Measuring non-Hermitian operators via weak values

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, the reality of an eigenvalue of some operator does not mean that it is necessarily Hermitian. There are examples of non-Hermitian operators that may admit real eigenvalues under some symmetry conditions. 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-semidefinite part of the non-Hermitian operator. We also prove an uncertainty relation for any two non-Hermitian operators and show that the fidelity of a quantum state under a 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 measuring the product of noncommuting projectors.

Figure 1

Plot of LHS and RHS of Eq. (14), given, respectively, by Eqs. (15) and (16), as a function of $s$. Both axes are dimensionless. Triangles show the LHS and circles the RHS. This figure clearly shows that the uncertainty relation given for non-Hermitian operators is satisfied by the creation and annihilation operators in phase states.

Figure 2

Lower and upper bounds on $\mathrm{\Delta }{E}_1\mathrm{\Delta }{E}_2$. Here both the axes are dimensionless. Blue squares show $\mathrm{\Delta }{E}_1\mathrm{\Delta }{E}_2$ and green triangles and the red solid line show the lower bound (25) and upper bound $\frac{1-F}{2}$, respectively on $\mathrm{\Delta }{E}_1\mathrm{\Delta }{E}_2$ as a function of $p$ at fixed values of $\theta =\pi /2$ and $\varphi =\pi /4$.