Only Hermitian Operators have been considered as Observables in Quantum Mechanics because not only they guarantee real eigenvalues but also ensures Existence of orthogonal eigen-basis. Expectation value of a non-Hermitian operator cannot be measured from the statistics of outcomes as the eigenbasis is not orthogonal. However, since expectation value of a non-Hermitian operator \(A\) is in general complex, if we can measure the weak value of \(R\) - the Hermitian component of the polar decomposition of \(A\) with preselected state \(|\psi\rangle\) and post-selected state \(|U^\dagger \psi\rangle\) \(U\) being the unitary component of the polar decomposition of \(A\), we can infer the expectation value of the non-Hermitian operator.
Another alternative is to infer the expectation value of the non Hermitian operator observing visibility and phase shift in a Mach Zehnder Interferometer in which we place the optical component corresponding to the \(R\) and \(U\) in each arm respectively. Then we can also infer the weak value of \(R\). The advantages of this method in simulating weak value in the region of amplification is experimentally demonstrated.
Alice in Wonderland had the Cheshire Cat - the grin the cat separated from its body. In physics, we have objects like neutrons and photons with properties like spin and polarization respectively. Can we create a Cheshire Neutron i.e. separate the spin of the electron the electron itself? In has been suggested that in an interferometric setup, if we choose pre-selected and post-selected states appropriately, the signature of spin of a neutron can be found in one the arm of the interferometer where the neutron is absent. Thus, in the duration between pre-selection and post-selection Cheshire neutron can be realized.
Over the last 5 years, there have been experiments claiming the demonstration of Quantum Cheshire Cat and at the same time papers refuting the idea of Quantum cheshire Cat as an illusion. Here, an appraisal of literature on Quantum Cheshire cat is presented where the experiments have been analyzed and found that because the experiments did not infer weak values from pointer shifts they are not loophole free. At the sametime, various criticisms of the Quantum Cheshire Cat as a phenomenon are rebutted.