrates its application to quantum noise models, providing a geometric approach to quantum error correction code design.

Findings

Nuclear numerical range is effective for block-diagonal noise models.

Geometric intersection methods can identify error correction codes.

Method generalizes beyond two-qubit systems.

Abstract

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 investigate models of quantum noise with block-diagonal structure of the corresponding Kraus operators. The problem of constructing a suitable quantum error correction code for this model can be restated as a geometric problem of finding intersection points of certain sets in the complex plane. This technique, worked out in the case of two-qubit systems, can be generalized for larger dimensions.