Fig. 4: Conceptual Feynman diagrams for the transition from \(\left|{{{\Psi }}}_{{\rm{i}}}\right\rangle\) to \(\left|{{{\Psi }}}_{{\rm{f}}}\right\rangle\).
From: Observing the quantum Cheshire cat effect with noninvasive weak measurement
The virtual paths defined by \({\hat{O}}_{k}\) can be arbitrarily set to satisfy \({\sum }_{k}{\hat{O}}_{k}=\hat{{\mathbb{I}}}\) so that \(\langle {{{\Psi }}}_{{\rm{f}}}| {{{\Psi }}}_{{\rm{i}}}\rangle ={\sum }_{k}\langle {{{\Psi }}}_{{\rm{f}}}| {\hat{O}}_{k}| {{{\Psi }}}_{{\rm{i}}}\rangle\). a \({\hat{O}}_{k}=\{{\hat{{{\Pi }}}}_{\text{u}}\otimes {\hat{{{\Pi }}}}_{\text{H}},{\hat{{{\Pi }}}}_{\text{u}}\otimes {\hat{{{\Pi }}}}_{\text{V}},{\hat{{{\Pi }}}}_{\text{l}}\otimes {\hat{{{\Pi }}}}_{\text{H}},{\hat{{{\Pi }}}}_{\text{l}}\otimes {\hat{{{\Pi }}}}_{\text{V}}\}\), b \({\hat{O}}_{k}=\{{\hat{{{\Pi }}}}_{\text{u}}\otimes \hat{{\mathbb{I}}}/2,{\hat{{{\Pi }}}}_{\text{u}}\otimes {\hat{\sigma }}_{\text{z}}/2,{\hat{{{\Pi }}}}_{\text{l}}\otimes \hat{{\mathbb{I}}}/2,{\hat{{{\Pi }}}}_{\text{l}}\otimes {\hat{\sigma }}_{\text{z}}/2\}\), where \({\hat{{{\Pi }}}}_{\text{u}}\) and \({\hat{{{\Pi }}}}_{\text{l}}\) represent the spatial modes of the state and \({\hat{{{\Pi }}}}_{\text{H}}\) and \({\hat{{{\Pi }}}}_{\text{V}}\) represent the polarization modes of the state. The transition amplitudes along the virtual paths in a and b are related with each other due to \(\hat{{\mathbb{I}}}={\hat{{{\Pi }}}}_{\text{H}}+{\hat{{{\Pi }}}}_{\text{V}}\) and \({\hat{\sigma }}_{\text{z}}={\hat{{{\Pi }}}}_{\text{H}}-{\hat{{{\Pi }}}}_{\text{V}}\). The line color shows the normalized transition amplitude of each virtual paths by the total transition amplitude 〈Ψf∣Ψi〉 for the initial and final states in Eqs. (1) and (2).
