Generalized Householder transformations

TL;DR

This paper generalizes the Householder transformation to include multiple negative eigenvalues and non-binary cases, providing new insights into operator-based contextuality in quantum theory.

Contribution

It introduces a generalized spectral decomposition of the Householder transformation, expanding its applicability to contextuality studies beyond binary observables.

Findings

Generalized the Householder transformation with multiple negative eigenvalues

Extended the framework to non-binary and operator-valued arguments

Discussed new forms of contextuality through operator functional relations

Abstract

The Householder transformation, allowing a rewrite of probabilities into expectations of dichotomic observables, is generalized in terms of its spectral decomposition. The dichotomy is modulated by allowing more than one negative eigenvalue or by abandoning binaries altogether, yielding generalized operator-valued arguments for contextuality. We also discuss a form of contextuality by the variation of the functional relations of the operators, in particular by additivity.