Faithful Pointer for qubit measurement

TL;DR

This paper introduces the concept of faithful pointers in qubit measurement, establishing conditions under which formal idealness aligns with operational idealness, thus improving the understanding of measurement fidelity.

Contribution

It derives a class of pointer states called faithful pointers where formal and operational idealness coincide, clarifying the relationship between theoretical and practical measurement conditions.

Findings

Mutual orthogonality does not guarantee real space distinguishability.

For Gaussian wavefunctions, formal idealness may not imply operational idealness.

Faithful pointers ensure formal and operational idealness are equivalent.

Abstract

In the context of von Neumann projective measurement scenario for a qubit system, it is widely believed that the mutual orthogonality between the post-interaction pointer states is the sufficient condition for achieving the ideal measurement situation. However, for experimentally verifying the observable probabilities, the real space distinction between the pointer distributions corresponding to post-interaction pointer states play crucial role. It is implicitly assumed that mutual orthogonality ensures the support between the post-interaction pointer distributions to be disjoint. We point out that mutual orthogonality (formal idealness) does \emph{not} necessarily imply the real space distinguishability (operational idealness), but converse is true. In fact, for the commonly referred Gaussian wavefunction, it is possible to obtain a measurement situation which is formally ideal but fully nonideal operationally. In this paper, we derive a class of pointer states, that we call faithful pointers, for which the degree of formal (non)idealness is equal to the operational (non)idealness. In other words, for the faithful pointers, if a measurement situation is formally ideal then it is operationally ideal and vice versa.