Fig. 1: Schematic of entanglement-controlled vectorial meta-holography (ECVMH)

a Schematic of method for retrieving phase of incident horizontally polarized states\(\,\left|{H}_{s}\right\rangle\). A holographic image (a color helical palette) is reconstructed through a four-step process that retrieves the phase of incident single polarization state \(\left(\left|{H}_{s}\right\rangle\right.\) and \(\left.\left|{V}_{s}\right\rangle\right)\), with its near-unity amplitude. b Schematic of method for retrieving amplitude ratio of incident cross-polarized states \(\left(\left|{H}_{s}\right\rangle\right.\) and \(\left.\left|{V}_{s}\right\rangle\right)\). The average intensities of holographic images from dual polarization states corresponds to their amplitude ratio, but the phase distribution remains random, even within the overlapped region. c Schematic of vectorial holography for retrieving both phase difference and amplitude ratio of two incident cross-polarized states by spatial multiplex of two metaholograms for \(\left|{L}_{s}\right\rangle\) and \(\left|{R}_{s}\right\rangle\) into a single metasurface. The inset shows the principle of the vectorial meta-holography. The phase difference is retrieved by the outer ring with spatially variant intensity distribution after the analyzer; while the complementary images are independent by letting their phases to be random, so their intensity ratio can derive the amplitude ratio of the two incident cross-polarized states. d The ECVMH is realized by the vectorial meta-holography and the incident quantum entangled photon-pairs \({|\Phi }_{\psi }^{+}\rangle\). When the signal photon is in \(|{\psi }_{s}^{+}\rangle\) state, the corresponding holographic image is shown in before and after the analyzer. The upper right inset shows the vectorial meta-holography when the signal photon collapses into arbitrary polarization state \(|{\psi }_{s}^{+}\rangle\), and the lower right inset displays its intensity distribution after the horizontally oriented analyzer. When \(|{\Phi }_{\psi }^{+}\rangle =|{\Phi }_{{LR}}^{+}\rangle\) and the idler photon in blue is triggered to a certain polarization state, the reconstructed meta-holography of signal photons after the analyzer in red accordingly changes, whose states are in MM, \(\left|{L}_{s}\right\rangle\), and \(\left|{R}_{s}\right\rangle\) states