For a wide range of applications, a fast, non-destructive, remote, and sensitive identification of samples with predefined characteristics is preferred instead of their full characterization. In this work, we report on the experimental implementation of a nonlocal quantum measurement scheme, which allows for differentiation among samples out of a predefined set of transparent and birefringent objects in a distant optical channel. The measurement is enabled by application of polarization-entangled photon pairs and is based on remote state preparation. On an example set of more than 80 objects characterized by different Mueller matrices, we show that only two coincidence measurements are already sufficient for successful discrimination. The number of measurements needed for sample differentiation is significantly decreased compared to a comprehensive polarimetric analysis. Our results demonstrate the potential of this polarization detection method for polarimetric applications in biomedical diagnostics, remote sensing, and other classification/detection tasks.

## I. INTRODUCTION

For a broad range of fundamental and applied research areas, the polarization of light serves as either a probe or a subject under study. Understanding the polarization response of an object of interest can reveal characteristic features in the sample’s structure and/or behavior, e.g., the birefringence of crystals and bio-tissues,^{1,2} polarization selectivity in reflection/transmission of periodic structures,^{3–5} and chemical composition via detection of chiral proteins, molecules, or their assemblies.^{6,7} The knowledge about the object under study that can be thus acquired is then widely employed for optical biomedical diagnostics,^{8–10} the control of nonlinear phenomena,^{11} fundamental studies,^{12} technical characterization,^{13,14} and remote sensing,^{15} as well as for currently expanding quantum technologies in the fields of communication, computing, and metrology.^{16,17}

A comprehensive characterization in terms of polarization is enabled by the well-acknowledged techniques of classical optics—Stokes and Mueller polarimetry^{18}—which are based on projective analysis of the polarization state or its change. Here, the retrieval of the complete information on the sample’s polarization behavior implies illumination of the specimen with different polarization states and interpretation of the output intensity modulation detected for several polarization projections. In the last few years, the realization of quantum polarimetry and particularly the application of non-classical states of light for polarization-based probing rapidly gained interest.^{19–21} Particular attention has been attracted by the possibility of nonlocal measurement configurations with the purpose of completely remote sample characterization, which has been already demonstrated in several configurations using classical and quantum correlations.^{22–26}

Recently, we have developed a theoretical model aiming at coincidence-based discrimination of polarization objects out of a predefined set, where the number of projective measurements for sample identification is significantly reduced and which can be performed remotely by employing polarization-entangled photon pairs.^{27} This measurement approach can be implemented with the optical arrangement depicted in Fig. 1. Here, the signal photons of the entangled pairs probe the sample and then undergo a particular but fixed polarization transformation. The required analysis with several different polarization projections is performed at the entangled idler photons in a different optical path. These photons do not interact with the sample, and the information transfer is enabled via the concept of remote state preparation.^{28} The coincidence counts between the signal and idler photons are measured for several projections of the idler photon and represent the response of the sample. After normalization, these coincidence counts can be then considered as coordinates and define a virtual space where the polarization response of the object is distinctively mapped. The coordinates of the sample in this measurement space are linked to its polarization properties, where the dimensionality of the space depends on the number of projections for which the idler photons need to be measured. A reduction of the total number of required measurements for the projective analysis is achieved by an optimized selection of the polarization projections in the idler arm and of the single fixed polarization transformation in the signal arm.^{27}

In this letter, we report on the experimental implementation of this non-destructive, remote, and sensitive polarization-based evaluation approach. Besides experimentally demonstrating the principle, the presented study aims at showing the potential of the method for realistic applications. Often, practically relevant tasks, e.g., identifying different types of tissue or detecting the tissue orientation,^{10} deal with specimens only slightly differing in the orientation of the polarization-dependent transmission axis or, in the case of birefringent samples, orientation of the optical axis. In order to mimic a set of such test samples, we use different orientations of a linear polarizer or a quarter-wave plate. Here, each orientation of either optical component results in a different Mueller matrix and is considered as a separate object to be differentiated from all other objects within the sample set. Using this test set, we show that only two projections in the idler arm and the corresponding coincidence measurements are already sufficient for successful differentiation between more than 80 objects with differences in the orientation of the optical axis down to just 2°. The needed number of measurements per sample is reduced in comparison to those needed for full characterization of the sample’s polarization properties. With this, the demonstrated approach of remote polarization-based detection becomes attractive for practical use for, e.g., biomedical diagnostics, communication link testing, and further remote sensing tasks with demand for real-time decision making.

