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Abstract
Coherence and steerability are two essential characteristics of quantum systems. For a two-qubit state, the first-order coherence and the maximal violation of linear steering inequality are used to operationally measure the degree of coherence and steerability, respectively. Recently, a complementary relation between first-order coherence and linear steerability has been proposed. In this paper, we report an experimental verification of the complementary relation by preparing biphoton polarization entangled states in an all-optical setup. We propose an operable method for experimental measurement of the first-order coherence and linear steerability and calculate the purity of the initial states by reconstructing the density matrices of them. The experimental results coincide with the theoretical predictions very well, which provides a valuable reference for the application of optical quantum technology.
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1. Introduction
Coherence, originally describing the interference capability of interacting fields and the characteristics of photon stream [1–3], is now widely regarded as a valuable resource in quantum information processing tasks [4–8]. On the operational side, various theoretical frameworks of measuring coherence have been proposed from different perspectives [9–13]. For a two-qubit state, the first-order coherence is used to measure the degree of coherence [12]. In recent years, some researchers have proposed the relationship between coherence and other nonlocal correlations [13–22], such as Bell nonlocality, intrinsic concurrence, and linear steerability, etc. In 2015, Svozilík et al. creatively established the relationship between coherence and Bell nonlocality, and proposed the conservation law for first-order coherence and the degree of violation of the CHSH inequality [13]. Soon after, Fan et al. proposed a definition of intrinsic concurrence for two-qubit states, and showed that intrinsic concurrence is always complementary to first-order coherence [14]. More recently, Du and Tong proposed a complementary relation between first-order coherence and linear steerability [15]. These research not only deepen the understanding of the coherence of quantum states, but also provide a reference for the application of quantum information.
On the other hand, based on the Einstein-Podolsky-Rosen (EPR) paradox [23], Schrödinger first proposed the concept of steering [24]. Assume Alice and Bob share a nonlocal two-qubit state, by performing suitable measurements only on her side, Alice has the ability to remotely steer Bob’s state [25]. Recently, steering was considered as a kind nonclassical correlation of quantum states [26] whose hierarchical relationship lies in between entanglement [27] and Bell nonlocality [28]. Soon, researches of steering have become a hot spot in the field of quantum information [29–34]. The so-called steering criterion is to detect whether a quantum state has steerability, and the well-established one is the linear steering inequality proposed by Cavalcanti et al. [29] in 2009. The maximal violation of linear steering inequality is used to operationally measure the degree of steerability [32]. With the development of theoretical research, verifications of steerability have also been widely reported experimentally [35–40]. In 2014, Sun et al. experimentally demonstrated an EPR steering game employing an "all-versus-nothing" criterion that strictly follows the practical concept of steering in an all-optical system [37]. Recently, Zhang et al. reported an experimental validation of quantum steering ellipsoids and tests of volume monogamy relations [38]. In addition, Yang et al. also reported many experimental researches on steering criterion and the related problems [39–41]. These experimental researches have promoted guidance from theory to practical application.
We are interested in the complementary relation between first-order coherence and linear steerability [15]. The aim of this paper is to verify the complementary relation in an all-optical setup by preparing biphoton polarization entangled states [42–49] with high fidelity. The rest of this paper is organized as follows: In Sec. II , we briefly review the theoretical framework of the complementary relation between first-order coherence and linear steerability. In Sec. III, we describe our experimental setup and method. In Sec. IV the experimental procedure and results are demonstrated in detail. Finally, a summary is given in Sec. V.
