Characterizing Turbulence-Induced Decay of Mutual Unbiasedness of Complementary Bases Relevant to Propagated Photonic Spatial-Mode States

Mutually complementary bases are crucial to secure quantum key distribution (QKD). By aiming at free-space QKD with use of spatially structured photons, the effect of turbulence on the mutual unbiasedness of two complementary bases relevant to photonic orbital-angular-momentum (OAM) modes is modeled theoretically, with novel insightful expressions and physical explanations yielded. For two complementary bases constructed from Laguerre-Gaussian (LG) modes with a fixed radial index of zero, it is shown that the superposed joint-two-LG-mode (JTLGM) correlation function plays a fundamental role in determining the turbulence-induced mutual-unbiasedness decay. The mutual information between the sent and detected photonic states is used as a metric to quantify the degree of turbulence-induced mutual-unbiasedness decay. It is found that the degree of turbulence-induced mutual-unbiasedness decay depends on the scaled atmospheric coherence width, anisotropy of turbulence and contents of the complementary bases, and it can become non-negligible in certain cases.