Operator on Operator Regression in Quantum Probability

TL;DR

This paper develops a quantum operator regression model where responses and predictors are operator-valued, introduces quantum $M$ estimators for scalar coefficients, and analyzes their large-sample properties based on eigenvalue observations.

Contribution

It proposes a novel quantum operator regression framework and introduces quantum $M$ estimators, extending classical regression techniques to quantum operator settings.

Findings

Quantum $M$ estimators are consistent under large samples.

Eigenvalue pairs serve as data for the quantum regression model.

Theoretical properties of estimators are derived for linear quantum models.

Abstract

This article introduces operator on operator regression in quantum probability. Here in the regression model, the response and the independent variables are certain operator valued observables, and they are linearly associated with unknown scalar coefficient (denoted by $\beta$), and the error is a random operator. In the course of this study, we propose a quantum version of a class of estimators (denoted by $M$ estimator) of $\beta$, and the large sample behaviour of those quantum version of the estimators are derived, given the fact that the true model is also linear and the samples are observed eigenvalue pairs of the operator valued observables.