2. Theoretical framework
Assume that Alice and Bob share a two-qubit state ${\rho _{AB}}$, and without loss of generality, ${\rho _{AB}}$ can be transformed into the following form by local unitary equivalence [50], that is,
On the other hand, suppose that Alice and Bob are both allowed to measure $n$ observables in their sites. The well-known linear steering inequality [29] for diagnosing steerability of a two-qubit state can be written as
More recently, Ref. [15] proposed a concise complementary relation between the first-order coherence and the maximum violation of three-measurement linear steering inequality [15], which is written as
where ${I_{AB}} = 2\textrm{Tr}(\rho _{AB}^2) - 1/2$, and $\textrm{Tr}(\rho _{AB}^2)$ represents the purity of the state. This formula is theoretically applicable to any two-qubit state. It reveals the relationship between first-order coherence and linear steerability.3. Experimental setup and method
To experimentally verify the complementary relation between coherence and steerability, we first prepare a family of two-qubit mixed states as
The schematic of the experimental setup is shown in Fig. 1. Part (a) is the polarization entanglement source, whose function is to prepare polarization-entangled photon pairs in the state $|\psi _\theta ^ + \rangle$ through the type-II spontaneous parametric down-conversion (SPDC) in the Sagnac interferometer [43]. The linearly polarized pump light produced by the 405-nm continuous-wave diode laser (Laser) passes through the half-wave plate (HWP) and the quarter-wave plate (QWP) to prepare the polarization superposition state $\cos \theta |H\rangle + {\textrm{e}^{\textrm{i}\phi }}\sin \theta |V\rangle$, in which the parameters $\theta$ can be set by the rotation of the HWP, and the phase angle $\phi$ can be cleared by adjusting the optical axis direction of the QWP. The Sagnac interferometer consists of a dual wavelength polarization beam splitter (DPBS), a dual wavelength half-wave plate (DHWP) oriented at $45^\circ$, two mirrors (MIRs), and a quasi-phase-matched periodically poled KTiOPO$_4$ (PPKTP). After passing through the dichromatic mirror (DM) and the DPBS, the horizontal pump light is focused on the PPKTP crystal to produce a clockwise down-converted 810-nm photon pair $|HV \rangle$, and then turn to $|VH \rangle$ after the DPBS. The vertical pump light is rotated into horizontal orientation by the DPBS, and then focused on the PPKTP crystal to produce a anticlockwise photon pair $|HV \rangle$. After these two pairs of photons pass through the DPBS again, the horizontal photons are transmitted and the vertical photons are reflected. The polarization entangled state state $|\psi _\theta ^ + \rangle$ is prepared by coupling the photons of path $A$ and $B$ into the single-mode fiber, respectively [44]. Part (b) is the unbalanced interferometer, which is placed in path $B$ and consists of two 50:50 nonpolarizing beam splitters (BSs), two mirrors (MIRs), two attenuators (ATTs), and a half-wave plate (HWP) set at $45^\circ$. The photons in path $B$ is separated into two paths by the first BS, and then classically mixed by the second BS. Here, the HWP is used to change the polarization direction of photons, i.e., $|H\rangle \leftrightarrows |V\rangle$, and the two ATTs are used to regulate the mixed weight $p$. In this way, we can prepare the mixed two-qubit state ${\rho _{AB}}(p,\theta )$ in Eq. (7). Part (c) is the measurement settings, which is symmetrically placed in path $A$ and $B$. The two 3-nm interference filters are used to filter stray photons, and the combinations of QWP+HWP+PBS are used to form any projective measurement bases [42]. The photons are detected by the four single-photon detectors (SPDs), and the signals are sent for coincidence measurements. The time of the coincidence window is 2ns, and the typical entanglement coincidence count rate is 1.8kHz/mW. Through the measuring measurement settings, the probability under different projective measurement bases can be obtained according to the coincidence counts. Moreover, we can reconstruct the optimal density matrix of quantum states by use of tomographically complete set of measurements [51].
Fig. 1. Schematic of the experimental setup. (a) The polarization entanglement source, whose function is to prepare polarization-entangled photon pairs in the state $|\psi _\theta ^ + \rangle$. (b) The unbalanced interferometer, whose function is to realize the classical proportional mixing of two Bell-like States, and finally prepare the required state in Eq. (7). (c) The measurement settings, which is used to realize the projective measurement and reconstruct the density matrix of the states. The main optical elements are as follows: QWP, quarter-wave plate; HWP, half-wave plate; MIR, mirror; DM, dichromatic mirror; DPBS, dual wavelength polarization beam splitter; DHWP, dual wavelength half-wave plate; PPKTP, quasi-phase-matched periodically poled KTiOPO4; BS, beam splitters; ATT, attenuators; PBS, polarization beam splitter; SPD, single-photon detectors.