## II. EXPERIMENTAL REALIZATION

The experimental implementation of the nonlocal polarization-based identification of objects is sketched in Fig. 1 and follows the concept described above.^{27}

A source of entangled photon pairs is realized by two periodically poled KTP (KTiOPO_{4}) crystals in type-II phase matching configuration, which are arranged with orthogonal optical axes within a polarization Mach–Zehnder interferometer.^{29} The wavelength-degenerate photons are generated at 810 nm via the process of spontaneous parametric down-conversion (SPDC) and are spatially separated into two channels: signal, where a sample is introduced later, and idler. Long pass filters (LPFs) ensure that no pump light enters the measurement arrangement. A polarization control module (not depicted in Fig. 1) consisting of two quarter-wave plates (QWPs) enclosing a half-wave plate is used to compensate for the minor temporal walk-off between the photons mainly related to the ambient conditions. The source nominally generates a Bell state in the form $|\Psi +\u232a=12(|H\u232a|V\u232a+|V\u232a|H\u232a)$. The measured density matrix of the experimentally realized state is provided in the inset of Fig. 1. It is characterized by the linear entropy of 0.13, a concurrence of 0.88, and fidelity with the nominal Bell state of 0.90.

Before being detected, the signal and idler photons are projected into different polarization states using a combination of QWP and linear polarizers (LPs) with different extinction ratios. In the signal arm, the fixed polarization transformation p^{s} is used; in the idler arm, several projections $pni$ (*n* = 1, 2, 3) are implemented. For detection of the transmitted photons, the fiber-coupled single photon counting modules are used and the coincidences between the channels are counted for each projection combination in a sequential manner.

As test samples, we utilized a conventional broadband linear polarizer (WP25M-VIS by Thorlabs) and an achromatic quarter-wave plate (AQWP05M-600 by Thorlabs), which were rotated around the light propagation direction. Although physically only two samples are employed, each orientation of either LP or QWP results in a different Mueller matrix and thus represents a different polarization object. Their Mueller matrices are *M*_{s} = *R*(*θ*)*M*_{LP,QWP}*R*(−*θ*), where *M*_{LP,QWP} are the Mueller matrices of a vertically oriented LP or QWP and *R*(*θ*) denotes the rotation around the direction of light propagation by angle *θ*. Hence, a substantial set of objects with different Mueller matrices can be mimicked. In the experiments, different orientations of LP and QWP have been obtained with an angular step of 1° within the range from 0° to 180° where the orientation angle is identified by the angle between the transmission axis for LP and fast axis for QWP to the global vertical orientation in the optical arrangement. The samples obtained by rotating the LP will be in the following referred to as LP subset and QWP subset for rotating the QWP, correspondingly.

In the experimental configuration described above, the signal photon undergoes polarization transformation *T*_{s} = p^{s}*M*_{s} (i.e., interacts with the sample and passes the p^{s} projector) and gets detected. At the instance of signal photon detection, the state of the idler photon, which does not interact with the sample, according to the concept of remote state preparation^{28} can be written as the partial trace over the signal photon state: $\rho r=trP[(Ts\u22971)\rho 0(Ts\u22971)\u2020]$, where † denotes the conjugate transpose. This relation points out the main effect of the fixed polarization projection p^{s} applied in the signal arm, as it demonstrates that the polarization transformation induced by the sample and the projection p^{s} influences the state of the idler photon. The specific role of the transformation p^{s} is to ensure that for all samples in the considered set, the reduced state *ρ*_{r} is different, which is not generally the case. Only in this case, every sample out of the set can be discriminated using suitable projective measurements for the idler photon. When the idler photon passes projection $pni$ (*n* = 1, 2, 3) and gets detected, the probability for recording a coincidence with the signal photon becomes $Pn=tr[(pni)\rho rtr(\rho r)(pni)\u2020]$,^{27} where division by tr(*ρ*_{r}) ensures normalization. Considering the distinct mapping of *ρ*_{r} in the polarization space, it is possible to find projections $pni$ so that the probabilities *P*_{n} also form unique combinations for every sample in a set. The number of coincidences *N*_{n} (*n* = 1, 2, 3) actually measured in the experiments within a fixed integration time of 1 s is proportional to *P*_{n} and thus forms the relative coordinates of the sample in a virtual n-dimensional space N^{n}.

