Measuring an entangled state of two particles is crucial to many quantum communication protocols. Yet Bellstate distinguishability using a finite apparatus obeying linear evolution and local measurement is theoretically limited. We extend known bounds for Bell-state distinguishability in one and two variables to the general case of entanglement in n two-state variables. We show that at most 2 n+1 − 1 classes out of 4 n hyper-Bell states can be distinguished with one copy of the input state. With two copies, complete distinguishability is possible. We present optimal schemes in each case.

This study demonstrates the feasibility of using OSI to measure breathing tidal volumes and breathing patterns with adequate accuracy. This is the first time that dynamic breathing tidal volume as well as breathing patterns is measured using optical surface imaging. The OSI-observed movement of the entire torso could serve as a new respiratory surrogate in the treatment room during radiation therapy.

The OSI-based technique provides an accurate measurement of tidal volume, airflow rate, and breathing pattern; all affect internal organ motion. This technique can be applied to various breathing patterns, including FB, BB, and CB. Substantial breathing irregularities and irreproducibility were observed and quantified with the OSI-based technique. These breathing parameters are useful to quantify breathing conditions, which could be used for effective tumor motion predictions.

Purpose To develop a physical, adaptive motion perturbation model to predict tumor motion using feedback from dynamic measurement of breathing conditions to compensate for breathing irregularities. Methods and Materials A novel respiratory motion perturbation (RMP) model was developed to predict tumor motion variations caused by breathing irregularities. This model contained 2 terms: the initial tumor motion trajectory, measured from 4-dimensional computed tomography (4DCT) images, and motion perturbation, calculated from breathing variations in tidal volume (TV) and breathing pattern (BP). The motion perturbation was derived from the patient-specific anatomy, tumor-specific location, and time-dependent breathing variations. Ten patients were studied, and 2 amplitude-binned 4DCT images for each patient were acquired within 2 weeks. The motion trajectories of 40 corresponding bifurcation points in both 4DCT images of each patient were obtained using deformable image registration. An in-house 4D data processing toolbox was developed to calculate the TV and BP as functions of the breathing phase. The motion was predicted from the simulation 4DCT scan to the treatment 4DCT scan, and vice versa, resulting in 800 predictions. For comparison, noncorrected motion differences and the predictions from a published 5-dimensional model were used. Results The average motion range in the superoinferior direction was 9.4 ± 4.4 mm, the average ΔTV ranged from 10 to 248 mm3 (−26% to 61%), and the ΔBP ranged from 0 to 0.2 (−71% to 333%) between the 2 4DCT scans. The mean noncorrected motion difference was 2.0 ± 2.8 mm between 2 4DCT motion trajectories. After applying the RMP model, the mean motion difference was reduced significantly to 1.2 ± 1.8 mm (P = .0018), a 40% improvement, similar to the 1.2 ± 1.8 mm (P = .72) predicted with the 5-dimensional model. Conclusions A novel physical RMP model was developed with an average accuracy of 1.2 ± 1.8 mm for interfraction motion prediction, similar to that of a published lung motion model. This physical RMP was analytically derived and is able to adapt to breathing irregularities. Further improvement of this RMP model is under investigation.

To automatically estimate average diaphragm motion trajectory (ADMT) based on four-dimensional computed tomography (4DCT), facilitating clinical assessment of respiratory motion and motion variation and retrospective motion study. We have developed an effective motion extraction approach and a machine-learning-based algorithm to estimate the ADMT. Eleven patients with 22 sets of 4DCT images (4DCT1 at simulation and 4DCT2 at treatment) were studied. After automatically segmenting the lungs, the differential volume-per-slice (dVPS) curves of the left and right lungs were calculated as a function of slice number for each phase with respective to the full-exhalation. After 5-slice moving average was performed, the discrete cosine transform (DCT) was applied to analyze the dVPS curves in frequency domain. The dimensionality of the spectrum data was reduced by using several lowest frequency coefficients (fv) to account for most of the spectrum energy (Σfv2). Multiple linear regression (MLR) method was then applied to determine the weights of these frequencies by fitting the ground truth—the measured ADMT, which are represented by three pivot points of the diaphragm on each side. The ‘leave-one-out’ cross validation method was employed to analyze the statistical performance of the prediction results in three image sets: 4DCT1, 4DCT2, and 4DCT1 + 4DCT2. Seven lowest frequencies in DCT domain were found to be sufficient to approximate the patient dVPS curves (R = 91%−96% in MLR fitting). The mean error in the predicted ADMT using leave-one-out method was 0.3 ± 1.9 mm for the left-side diaphragm and 0.0 ± 1.4 mm for the right-side diaphragm. The prediction error is lower in 4DCT2 than 4DCT1, and is the lowest in 4DCT1 and 4DCT2 combined. This frequency-analysis-based machine learning technique was employed to predict the ADMT automatically with an acceptable error (0.2 ± 1.6 mm). This volumetric approach is not affected by the presence of the lung tumors, providing an automatic robust tool to evaluate diaphragm motion.