In order to measure the first order coherence and the maximum violation of three-measurement linear steering inequality of ${\rho _{AB}}(p,\theta )$ experimentally, we first need to construct projective measurement basis. We use $|\psi _i^n\rangle \; (i = 1,2,3; \; n = 1,2)$ to represent the eigenstates of the Pauli operator ${\sigma _i}$, so the projective measurement basis can be expressed as $\Pi _i^n = |\psi _i^n\rangle \langle \psi _i^n|$, which can be realized by adjusting the optical axis angles of HWPs and QWPs in Fig. 1(c). In the experiment, we perform $\Pi _i^m \; (i = 1,2,3; \; m = 1,2)$ and $\Pi _i^n \; (i = 1,2,3; \; n = 1,2)$ on the photons of path $A$ and path $B$, respectively. The coincidence number in this case is represented by $N_i^{mn}$, and the measurement probability can be calculated as
Further, we can calculate ${D_{AB}}$ and ${F_{AB}}$ according to Eqs. (2) and (5). On the other hand, by reconstructing the density matrix of ${\rho _{AB}}(p,\theta )$, we can directly calculate the purity and obtain the value of ${I_{AB}}$.
4. Experimental procedure and results
To verify the complementary relation between coherence and steerability, we adopt single variable method and prepare four kinds of 44 initial states, i.e., ${\rho _{AB}}(\theta ){|_{p = 1}}$, ${\rho _{AB}}(\theta ){|_{p = 0.5}}$, ${\rho _{AB}}(p){|_{\theta = 0^\circ }}$, and ${\rho _{AB}}(p){|_{\theta = 45^\circ }}$. In our experiments, the angle parameter $\theta$ can be precisely set by the HWP’s optical axis direction in Fig. 1(a), and we set $\theta$ as $0^\circ, 10^\circ,20^\circ, 30^\circ, 40^\circ, 45^\circ, 50^\circ, 60^\circ, 70^\circ, 80^\circ, 90^\circ$. The mixed parameter $p$ can be adjusted by the relative amplitudes between these two arms in the unbalanced interferometer, and we set $p$ as $0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1$.
The density matrices of all the prepared initial states are reconstructed by tomography [51,52]. In this way, we can acquire the measured value of ${I_{AB}}$, as well as the fidelity [3] of the state ${\rho _{AB}}$ with
where ${\rho _0}$ is the goal state to be prepared. The average fidelities of the four kinds of quantum states are $\bar {\mathscr{F}}[{\rho _{AB}}(\theta ){|_{p = 1}}] = 0.9986 \pm 0.0012$, $\bar {\mathscr{F}}[{\rho _{AB}}(\theta ){|_{p = 0.5}}] = 0.9987 \pm 0.0027$, $\bar {\mathscr{F}}[{\rho _{AB}}(p){|_{\theta = 0^\circ }}] = 0.9991 \pm 0.0002$, and $\bar {\mathscr{F}}[{\rho _{AB}}(p){|_{\theta = 45^\circ }}] = 0.9989 \pm 0.0038$, respectively. Here, the error bars are estimated according to the Poissonian counting statistics [52,53]. As an example, for a set of original coincidence counts, we numerically generate 50 sets of new counts, each draw randomly from a Poissonian distribution with mean equal to the original counts. So that we can generate 50 new density matrices of the goal state and calculate 50 fidelity values. The error bar refers to the standard deviation of fidelity. The high fidelity not only reflects the preparation quality of quantum states, but also ensures the credibility of experimental results. Besides, we select four typical quantum states, ${\rho _1} = {\rho _{AB}}{|_{p = 1,\theta = 45^\circ }} = |{\psi ^ + }\rangle \langle {\psi ^ + }|$, ${\rho _2} = {\rho _{AB}}{|_{p = 1/2,\theta = 45^\circ }} = (|{\psi ^ + }\rangle \langle {\psi ^ + }| + |{\phi ^ + }\rangle \langle {\phi ^ + }|)/2$, ${\rho _3} = {\rho _{AB}}{|_{p = 1/2,\theta = 0^\circ }} = (|HH\rangle \langle HH| + |HV\rangle \langle HV|)/2$, and ${\rho _4} = {\rho _{AB}}{|_{p = 0,\theta = 45^\circ }} = |{\phi ^ + }\rangle \langle {\phi ^ + }|$. Note that $|{\psi ^ + }\rangle = (|HV\rangle + |VH\rangle )/\sqrt 2$ and $|{\phi ^ + }\rangle = (|HH\rangle + |VV\rangle )/\sqrt 2$ are two Bell states. The graphical representation tomographic results of their density matrices are graphically shown in Fig. 2.Fig. 2. Graphical representation of the tomographic results for the four states: (a) Re(${\rho _1}$), (b) Re(${\rho _2}$), (c) Re(${\rho _3}$) and (d) Re(${\rho _4}$) represent the real parts of the density matrices. (a$'$) Im(${\rho _1}$), (b$'$) Im(${\rho _2}$), (c$'$) Im(${\rho _3}$) and (d$'$) Im(${\rho _4}$) represent their imaginary parts.
