# Polarization Sensitive Imaging with Qubits
## 3 Results  
A transparent 3M scotch tape is applied to a standard microscope slide and the density matrix of the entangled states are compared for photons propagating through the slide glass with and without the birefringent tape. Figure 2a,c show the real part of the DQST, direct tomographic density matrix measured using coincidence counts and Equation (2) with and without the birefringent tape, respectively. The density matrix obtained through the inverse numerical optimization technique, IQST, Equation (7), for the investigated sample with and without the birefringent tape are shown in Figure 2b,d, respectively. The birefringent material changes the phase and polarization of entangled photons in the sample arm, and the density matrix of the entangled states reflects these changes, as evident in Figure 2a-d. Concurrence and the entanglement levels are computed using Equations (8) and (11), for DQST and IQST methods, respectively, from the density matrices. DQST measurements of concurrence and entanglement without tape are 82.99% \(\pm\) 0.3101 and 76.2% \(\pm\) 0.8355, while IQST estimation of concurrence is 91.04% \(\pm\) 0.0102 and 87.27% \(\pm\) 0.0284 for entanglement. The same measurements of concurrence and entanglement with the birefringent tape are 14.57% \(\pm\) 0.112 and 4.77% \(\pm\) 0.124 for DQST, and IQST estimation of concurrence is  
Figure 1: Experimental setup for quantum state tomography imaging.  
49.57% \(\pm\) 0.005 and 34.99% \(\pm\) 0.0109 of entanglement. Errors associated with entanglement and concurrence calculations represent the shot noise error in the measured coincidence counts and the uncertainty in the settings of the angles of the wave plates to set projection operators. The results for the birefringent tape appear different in the reconstructed density matrices in Figure 2c,d, between direct and the inverse technique. We attribute these differences to the superior accuracy of the inverse numerical optimization, IQST, compared to direct quantum state tomography, DQST.  
The transparent 3M scotch tape is applied to a standard microscope slide and scanned in a transverse plane with 10 \(\upmu\)m resolution. The photo of the tape is shown in Figure 3a, and the same section of the tape between two cross polarizers is shown in Figure 3b. The density matrix, concurrence, and entanglement were recorded for each pixel. The 2D DQST images of concurrence and entanglement are shown in Figure 3c,e, (Equations (2), (8) and (11)). The inverse numerical optimization technique, IQST is used to image levels of concurrence and entanglement, and are shown in Figure 3d,f, respectively (Equations (7), (8) and (11)). The SNR in the IQST Figure 3f was measured at 48 db while in the direct-QST, Figure 3e, it was measured at 8 db. In our work, we define the SNR as the ratio of average and standard deviation of pixels measured in a region of the glass slide. The factor that most influences the SNR is the acquisition time. Coincidence counts were measured for each pixel over one second. Reduction of experimental noise is clearly evident from Figure 3e,f. The image enhancement in Figure 3f is a result of an inverse numerical optimization technique that reduces experimental noise and provides positive, semidefinite unit trace density matrix [23]. The optimization technique improves overall image quality and detail resolution. Average entanglement of DQST pixels without the birefringent tape is 62.32% with standard deviation of 7.96%, while pixel distribution in the IQST technique image is 86.36% with standard deviation of 1.79%.  
Figure 2: (**a**) DQST density matrix without the tape, (**b**) IQST density matrix without the tape, (**c**) DQST density matrix with the tape, (**d**) IQST density matrix with the tape.