Fig. 1: The Role of RFs and the Impact of Their Misalignment in Quantum Information Processing.

a Demonstration of RF Misalignment for Classical and Quantum Objects. The left side shows objects defined in a local RF at Site-i, while the right side shows the same objects as viewed from a common RF. For a classical object (upper), e.g., a cat, a misalignment between frames is described by a 3D rotation g_i ∈ SO(3). An orientation vector \({\overrightarrow{{{{\bf{r}}}}}}_{i}\) in the local frame is perceived as \({g}_{i}{\overrightarrow{{{{\bf{r}}}}}}_{i}\) in the common frame. For a quantum object (down), such as a photon in a horizontally-polarized state \(| {H}_{i}\rangle\), the same physical rotation corresponds to a transformation \({\widehat{V}}_{i}({g}_{i})\in SU(2)\) on its quantum state. This results in a different state, such as an elliptical polarization, in the common frame. b Demonstration of the Effects of Drifting RFs on the State Shared in Two Sites. From the view of the common reference point, the coordination of site A and B is changing with g_A(t), g_B(t) ∈ SO(3). The corresponding shared state ρ_AB is experiencing the unitary rotation given \({\widehat{V}}_{A}({g}_{A})\otimes {\widehat{V}}_{B}({g}_{B})\). The average effect over the measuring time is the G-twirling. c Demonstration of k-copy Quantum States Shared in N sites. The connected dots represent an N-qudit state shared in N sites. And the shared k copies mean that kN-qudit states are included. At each site, there are k qudits (k balls), one for each copy.