Figure 2

Example of a THA against the IM and the PM of Alice (Bob). For simplicity, we assume that Eve sends Alice (Bob) two high intensity single-mode coherent pulses, each of which is denoted by \(\left| {\beta _{\text{E}}{e^{i\theta _{\text{E}}}}} \right\rangle \). One of them targets the IM and the other one targets the PM. We further assume also for simplicity that the back-reflected light from the IM and the PM to Eve is in a product state of two coherent states. One comes from the IM, which we denote by \(\left| {{\beta _{r}}{e^{i{\theta _{r}}}}} \right\rangle \), and the other comes from the PM, which has the form \(\left| {{\sqrt{I_{\text{max}}} }{e^{i{\theta _{\chi }}}}} \right\rangle \), where r and \(\chi \) refer to the intensity setting and basis choice, respectively, with \(r \in \{\text{s,v,w}\}\) and \(\chi \in \{\text{Z,X}\}\). Eve can learn partial information about the intensity settings and the basis choices by separately measuring the states \(\left| {{\beta _{r}}{e^{i{\theta _{r}}}}} \right\rangle \) and \(\left| {{\sqrt{I_{\text{max}}} }{e^{i{\theta _{\chi }}}}} \right\rangle \).