Inequality relations for the hierarchy of quantum correlations in two-qubit systems

doi:10.1007/s11467-022-1222-x

Front. Phys.

2023, Vol. 18

Issue (1) : 11301 https://doi.org/10.1007/s11467-022-1222-x

RESEARCH ARTICLE

Inequality relations for the hierarchy of quantum correlations in two-qubit systems

Xiao-Gang Fan¹, Fa Zhao¹, Huan Yang², Fei Ming³, Dong Wang¹, Liu Ye¹(

)

¹. School of Physics and optoelectronics engineering, Anhui University, Hefei 230601, China
². Department of Experiment and Practical Training Management, West Anhui University, Lu’an 237012, China
³. Institute of Advanced Manufacturing Engineering, Hefei University, Hefei 230022, China

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Abstract

Entanglement, quantum steering and Bell nonlocality can be used to describe the distinct quantum correlations of quantum systems. Because of their different characteristics and application fields, how to divide them quantitatively and accurately becomes particularly important. Based on the sufficient and necessary criterion for quantum steering of an arbitrary two-qubit T-state, we derive the inequality relations between quantum steering and entanglement as well as between quantum steering and Bell nonlocality for the T-state. Additionally, we have verified those relations experimentally.

Keywords entanglement quantum steering Bell nonlocality inequality relation

Corresponding Author(s): Liu Ye

Issue Date: 08 February 2023

Cite this article:

Xiao-Gang Fan,Fa Zhao,Huan Yang, et al. Inequality relations for the hierarchy of quantum correlations in two-qubit systems[J]. Front. Phys. , 2023, 18(1): 11301.

URL:

https://academic.hep.com.cn/fop/EN/10.1007/s11467-022-1222-x
https://academic.hep.com.cn/fop/EN/Y2023/V18/I1/11301

Fig.1 A diagrammatic sketch of experimental setup: (a) source stage, (b) unbalanced Mach−Zehnder device stage and (c) tomography stage. OM is short for optical maser, HWP for half-wave plate, PBS for polarizing beam splitter, BS for beam splitter, BBO for type-I

β

-barium borate crystal, RP for reflecting prism, ATT for attenuator, QWP for quarter wave plater, IF for 3-nm interference filter and SPD for single photon detector.

	$q =$ 0	$q =$ 0.1	$q =$ 0.2

$p = 0$	$0.9976 ± 0.0005$	$0.9979 ± 0.0018$	$0.9984 ± 0.0014$
$p = 0.05$	$0.9970 ± 0.0001$	$0.9970 ± 0.0010$	$0.9993 ± 0.0005$
$p = 0.1$	$0.9971 ± 0.0002$	$0.9969 ± 0.0021$	$0.9996 ± 0.0002$
$p = 0.15$	$0.9970 ± 0.0001$	$0.9994 ± 0.0003$	$0.9990 ± 0.0003$
$p = 0.2$	$0.9966 ± 0.0002$	$0.9986 ± 0.0007$	$0.9994 ± 0.0003$
	$q =$ 0.3	$q =$ 0.4	$q =$ 0.5

$p = 0$	$0.9986 ± 0.0013$	$0.9992 ± 0.0007$	$0.9988 ± 0.0008$
$p = 0.05$	$0.9976 ± 0.0009$	$0.9968 ± 0.0026$	$0.9988 ± 0.0008$
$p = 0.1$	$0.9987 ± 0.0004$	$0.9953 ± 0.0017$	$0.9986 ± 0.0006$
$p = 0.15$	$0.9987 ± 0.0004$	$0.9987 ± 0.0006$	$0.9990 ± 0.0004$
$p = 0.2$	$0.9980 ± 0.0005$	$0.9990 ± 0.0003$	$0.9993 ± 0.0002$

Tab.1 The fidelities of experimental prepared states.

Fig.2 EPR steering

F (ρ)

versus concurrence

E (ρ)

for two-qubit T-states

ρ

. The upper bound (red solid line) can be achieved by the 2-rank T-states. And the lower bound (blue solid line) can be achieved by the Werner states. EPR steering

F (ρ)

along the Y axis, and concurrence

E (ρ)

along the X axis for

5 × 104

randomly generated two-qubit T-states, where we show them with gray dots, by using a specific Mathematica package. The green dots are obtained by the experimental process with a set of Bell-diagonal states.