The experimental results and theoretical predictions for the complementary relation between coherence and steerability are shown in Figs. 3 and 4. The $x$ axis in Fig. 3 and Fig. 4 represent the angle parameter $\theta$ and the mixed parameter $p$, respectively. The brown diamonds, purple squares, and orange circulars represent the measured values of ${I_{AB}}$, $D_{AB}^2$ and $F_{AB}^2/2$, respectively. The gray triangles denote the difference between left and right forms of Eq. (6), i.e., $L - R = D_{AB}^2 + F_{AB}^2/2 - {I_{AB}}$. The error bars represent the corresponding standard deviations according to the Poissonian counting statistics. The solid line of the corresponding color represents the theoretical calculation value. For the initial state shown in Eq. (2), we can calculate the corresponding theoretical value as follows,
Fig. 3. The experimental results and theoretical predictions for the complementary relation between coherence and steerability with the input states: (a) ${\rho _{AB}}(\theta ){|_{p = 1}}$, and (b) ${\rho _{AB}}(\theta ){|_{p = 0.5}}$. The $x$ axis represents the angle parameter of the initial Bell-like state. The $y$ axis represents the values of the corresponding measured terms. The brown diamonds, purple squares, and orange circulars represent the measured values of ${I_{AB}}$, $D_{AB}^2$ and $F_{AB}^2/2$, respectively. The gray triangles denote the difference between left and right forms of Eq. (6), i.e., $L - R = D_{AB}^2 + F_{AB}^2/2 - {I_{AB}}$. The solid line of the corresponding color represents the theoretical calculation value.
Fig. 4. The experimental results and theoretical predictions for the input states: (a) ${\rho _{AB}}(p){|_{\theta = 0^\circ }}$, and (b) ${\rho _{AB}}(p){|_{\theta = 45^\circ }}$. The $x$ axis represents the mixed parameter, and the other legends are consistent with Fig. 3. Note that the purple and gray solid lines, the brown and orange solid lines in (b) actually coincide in theoretical calculation.
From these figures, it can be seen that the experimental results well coincide with the theoretical predictions, since the quantum states we prepared have relatively high fidelity. The measured values of the difference between left and right forms of Eq. (6) is approximately zero, so the correctness of the equation is verified experimentally. Note that ${I_{AB}}$ in Fig. 3 is a horizontal line, which shows that the sum of coherence and steerability is conserved. Due to the change of angle parameter of the initial state, the coherence decreases with the increase of steerability and vice versa. We know that the dynamics of a closed quantum system are described by a unitary transform, and the purity of state remains unchanged under unitary transform. Therefore, this conservation relation is also valid under unitary transformation. Since the first-order coherence of Bell diagonal states ${\rho _{AB}}(p){|_{\theta = 45^\circ }}$ is zero, the purple and gray solid lines shown in Fig. 4(b) are actually coincident in theory, and the same is true for the brown and orange solid lines.
