## 2 Materials and Methods  
### Experiment  
The experimental setup is shown in Figure 1. The desired partially polarized states are generated by a Mach-Zehnder interferometer, where a polarizing beamsplitter generates beams of horizontal and vertical polarizations, which are then recombined on a 50:50 non-polarizing beamsplitter (BS), such that they are separated by a small distance and not coaxially superposing. This allows for control over the DOP by adjusting relative intensity of each arm of the interferometer [21; 32]. This is done by attenuating one arm by tuning a variable neutral-density filter. To change the basis from horizontal/vertical to an arbitrary basis, the light is passed through a quarter-wave plate (QWP), half-wave plate (HWP), and another QWP. The light passes through approximately 1 m of fiber prior to entering free-space. The preparation of states requires approximately 1 m of propagation through free-space. Once in the desired polarization state, the light is transmitted through free-space for approximately 3 m. When measuring the effects of turbulence, the light passes through the entire length of the water tank, which is 30 cm long and holds approximately 10 L of water when full (the tank measures approximately 30 cm by 15 cm by 20 cm and is made out of uncoated glass). Because the propagation distance is on the order of meters, we assume any losses related to other link parameters will be negligible. We generate underwater turbulence tank with submerged bubblers. As mentioned previously, underwater bubbles are known to cause fluctuations in the intensity of propagating light [27; 28; 29; 30]. While we choose to use bubbles as a means of generating turbulence, which produces, changes in indices of refraction when the light passes through the surface interfaces, there has also been significant research in turbulence generated with fog chambers [33; 34], which create continuously varying indices of refraction. These fluctuations can be characterized using the scintillation index [9]. \[\sigma_{I}^{2}=\frac{\langle I^{2}\rangle-\langle I\rangle^{2}}{\langle I \rangle^{2}}, \tag{3}\]  
where \(I\) is the experimentally measured intensity and \(\langle\cdot\rangle\) denotes the temporal average. The photodiode detector used is a 'bucket detector'; it only measures the intensity. No wavefront measurements are conducted. The scintillation index for light passing through underwater bubbles varies depending on the size and number of bubbles, but typical values are between 0.1 and 1.0 [28; 29]. Over the course of the experiment and including all intensity projection measurements, we observe an average \(\sigma_{i}^{2}=0.5\pm 0.1\). Figure 1: Experimental setup used to generate different polarization states, pass them through experimentally generated turbulence, and perform tomography measurements. Abbreviations: PBS = polarizing beamsplitter, QWP = quarter-wave plate, HWP = half-wave plate, P = linear polarizer, VND = variable neutral density filter, BS = 50:50 non-polarizing beamsplitter, Det. = photodetector. To calculate the resulting power at the detector, \(P_{rec}\), we can summarize the free-space link budget as follows  
\[P_{rec}=P_{trans}+G_{trans}-L_{trans}-L_{fs}-L_{turb}+G_{rec}-G_{rec}. \tag{4}\]  
The approximate link parameters are expressed in the Table 1.  
The transmitter loss comes from the fiber optic cable and state preparation. The loss from the fiber is uniform across all measurements and the values are not expected to have any significant impact on the data. The variability in the transmission loss comes from the VND filter, which alters the SOP. While there may be very small loss associated with the free-space propagation, it is negligible in comparison to the loss due to the turbulence and electronic noise of our detectors. Loss due to turbulence varies slightly between trials. Loss at the receiver is variable due to the nature of our setup; depending on the SOP and the orientation of the QWP and LP, we measure different intensities as expected.