Fig.3 Bell nonlocality

G (ρ)

versus EPR steering

F (ρ)

for two-qubit T-states

ρ

. The upper bound (red solid line) can be achieved by the 2-rank T-states. The lower bound (blue solid line) can be achieved by the Werner states. Bell nonlocality

G (ρ)

along the Y axis, and EPR steering

F (ρ)

along the X axis for

5 × 104

1	Einstein A., Podolsky B., Rosen N.. Can quantum-mechanical description of physical reality be considered complete. Phys. Rev., 1935, 47(10): 777 https://doi.org/10.1103/PhysRev.47.777
2	Schrödinger E.. Discussion of relations between separated systems. Math. Proc. Camb. Philos. Soc., 1935, 31(4): 555 https://doi.org/10.1017/S0305004100013554
3	Horodecki R., Horodecki P., Horodecki M., Horodecki K.. Quantum entanglement. Rev. Mod. Phys., 2009, 81(2): 865 https://doi.org/10.1103/RevModPhys.81.865
4	K. Wootters W.. Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett., 1998, 80(10): 2245 https://doi.org/10.1103/PhysRevLett.80.2245
5	Piani M.. Relative entropy of entanglement and restricted measurements. Phys. Rev. Lett., 2009, 103(16): 160504 https://doi.org/10.1103/PhysRevLett.103.160504
6	Miranowicz A., Grudka A.. Ordering two-qubit states with concurrence and negativity. Phys. Rev. A, 2004, 70(3): 032326 https://doi.org/10.1103/PhysRevA.70.032326
7	Ekert A., Jozsa R.. Quantum computation and Shor’s factoring algorithm. Rev. Mod. Phys., 1996, 68(3): 733 https://doi.org/10.1103/RevModPhys.68.733
8	H. Bennett C., Brassard G., Crépeau C., Jozsa R., Peres A., K. Wootters W.. Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Phys. Rev. Lett., 1993, 70(13): 1895 https://doi.org/10.1103/PhysRevLett.70.1895
9	Gisin N., Ribordy G., Tittel W., Zbinden H.. Quantum cryptography. Rev. Mod. Phys., 2002, 74(1): 145 https://doi.org/10.1103/RevModPhys.74.145
10	H. Bennett C., P. DiVincenzo D., Smolin J., K. Wootters W.. Mixed-state entanglement and quantum error correction. Phys. Rev. A, 1996, 54(5): 3824 https://doi.org/10.1103/PhysRevA.54.3824
11	M. Wiseman H., J. Jones S., C. Doherty A.. Steering, entanglement, nonlocality, and the Einstein−Podolsky−Rosen paradox. Phys. Rev. Lett., 2007, 98(14): 140402 https://doi.org/10.1103/PhysRevLett.98.140402
12	Uola R., C. S. Costa A., C. Nguyen H., Gühne O.. Quantum steering. Rev. Mod. Phys., 2020, 92(1): 015001 https://doi.org/10.1103/RevModPhys.92.015001
13	S. Bell J.. On the Einstein−Podolsky−Rosen paradox. Physics, 1964, 1(3): 195 https://doi.org/10.1103/PhysicsPhysiqueFizika.1.195
14	Brunner N., Cavalcanti D., Pironio S., Scarani V., Wehner S.. Bell nonlocality. Rev. Mod. Phys., 2014, 86(2): 419 https://doi.org/10.1103/RevModPhys.86.419
15	X. Zhong W., L. Cheng G., M. Hu X.. One-way Einstein−Podolsky−Rosen steering via atomic coherence. Opt. Express, 2017, 25(10): 11584 https://doi.org/10.1364/OE.25.011584
16	Branciard C., G. Cavalcanti E., P. Walborn S., Scarani V., M. Wiseman H.. One-sided device-independent quantum key distribution: Security, feasibility, and the connection with steering. Phys. Rev. A, 2012, 85(1): 010301(R) https://doi.org/10.1103/PhysRevA.85.010301