5. Conclusions
In conclusion, we have experimentally investigated the relationship between first-order coherence and linear steerability via an all-optical platform. In our experiment, we prepared the initial high fidelity biphoton polarization entangled states which are mixed by two kinds of Bell-like states. We reconstructed the density matrix of quantum states by quantum tomography, and then calculated the purity of quantum states. In addition, we proposed an operable method for experimental measurement of the first-order coherence and linear steerability. It can be seen that our experimental results coincide well with the theoretical predictions. We believe that our experimental results would deepen the understanding of the relationship between first-order coherence and linear steerability, and these experimental methods can be applied to the scene of optical experiments.
Funding
Natural Science Foundation of Anhui Province (2008085QA47, 2108085MA18); National Natural Science Foundation of China (11605028); Natural Science Research Project of Education Department of Anhui Province of China (KJ2019A0531, KJ2020A0527); Excellent Young Talents Fund Program of Higher Education Institutions of Anhui Province (gxyqZD2019042); Research Centre for Quantum Information Technologyf of Fuyang Normal University (kytd201706).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
2. Z. Ficek and S. S. Swain, Quantum Interference and Coherence-Theory and Experiments (Springer Series in Optical Sciences, 2004).
3. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).
4. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2010).
5. A. Streltsov, G. Adesso, and M. B. Plenio, “Colloquium: Quantum coherence as a resource,” Rev. Mod. Phys. 89(4), 041003 (2017). [CrossRef]
6. M.-L. Hu, X. Hu, J. Wang, Y. Peng, Y.-R. Zhang, and H. Fan, “Quantum coherence and geometric quantum discord,” Phys. Rep. 762-764, 1–100 (2018). [CrossRef]
7. A. Winter and D. Yang, “Operational resource theory of coherence,” Phys. Rev. Lett. 116(12), 120404 (2016). [CrossRef]
8. E. Chitambar and G. Gour, “Quantum resource theories,” Rev. Mod. Phys. 91(2), 025001 (2019). [CrossRef]
9. T. Baumgratz, M. Cramer, and M. B. Plenio, “Quantifying coherence,” Phys. Rev. Lett. 113(14), 140401 (2014). [CrossRef]
10. D. Girolami, “Observable measure of quantum coherence in finite dimensional systems,” Phys. Rev. Lett. 113(17), 170401 (2014). [CrossRef]
11. X.-D. Yu, D.-J. Zhang, G. F. Xu, and D. M. Tong, “Alternative framework for quantifying coherence,” Phys. Rev. A 94(6), 060302 (2016). [CrossRef]
12. K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photonics 7(1), 72–78 (2013). [CrossRef]
13. J. Svozilík, A. Vallés, J. Perina Jr., and J. P. Torres, “Revealing hidden coherence in partially coherent light,” Phys. Rev. Lett. 115(22), 220501 (2015). [CrossRef]
14. X.-G. Fan, W.-Y. Sun, Z.-Y. Ding, F. Ming, H. Yang, D. Wang, and L. Ye, “Universal complementarity between coherence and intrinsic concurrence for two-qubit states,” New J. Phys. 21(9), 093053 (2019). [CrossRef]
15. M.-M. Du and D. M. Tong, “Relationship between first-order coherence and the maximum violation of the three-setting linear steering inequality for a two-qubit system,” Phys. Rev. A 103(3), 032407 (2021). [CrossRef]