Figure 2 provides a representative measurement outcome when three projections are applied in the idler arm (see Sec. III and Appendix B for details on the used p^{s} and $pni$). The coincidences *N*_{n} between channels for each $pni$ were counted in a sequential manner for LP [Fig. 2(a)] and QWP [Fig. 2(b)] rotated with 1° angular step. The mean coincidence counts (solid curves) and 95% confidence intervals (shaded areas) are evaluated over eight repetitions of the experimental run, while Student’s coefficient is employed to account for the limited number of experimental runs. To translate the measured counts to relative coordinates in the virtual space N^{3}, the measured data are scaled via dividing all the counts by the global maximal value in the dataset.

The samples’ responses are then allocated in the n-dimensional space N^{n} using the obtained relative coordinates, and their distribution serves for distinguishing the samples. In the following, we provide the experimental results for the cases of two and three projections in the idler arm.

## III. RESULTS

### A. Nonlocal differentiation with three coincidence measurements

Following our theoretical investigations,^{27} we first studied the feasibility of the experimental nonlocal differentiation between polarization objects using three coincidence measurements between the signal photon carrying the sample information and the idler photon analyzed at three different polarization projections, correspondingly. For this, we performed a set of experiments with the set of test samples described in Sec. II. The actual implementations of both p^{s} and $pni$ have been realized by combinations of conventional QWP and LP placed before the detection system, as shown in Fig. 1. The orientation angles of QWP and LP for every projection are provided in Table I.

. | p^{s}
. | $p1i$ (deg) . | $p2i$ (deg) . | $p3i$ (deg) . |
---|---|---|---|---|

QWP | 62 | 170 | 18 | 45 |

LP | 90 | 7.5 | 110 | 34 |

. | p^{s}
. | $p1i$ (deg) . | $p2i$ (deg) . | $p3i$ (deg) . |
---|---|---|---|---|

QWP | 62 | 170 | 18 | 45 |

LP | 90 | 7.5 | 110 | 34 |

The obtained experimental results are summarized in Fig. 3. We show three-dimensional space N^{3} defined by the coordinates *N*_{n} (*n* = 1, 2, 3), which correspond to the coincidences counted between the signal after manipulation p^{s} and idler photons after the projections $pni$.

Each point marked as a circle (QWP subset) or diamond (LP subset) depicts the mean value of the coincidence-based location obtained over at least eight repetitions of the experiment. The measurements form two loops corresponding to the samples realized by the QWP (outer loop, pointed out with a “QWP” label) and the LP (inner loop with “LP” label). For both subsets, the filled black circles define the 0-th degree of orientation of the fast axis for the QWP and the transmission axis for the LP. These loops are well separated, indicating that the two sample subsets can be distinguished for all samples.

The measured mean coordinates for all samples also appear separated. However, it has been found that for a rotation step of 1°, the uncertainty regions for the coordinate measurements are overlapping so that a reliable distinction between all objects is not possible. However, a reduced but sizable subset of the measured objects can still be distinguished. The locations of these objects are marked by the filled circles for the QWP and filled diamonds for the LP samples, respectively, whereas empty symbols denote the objects that cannot be differentiated. The uncertainty regions for the distinguishable objects are marked by the shaded ellipsoids in Fig. 3. These ellipsoids represent the 95% confidence regions for the sample positions, where semi-axes in each direction are equal to the 95% confidence intervals for the corresponding coordinate. The color of the ellipsoids represents the largest value of the standard deviation over three coordinates for a single sample over all repetitions of the experiment.

Overall, 42 samples in the LP subset and 60 samples in the QWP subset can be reliably separated from each other. However, since the volume and elongation of the uncertainty regions as well as the mean location points are unevenly distributed for the different angles of the LP and QWP subsets, the samples that can be distinguished are not evenly distributed as well. The median step in the optical axis orientation required to obtain separable samples is 4° for LP and 2° for QWP with corresponding maximal values reaching no more than 8° and 9°. For better visibility, one of the domains with a relatively large spread of the confidence regions (highlighted with the black dashed rectangle) is shown zoomed in the inset of Fig. 3. At the same time, for a large range of angles, the distinguishable samples have been obtained already with just 2° rotation difference for both LP and QWP.