17	Opanchuk B., Arnaud L., D. Reid M.. Detecting faked continuous-variable entanglement using one-sided device independent entanglement witnesses. Phys. Rev. A, 2014, 89(6): 062101 https://doi.org/10.1103/PhysRevA.89.062101
18	Walk N., Hosseini S., Geng J., Thearle O., Y. Haw J., Armstrong S., M. Assad S., Janousek J., C. Ralph T., Symul T., M. Wiseman H., K. Lam P.. Experimental demonstration of Gaussian protocols for one-sided device-independent quantum key distribution. Optica, 2016, 3(6): 634 https://doi.org/10.1364/OPTICA.3.000634
19	M. Zhang C., Li M., W. Li H., Q. Yin Z., Wang D., Z. Huang J., G. Han Y., L. Xu M., Chen W., Wang S., Treeviriyanupab P., C. Guo G., F. Han Z.. Decoy-state measurement-device independent quantum key distribution based on the Clauser−Horne−Shimony−Holt inequality. Phys. Rev. A, 2014, 90(3): 034302 https://doi.org/10.1103/PhysRevA.90.034302
20	Brukner Č., Żukowski M., W. Pan J., Zeilinger A.. Bell’s inequalities and quantum communication complexity. Phys. Rev. Lett., 2004, 92(12): 127901 https://doi.org/10.1103/PhysRevLett.92.127901
21	Pironio S., Acín A., Massar S., B. de la Giroday A., N. Matsukevich D., Maunz P., Olmschenk S., Hayes D., Luo L., A. Manning T., Monroe C.. Random numbers certified by Bell’s theorem. Nature, 2010, 464(7291): 1021 https://doi.org/10.1038/nature09008
22	T. Quintino M., Vertesi T., Cavalcanti D., Augusiak R., Demianowicz M., Acín A., Brunner N.. Inequivalence of entanglement, steering, and Bell nonlocality for general measurements. Phys. Rev. A, 2015, 92(3): 032107 https://doi.org/10.1103/PhysRevA.92.032107
23	G. Fan X., Yang H., Ming F., Wang D., Ye L.. Constraint relation between steerability and concurrence for two-qubit states. Ann. Phys., 2021, 533(8): 2100098 https://doi.org/10.1002/andp.202100098
24	Chen C., L. Ren C., J. Ye X., L. Chen J.. Mapping criteria between nonlocality and steerability in qudit−qubit systems and between steerability and entanglement in qubit-qudit systems. Phys. Rev. A, 2018, 98(5): 052114 https://doi.org/10.1103/PhysRevA.98.052114
25	Das D., Sasmal S., Roy S.. Detecting Einstein−Podolsky−Rosen steering through entanglement detection. Phys. Rev. A, 2019, 99(5): 052109 https://doi.org/10.1103/PhysRevA.99.052109
26	Verstraete F., M. Wolf M.. Entanglement versus bell violations and their behavior under local filtering operations. Phys. Rev. Lett., 2002, 89(17): 170401 https://doi.org/10.1103/PhysRevLett.89.170401
27	Bartkiewicz K., Horst B., Lemr K., Miranowicz A.. Entanglement estimation from Bell inequality violation. Phys. Rev. A, 2013, 88(5): 052105 https://doi.org/10.1103/PhysRevA.88.052105
28	Horst B., Bartkiewicz K., Miranowicz A.. Two-qubit mixed states more entangled than pure states: Comparison of the relative entropy of entanglement for a given nonlocality. Phys. Rev. A, 2013, 87(4): 042108 https://doi.org/10.1103/PhysRevA.87.042108
29	Bartkiewicz K., Lemr K., Černoch A., Miranowicz A.. Bell nonlocality and fully entangled fraction measured in an entanglement-swapping device without quantum state tomography. Phys. Rev. A, 2017, 95(3): 030102(R) https://doi.org/10.1103/PhysRevA.95.030102
30	F. Su Z., S. Tan H., Y. Li X.. Entanglement as upper bound for the nonlocality of a general two-qubit system. Phys. Rev. A, 2020, 101(4): 042112 https://doi.org/10.1103/PhysRevA.101.042112
