High-dimensional entangled states are promising candidates for increasing the security and encoding capacity of quantum systems. While it is possible to witness and set bounds for the entanglement, precisely quantifying the dimensionality and purity in a fast and accurate manner remains an open challenge. Here, we report an approach that simultaneously returns the dimensionality and purity of high-dimensional entangled states by simple projective measurements. We show that the outcome of a conditional measurement returns a visibility that scales monotonically with state dimensionality and purity, allowing for quantitative measurements for general photonic quantum systems. We illustrate our method using two separate bases, the orbital angular momentum and pixels bases, and quantify the state dimensionality by a variety of definitions over a wide range of noise levels, highlighting its usefulness in practical situations. Importantly, the number of measurements needed in our approach scale linearly with dimensions, reducing data acquisition time significantly. Our technique provides a simple, fast and direct measurement approach.

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Three-dimensional structured illumination microscopy (3D-SIM) is a wide-field technique that can provide doubled resolution and improved image contrast. In this work, we demonstrate a simple approach to 3D-SIM - using three-beam interference with circular polarization to generate the pattern of structured illumination, so that the modulation contrast is routinely maintained at all orientations without a complicated polarization rotator or mechanical motion. We derive the resultant intensity distribution of the interference pattern to confirm the modulation contrast independent of orientation, and compare the result with those using interfering beams of linear polarization. To evaluate the influence of the modulation contrast on imaging, we compare the simulated SIM images of 100-nm beads. Experimental results are presented to confirm the simulations. Our approach requires merely a λ/4-wave plate to alter the interfering beams from linear to circular polarization. This simplicity together with the use of a spatial light modulator to control the interference pattern facilitates the implementation of a 3D-SIM system and should broaden its application.

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