Measuring Non-Hermitian Operators via Weak Values

TL;DR

This paper introduces a practical protocol for measuring the expectation values of non-Hermitian operators in quantum systems using weak measurements, expanding the tools for quantum measurement and uncertainty analysis.

Contribution

It presents a novel method to measure non-Hermitian operators via weak values, along with new uncertainty relations and applications in quantum fidelity and operator products.

Findings

Method to measure non-Hermitian operators using weak values.

Derived a new uncertainty relation for non-Hermitian operators.

Application to quantum fidelity and testing mathematical formulas.

Abstract

In quantum theory, a physical observable is represented by a Hermitian operator as it admits real eigenvalues. This stems from the fact that any measuring apparatus that is supposed to measure a physical observable will always yield a real number. However, reality of eigenvalue of some operator does not mean that it is necessarily Hermitian. There are examples of non-Hermitian operators which may admit real eigenvalues under some symmetry conditions. However, in general, given a non-Hermitian operator, its average value in a quantum state is a complex number and there are only very limited methods available to measure it. Following standard quantum mechanics, we provide an experimentally feasible protocol to measure the expectation value of any non-Hermitian operator via weak measurements. The average of a non-Hermitian operator in a pure state is a complex multiple of the weak value of the positive semi-definite part of the non-Hermitian operator. We also prove a new uncertainty relation for any two non-Hermitian operators and show that the fidelity of a quantum state under quantum channel can be measured using the average of the corresponding Kraus operators. The importance of our method is shown in testing the stronger uncertainty relation, verifying the Ramanujan formula and in measuring the product of non commuting projectors.