31	Li M., J. Zhao M., M. Fei S., X. Wang Z.. Experimental detection of quantum entanglement. Front. Phys., 2013, 8(4): 357 https://doi.org/10.1007/s11467-013-0355-3
32	R. Zhong Z., Wang X., Qin W.. Towards quantum entanglement of micromirrors via a two-level atom and radiation pressure. Front. Phys., 2018, 13(5): 130319 https://doi.org/10.1007/s11467-018-0824-9
33	Dong Q., J. Torres-Arenas A., H. Sun G., C. Qiang W., H. Dong S.. Entanglement measures of a new type pseudo-pure state in accelerated frames. Front. Phys., 2019, 14(2): 21603 https://doi.org/10.1007/s11467-018-0876-x
34	Zhang P.. Quantum entanglement in the Sachdev−Ye−Kitaev model and its generalizations. Front. Phys., 2022, 17(4): 43201 https://doi.org/10.1007/s11467-022-1162-5
35	Y. Yang Y., Y. Sun W., N. Shi W., Ming F., Wang D., Ye L.. Dynamical characteristic of measurement uncertainty under Heisenberg spin models with Dzyaloshinskii−Moriya interactions. Front. Phys., 2019, 14(3): 31601 https://doi.org/10.1007/s11467-018-0880-1
36	Y. Cheng L., Ming F., Zhao F., Ye L., Wang D.. The uncertainty and quantum correlation of measurement in double quantum-dot systems. Front. Phys., 2022, 17(6): 61504 https://doi.org/10.1007/s11467-022-1178-x
37	Cao Y., Wang D., G. Fan X., Ming F., Y. Wang Z., Ye L.. Complementary relation between quantum entanglement and entropic uncertainty. Commum. Theor. Phys., 2021, 73(1): 015101 https://doi.org/10.1088/1572-9494/abc46f
38	C. Nguyen H., Vu T.. Necessary and sufficient condition for steerability of two-qubit states by the geometry of steering outcomes. Europhys. Lett., 2016, 115(1): 10003 https://doi.org/10.1209/0295-5075/115/10003
39	C. Nguyen H., V. Nguyen H., Gühne O.. Geometry of Einstein−Podolsky−Rosen correlations. Phys. Rev. Lett., 2019, 122(24): 240401 https://doi.org/10.1103/PhysRevLett.122.240401
40	G. Fan X., Y. Sun W., Y. Ding Z., Yang H., Ming F., Wang D., Ye L.. Universal complementarity between coherence and intrinsic concurrence for two-qubit states. New J. Phys., 2019, 21(9): 093053 https://doi.org/10.1088/1367-2630/ab41b1
41	Jevtic S., J. W. Hall M., R. Anderson M., Zwierz M., M. Wiseman H.. Einstein−Podolsky−Rosen steering and the steering ellipsoid. J. Opt. Soc. Am. B, 2015, 32(4): A40 https://doi.org/10.1364/JOSAB.32.000A40
42	G. Fan X.Yang H.Ming F.K. Song X.Wang D. Ye L., Necessary and sufficient criterion of steering for two-qubit T states, arXiv: 2103.04280v1 (2021)
43	Horodecki R., Horodecki P., Horodecki M.. Violating bell inequality by mixed spin-1/2 states: Necessary and sufficient condition. Phys. Lett. A, 1995, 200(5): 340 https://doi.org/10.1016/0375-9601(95)00214-N
44	G. Kwiat P., Waks E., G. White A., Appelbaum I., H. Eberhard P.. Ultrabright source of polarization-entangled photons. Phys. Rev. A, 1999, 60(2): R773 https://doi.org/10.1103/PhysRevA.60.R773
45	Aiello A., Puentes G., Voigt D., P. Woerdman J.. Maximally entangled mixed-state generation via local operations. Phys. Rev. A, 2007, 75(6): 062118 https://doi.org/10.1103/PhysRevA.75.062118

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