16. W.-Y. Sun, D. Wang, B.-L. Fang, Z.-Y. Ding, H. Yang, F. Ming, and L. Ye, “Intrinsic relations of bipartite quantum resources in tripartite systems,” Ann. Phys. 531(2), 1800358 (2019). [CrossRef]
17. A. Cernoch, K. Bartkiewicz, K. Lemr, and J. Soubusta, “Experimental tests of coherence and entanglement conservation under unitary evolutions,” Phys. Rev. A 97(4), 042305 (2018). [CrossRef]
18. X.-G. Fan, H. Yang, F. Ming, D. Wang, and L. Ye, “Constraint relation between steerability and concurrence for two-qubit states,” Ann. Phys. 533(8), 2100098 (2021). [CrossRef]
19. M.-L. Hu and H. Fan, “Relative quantum coherence, incompatibility, and quantum correlations of states,” Phys. Rev. A 95(5), 052106 (2017). [CrossRef]
20. M.-L. Hu and H. Fan, “Nonlocal advantage of quantum coherence in high-dimensional states,” Phys. Rev. A 98(2), 022312 (2018). [CrossRef]
21. M.-L. Hu, X.-M. Wang, and H. Fan, “Hierarchy of the nonlocal advantage of quantum coherence and bell nonlocality,” Phys. Rev. A 98(3), 032317 (2018). [CrossRef]
22. M.-L. Hu, Y.-H. Zhang, and H. Fan, “Nonlocal advantage of quantum coherence in a dephasing channel with memory,” Chin. Phys. B 30(3), 030308 (2021). [CrossRef]
23. A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47(10), 777–780 (1935). [CrossRef]
24. E. Schrödinger and M. Born, “Discussion of probability relations between separated systems,” Math. Proc. Cambridge Philos. Soc. 31(4), 555–563 (1935). [CrossRef]
25. R. Uola, A. C. S. Costa, H. Chau Nguyen, and O. Gühne, “Quantum steering,” Rev. Mod. Phys. 92(1), 015001 (2020). [CrossRef]
26. H. M. Wiseman, S. J. Jones, and A. C. Doherty, “Steering, entanglement, nonlocality, and the einstein-podolsky-rosen paradox,” Phys. Rev. Lett. 98(14), 140402 (2007). [CrossRef]
27. R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81(2), 865–942 (2009). [CrossRef]
28. N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, “Bell nonlocality,” Rev. Mod. Phys. 86(2), 419–478 (2014). [CrossRef]
29. E. G. Cavalcanti, S. J. Jones, H. M. Wiseman, and M. D. Reid, “Experimental criteria for steering and the einstein-podolsky-rosen paradox,” Phys. Rev. A 80(3), 032112 (2009). [CrossRef]
30. S. P. Walborn, A. Salles, R. M. Gomes, F. Toscano, and P. H. Souto Ribeiro, “Revealing hidden einstein-podolsky-rosen nonlocality,” Phys. Rev. Lett. 106(13), 130402 (2011). [CrossRef]
31. Q. Y. He and M. D. Reid, “Genuine multipartite einstein-podolsky-rosen steering,” Phys. Rev. Lett. 111(25), 250403 (2013). [CrossRef]
32. A. C. S. Costa and R. M. Angelo, “Quantification of einstein-podolski-rosen steering for two-qubit states,” Phys. Rev. A 93(2), 020103 (2016). [CrossRef]
33. A. C. S. Costa, R. Uola, and O. Gühne, “Steering criteria from general entropic uncertainty relations,” Phys. Rev. A 98(5), 050104 (2018). [CrossRef]
34. T. Kriváchy, F. Fröwis, and N. Brunner, “Tight steering inequalities from generalized entropic uncertainty relations,” Phys. Rev. A 98(6), 062111 (2018). [CrossRef]
35. J. Li, C.-Y. Wang, T.-J. Liu, and Q. Wang, “Experimental verification of steerability via geometric bell-like inequalities,” Phys. Rev. A 97(3), 032107 (2018). [CrossRef]
36. T. Pramanik, Y.-W. Cho, S.-W. Han, S.-Y. Lee, Y.-S. Kim, and S. Moon, “Revealing hidden quantum steerability using local filtering operations,” Phys. Rev. A 99(3), 030101 (2019). [CrossRef]