### B. Nonlocal differentiation with just two coincidence measurements

After showing experimentally the feasibility of three-projection nonlocal probing of polarization samples, we studied whether just two coincidence measurements can be sufficient for distinguishing the samples. For these experiments, the same set of test samples was employed. In the idler channel, two of the previously employed polarization projections were used (see Table II). However, in this case, the optimal fixed polarization transformation in the signal arm projects on a partially polarized state (see Appendix B for details). To implement this, a custom LP with a very low extinction ratio of 4:1 was used. The measurement results with this fixed signal transformation and the two projections in the idler are summarized over at least eight measurement runs in Fig. 4.

. | p^{s} (deg)
. | $p1i$ (deg) . | $p2i$ (deg) . |
---|---|---|---|

QWP | 62 | 18 | 45 |

LP | 90 | 110 | 34 |

. | p^{s} (deg)
. | $p1i$ (deg) . | $p2i$ (deg) . |
---|---|---|---|

QWP | 62 | 18 | 45 |

LP | 90 | 110 | 34 |

Analogous to the previous case, Fig. 4 shows the space N^{2}, which now is just two-dimensional based on the two projections $[p1i,p2i]$ in the idler arm. Similar coding as in Fig. 3 is used to represent the measurement statistics. Accounting for the two-dimensional representation, the 95% confidence regions are depicted here as ellipses. Here as well, the mean values of the corresponding samples’ locations are marked as diamonds for LP and circles for QWP subsets, respectively, where empty symbols correspond to unresolved samples and filled symbols denote the samples that can be successfully differentiated.

The change of the polarization transformation implemented in the signal arm has resulted in a changed shape of the overall distribution of the samples in the virtual space. In particular, for the QWP, the measured points are overlapping in the range from ∼60°–110° of the fast axis orientation. Here, only three samples can be separated from each other. This region is highlighted with a black dashed rectangle labeled “(a)” and shown zoomed in the corresponding inset. The mentioned separable samples are marked with numbers. The overall number of distinguishable samples from the QWP subset with just two coincidence measurements is decreased to 34. The median step was found to be smaller than for the three-projection measurement and is equal to 3°. The same median step required to separate different samples has been detected for the LP subset, where the distribution of measured responses was found similar to the previous case. The points are allocated along a single-loop curve with no self-crossing points and no exceptional regions, such as those that appeared in the case of a QWP subset. With this, the full range of angles with 54 different samples could be separably projected in the N^{2} space of relative coincidence counts. For both LP and QWP subsets, the maximum step in the rotation angle was defined as 8°, where the exceptional range from inset “(a)” of Fig. 4 was excluded. The minimal sufficient step is again 2°. However, the distribution of the sample responses is characterized by one more particular region, which is highlighted with a black dashed rectangle labeled “(b)” and shown zoomed in the respective inset. It has been found that the confidence regions of three samples from the QWP subset initially found as distinguishable (pointed out with numbers in the inset) coincide with the confidence regions of the closely allocated samples from the LP subset. Although the mean values of all these samples are still separable, three mentioned samples from the QWP set have been excluded from the list of distinguishable objects, and thus, the corresponding points have been marked with empty circles. Nevertheless, 84 samples in total can be reliably separated under the implemented experimental conditions, which significantly exceeds a typical set size for various applications and shows the sufficiency of just two coincidence measurements for practical differentiation between polarization objects.

## IV. DISCUSSION AND CONCLUSIONS

The presented results have revealed several points worth of particular attention.

The first one relates to the spread of the sample responses in the space N^{3} defined by coincidence measurements for three projections in the idler arm. The measured loops have not been projected in one plane, and the resolution in terms of orientation angle could not be achieved better than 2°. This can be explained by the combination of several factors. On one hand, the polarization transformations for the idler arm have been realized using combinations of conventional QWP and LP, which were close to optimal projections but still not most advantageous. This issue can be mitigated by the use of projections to partially polarized states. Using customized polarization manipulators that result in partially polarized states instead of the conventional LP and QWP used in our feasibility demonstration, the distance of the measured sample locations can be enlarged.^{27} On the other hand, the level of entanglement of the two-photon state employed in the experiment and decoherence occurring in the samples have to be accounted for as well. For our experiments, the target polarization transformation was optimized, considering the nominal Bell state in the form $|\Psi +\u232a=12(|H\u232a|V\u232a+|V\u232a|H\u232a)$ (refer to Appendix B for details). At the same time, the concurrence of the utilized polarization-entangled photon pairs during the reported experiments has been observed at the level of 0.88 (for details on source characterization, refer to Appendix A). The latter is supposed to cause slight shrinkage of the loops of LP and QWP in terms of the reduced state of the idler photon and thus changed projections in the space of coincidences.^{27} For example, it is expected that providing the projections in the signal and idler arms would be optimized to expand the distribution of the outer loop of points (QWP subset) to the limits of the coordinate system (i.e., from maximal coordinate values of around 0.8–1.0) without an AN increase of the experimental error; the QWP subset of distinguishable samples could be enlarged from 60 to around 75. This, in turn, would allow one to enhance resolution in terms of principal axis orientation down to 1° in a portion of the angular range.

The second point to discuss is related to the experiment with two projections in the idler arm. As discussed above, a combination of conventional LP and QWP has been initially used in the signal arm as well (see Sec. III A and Appendix B). Although being again not optimal, the resulting polarization transformation p^{s} performed sufficiently well to enable differentiation between test samples with three coincidence measurements. It appeared, however, to be not suitable for the case of only two projections. To introduce the necessary partial polarization contribution, we replaced the conventional LP with a custom linear polarizer of a tailored extinction ratio (see Appendix C). This allowed us to distribute the sample responses over the N^{2} space of relative coincidence counts in a more advantageous manner. At the same time, this resulted in a slightly changed shape of the loop for samples within the QWP subset and the appearance of the exceptional region where fewer samples could be reliably separated [see Sec. III B and inset “(b)”of Fig. 4]. This issue can be addressed in the future with a custom phase retarder instead of the conventional QWP as well as further refinement of the numerical procedure for optimizing the polarization projectors in both signal and idler arms.

The advancements discussed above are expected to enable broader flexibility for realization of the optimal polarization transformations p^{s} and $pni$. This, in turn, would eliminate the partially overlapped and exceptional domains from insets of Fig. 4, but would also allow us to experimentally perform differentiation between even larger sets of samples and/or samples with smaller changes in polarization behavior. At the same time, the results presented in this Letter prove, in practice, the feasibility of the proposed method for nonlocal differentiation between objects without complete polarization analysis after the sample and the reduced total number of the required measurements.

Besides this, the advantages of the suggested approach are expected to include the high sensitivity due to the inherently high signal-to-noise ratio of the coincidence-based detection and non-destructiveness in the context of light-induced damage of the sample. The latter is a straightforward result of the nonlocal configuration, but can be additionally engineered by using the non-degenerate polarization-entangled photon pairs, where the sample could be probed at the least harmful spectral range and the idler photon could be analyzed in the range where highly efficient optics and detectors are available.

To sum up, in this Letter, we report on the first experimental demonstration of nonlocal differentiation between polarization objects using polarization-entangled photon pairs. With just two polarization projections in the remote optical channel, a single fixed polarization transformation in the local channel, and respective coincidence measurements between them, we demonstrated successful differentiation of more than 80 samples. While the proposed method substantially decreases the required number of measurements in comparison to full characterization with either classical or non-classical states of light, we were able to distinguish objects characterized by the difference in the principal optical axis orientation down to just 2°. Taking into account further advantages of nonlocal coincidence based probing, such as non-destructiveness and low noise, as well as recent advances in optics miniaturization^{30} and active research toward practical ghost measurement configurations,^{31} there is a high potential for practical implementation of the approach, particularly for applications in biomedical diagnostics, remote sensing, and other classification/detection tasks with the demand for highly accurate real-time decision making.