37. K. Sun, J.-S. Xu, X.-J. Ye, Y.-C. Wu, J.-L. Chen, C.-F. Li, and G.-C. Guo, “Experimental demonstration of the einstein-podolsky-rosen steering game based on the all-versus-nothing proof,” Phys. Rev. Lett. 113(14), 140402 (2014). [CrossRef]
38. C. Zhang, S. Cheng, L. Li, Q.-Y. Liang, B.-H. Liu, Y.-F. Huang, C.-F. Li, G.-C. Guo, M. J. W. Hal, H. M. Wiseman, and G. J. Pryde, “Experimental validation of quantum steering ellipsoids and tests of volume monogamy relations,” Phys. Rev. Lett. 122(7), 070402 (2019). [CrossRef]
39. H. Yang, Z.-Y. Ding, D. Wang, H. Yuan, X.-K. Song, J. Yang, C.-J. Zhang, and L. Ye, “Experimental certification of the steering criterion based on a general entropic uncertainty relation,” Phys. Rev. A 101(2), 022324 (2020). [CrossRef]
40. H. Yang, F. Zhao, X.-G. Fan, Z.-Y. Ding, D. Wang, X.-K. Song, H. Yuan, C.-J. Zhang, and L. Ye, “Estimating quantum steering and bell nonlocality through quantum entanglement in two-photon systems,” Opt. Express 29(17), 26822–26830 (2021). [CrossRef]
41. H. Yang, Z.-Y. Ding, X.-K. Song, H. Yuan, D. Wang, J. Yang, C.-J. Zhang, and L. Ye, “Verification of complementarity relations between quantum steering criteria using an optical system,” Phys. Rev. A 103(2), 022207 (2021). [CrossRef]
42. J.-W. Pan, Z.-B. Chen, C.-Y. Lu, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84(2), 777–838 (2012). [CrossRef]
43. T. Kim, M. Fiorentino, and F. Wong, “Phase-stable source of polarization-entangled photons using a polarization sagnac interferometer,” Phys. Rev. A 73(1), 012316 (2006). [CrossRef]
44. A. Fedrizzi, T. Herbst, A. Poppe, T. Jennewein, and A. Zeilinger, “A wavelength-tunable fiber-coupled source of narrowband entangled photons,” Opt. Express 15(23), 15377–15386 (2007). [CrossRef]
45. Z.-Y. Zhou, S.-L. Liu, Y. Li, D.-S. Ding, W. Zhang, S. Shi, M.-X. Dong, B.-S. Shi, and G.-C. Guo, “Orbital angular momentum-entanglement frequency transducer,” Phys. Rev. Lett. 117(10), 103601 (2016). [CrossRef]
46. W.-H. Zhang, G. Chen, X.-X. Peng, X.-J. Ye, P. Yin, Y. Xiao, Z.-B. Hou, Z.-D. Cheng, Y.-C. Wu, J.-S. Xu, C.-F. Li, and G.-C. Guo, “Experimentally robust self-testing for bipartite and tripartite entangled states,” Phys. Rev. Lett. 121(24), 240402 (2018). [CrossRef]
47. J. Gao, L. F. Qiao, Z. Q. Jiao, Y. C. Ma, C. Q. Hu, R. J. Ren, A. L. Yang, H. Tang, M. H. Yung, and X. M. Jin, “Experimental machine learning of quantum states,” Phys. Rev. Lett. 120(24), 240501 (2018). [CrossRef]
48. Z.-Y. Ding, H. Yang, D. Wang, H. Yuan, J. Yang, and L. Ye, “Experimental investigation of entropic uncertainty relations and coherence uncertainty relations,” Phys. Rev. A 101(3), 032101 (2020). [CrossRef]
49. Z.-Y. Ding, H. Yang, H. Yuan, D. Wang, J. Yang, and L. Ye, “Experimental investigation of linear-entropy-based uncertainty relations in all-optical systems,” Phys. Rev. A 101(2), 022116 (2020). [CrossRef]
50. S. Luo, “Quantum discord for two-qubit systems,” Phys. Rev. A 77(4), 042303 (2008). [CrossRef]
51. D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64(5), 052312 (2001). [CrossRef]
52. J. B. Altepeter, E. R. Jeffrey, and P. G. Kwiat, “Photonic state tomography,” Adv. At., Mol., Opt. Phys. 52, 105–159 (2005). [CrossRef]
53. Y. Xiao, X.-J. Ye, K. Sun, J.-S. Xu, C.-F. Li, and G.-C. Guo, “Demonstration of multisetting one-way einstein-podolsky-rosen steering in two-qubit systems,” Phys. Rev. Lett. 118(14), 140404 (2017). [CrossRef]