## ACKNOWLEDGMENTS

The fabrication of the partial polarizer samples within this work was partly carried out by the microstructure technology team at IAP Jena. The authors would like to thank them for providing the fabrication facilities, carrying out processes, and providing support.

The authors also thank Johannes Kretzschmar, FSU Jena, for his help with artistic visualization of the experimental arrangement.

This work was funded by the German Ministry of Education and Research (“QuantIm4Life” project, Grant No. FKZ 13N14877). V.B. acknowledges funding of this work also through the Pro-Chance-career program (Grant No. AZ 2.11.3-A1/2022-01) of Friedrich Schiller University Jena. This project (Grant No. 20FUN02 POLight) received funding from the EMPIR program co-financed by the Participating States and from the European Union’s Horizon 2020 Research and Innovation Program. A.A.S. also acknowledges support from the UA-DAAD exchange scheme and Australian Research Council (Grant Nos. DP190101559, CE200100010). The authors also acknowledge support by the German Research Foundation Projekt-Nr. 512648189 and the Open Access Publication Fund of the Thueringer Universitaets- und Landesbibliothek Jena.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

V.R.B. designed and conducted the experiments, collected and analyzed the data, wrote the manuscript, L.Z. supported construction and alignment of the experimental arrangement, A.V. developed the used theoretical model and performed simulations, P.S.C. and F.S. designed and constructed the source of polarization-entangled photon pairs, T.S. designed and manufactured the custom partial polarizer, wrote the manuscript, F.S. wrote the manuscript, A.A.S. and F.S. conceived the study, T.P., A.A.S., and F.S. supervised the research. All authors reviewed the manuscript.

**Vira R. Besaga**: Data curation (lead); Formal analysis (lead); Funding acquisition (supporting); Investigation (lead); Methodology (equal); Software (lead); Validation (equal); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead). **Luosha Zhang**: Data curation (supporting); Investigation (supporting); Writing – review & editing (supporting). **Andres Vega**: Methodology (equal); Software (supporting); Writing – review & editing (supporting). **Purujit Singh Chauhan**: Data curation (supporting); Resources (equal); Writing – review & editing (supporting). **Thomas Siefke**: Funding acquisition (supporting); Methodology (supporting); Resources (equal); Writing – review & editing (supporting). **Fabian Steinlechner**: Resources (supporting); Writing – review & editing (supporting). **Thomas Pertsch**: Resources (lead); Writing – review & editing (supporting). **Andrey A. Sukhorukov**: Conceptualization (lead); Funding acquisition (supporting); Methodology (equal); Writing – review & editing (supporting). **Frank Setzpfandt**: Conceptualization (lead); Funding acquisition (lead); Project administration (lead); Resources (equal); Supervision (lead); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

### APPENDIX A: CHARACTERIZATION OF THE SOURCE OF POLARIZATION-ENTANGLED PHOTON PAIRS

Prior to every experiment on the nonlocal differentiation between samples, the photon state is characterized via the quantum state tomography.^{32} For this, a combination of conventional QWP and LP in each channel (in place of fixed and changing projections in Fig. 1) is used to realize 16 polarization projection combinations in the |*H*⟩, |*V*⟩; |*D*⟩, |*A*⟩; and |*L*⟩, |*R*⟩ bases.^{33} These notations stand for horizontal, vertical, diagonal (+45°), anti-diagonal (−45°), right-circular, and left-circular polarization states. The coincidences are counted using a time tagger with 1 ps temporal resolution within a 3 ns coincidence window over 10 s integration time. The raw coincidence counts are corrected for accidental counts, detector efficiency, and intensity drift.^{32} The corrected values of coincidence counts are then used to reconstruct the density matrix of the state by the maximum likelihood method.^{33}

### APPENDIX B: REALIZATION OF POLARIZATION PROJECTIONS IN THE SIGNAL AND IDLER ARMS

An optimal polarization transformation in the signal arm is responsible for ensuring the distinct response of different samples in terms of polarization. Here, the resolution of the differentiation between the samples and thus a maximum number of distinguishable samples are defined, in the first place, by the maximally achievable distance between the points associated with the samples in the virtual measurement space N^{n}. Within the current study, each sample within the predefined set corresponds to a unique Mueller matrix realized by a certain rotation angle of either LP or QWP in the optical path of the signal photon. In practice, the resolution (here, the minimum angular increment of LP or QWP rotation required to realize separable samples) decreases, as discussed in Sec. IV, due to the experimental error in determining the coordinates of the mentioned points. When designing our experiments, we started off with numerical search for the optimal transformation p^{s} considering the nominal state of our source of polarization-entangled photon pairs. Here, the objective function of the optimization process based on the Matlab^{®} *optimset* function was following our previous theoretical model^{27} and was not restricted to feasibility of its implementation with conventional optical components. An example of the Mueller matrix of such a versatile solution, which could be realized in a form of a metasurface, is represented in Table III. Using this transformation as a reference, we searched for the closest experimentally feasible solutions, which could be realized with phase retarders and linear polarizers only. Here, the combination of optical elements in the optical path and angles of their orientation were parameters for optimization. As a result, we were able to find the respective transformations when combinations of a QWP and either a commercial high-extinction ratio or custom LP with tailored extinction ratio were used. The corresponding Mueller matrices, which were realized in the experiments with three and two coincidence measurements, are represented in Table III as well. For this, the QWP was oriented with its fast axis at 62° and LP with its transmission axis at 90° to the global vertical orientation in the optical arrangement. A similar approach has been used when implementing polarization transformations for the idler photon. An overview of the orientation angles of the QWP and LP used in the experiments is provided in Sec. III in Tables I and II.

p^{s} optimal | 0.6363 | −0.2086 | 0.1488 | −0.2716 |

−0.2086 | 0.3549 | 0.2452 | 0.3505 | |

0.1488 | 0.2452 | 0.1604 | −0.4491 | |

0.2716 | −0.3505 | 0.4491 | −0.1210 | |

p^{s} for | 0.5000 | −0.5000 | 0 | 0 |

N^{3} ≔ [N_{1}, N_{2}, N_{3}] | −0.1563 | 0.1563 | 0 | 0 |

0.2318 | −0.2318 | 0 | 0 | |

0.4145 | −0.4145 | 0 | 0 | |

p^{s} for | 0.6351 | −0.3649 | 0 | 0 |

N^{2} ≔ [N_{1}, N_{2}] | −0.1141 | 0.1986 | −0.2410 | 0.4310 |

0.1691 | −0.2944 | 0.3573 | 0.2907 | |

0.3025 | −0.5266 | −0.2907 | 0 |

p^{s} optimal | 0.6363 | −0.2086 | 0.1488 | −0.2716 |

−0.2086 | 0.3549 | 0.2452 | 0.3505 | |

0.1488 | 0.2452 | 0.1604 | −0.4491 | |

0.2716 | −0.3505 | 0.4491 | −0.1210 | |

p^{s} for | 0.5000 | −0.5000 | 0 | 0 |

N^{3} ≔ [N_{1}, N_{2}, N_{3}] | −0.1563 | 0.1563 | 0 | 0 |

0.2318 | −0.2318 | 0 | 0 | |

0.4145 | −0.4145 | 0 | 0 | |

p^{s} for | 0.6351 | −0.3649 | 0 | 0 |

N^{2} ≔ [N_{1}, N_{2}] | −0.1141 | 0.1986 | −0.2410 | 0.4310 |

0.1691 | −0.2944 | 0.3573 | 0.2907 | |

0.3025 | −0.5266 | −0.2907 | 0 |

### APPENDIX C: MANUFACTURING OF A PARTIAL POLARIZER

The partial polarizer has been created on a fused silica substrate (Siegert) that is initially cleansed using a piranha solution. Following this, 25 nm of chromium is added using ion beam sputter deposition (Ionfab 300LC (OIPT)). Next, OEBR-CAN038 AE 2.0CP (Tokyo Ohka Kogyo Co. LTD) EBL (electron beam lithography) resist is applied and patterned using a Vistec 350OS electron beam writer with character projection apertures.^{34} The resultant grating pattern has a period of 300 nm, which is transferred into the chromium layer through ion beam etching using the Ionfab 300LC (OIPT). Following this, the remaining resist is removed using oxygen plasma etching before the substrate is diced to size. The extinction ratio of the manufactured polarizer has been measured at 810 nm as 3.7 to 1 using a Perkin Elmer Lambda 950 spectrophotometer.

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