Local quantum Fisher information and quantum correlation in the mixed-spin Heisenberg <i>XXZ</i> chain

doi:10.1007/s11467-023-1336-9

Front. Phys.

2024, Vol. 19

Issue (2) : 21201 https://doi.org/10.1007/s11467-023-1336-9

RESEARCH ARTICLE

Local quantum Fisher information and quantum correlation in the mixed-spin Heisenberg XXZ chain

Peng-Fei Wei¹, Qi Luo¹, Huang-Qiu-Chen Wang¹, Shao-Jie Xiong², Bo Liu³(

), Zhe Sun^1,⁴(

)

¹. School of Physics, Hangzhou Normal University, Hangzhou 310036, China
². Zhejiang Institute of Modern Physics and School of Physics, Zhejiang University, Hangzhou 310027, China
³. School of Information and Electrical Engineering, Hangzhou City University, Hangzhou 310015, China
⁴. Zhejiang Provincial Key Laboratory of Urban Wetlands and Regional Change, Hangzhou 311121, China

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Abstract

We study the local quantum Fisher information (LQFI) in the mixed-spin Heisenberg XXZ chain. Both the maximal and minimal LQFI are studied and the former is essential to determine the accuracy of the quantum parameter estimation, the latter can be well used to characterize the discord-type quantum correlations. We investigate the effects of the temperature and the anisotropy parameter on the maximal LQFI and thus on the accuracy of the parameter estimation. Then we make use of the minimal LQFI to study the discord-type correlations of different site pairs. Different dimensions of the subsystems cause different values of the minimal LQFI which reflects the asymmetry of the discord-type correlation. In addition, the site pairs at different positions of the spin chains have different minimal LQFI, which reveals the influence of the surrounding spins on the bipartite quantum correlation. Our results show that the LQFI obtained through a simple calculation process provides a convenient way to investigate the discord-type correlation in high-dimensional systems.

Keywords local quantum Fisher information quantum correlation mixed-spin Heisenberg XXZ chain

Corresponding Author(s): Bo Liu,Zhe Sun

Issue Date: 27 September 2023

Cite this article:

Peng-Fei Wei,Qi Luo,Huang-Qiu-Chen Wang, et al. Local quantum Fisher information and quantum correlation in the mixed-spin Heisenberg XXZ chain[J]. Front. Phys. , 2024, 19(2): 21201.

URL:

https://academic.hep.com.cn/fop/EN/10.1007/s11467-023-1336-9
https://academic.hep.com.cn/fop/EN/Y2024/V19/I2/21201

Fig.1 For the two-site case of the Hamiltonian in Eq. (9),

Q A max

and

Q B max

vary with temperature

T

. We choose the spin length

s = 1 / 2

for the subsystem

A

and

S = 1

(a, d, g),

S = 3 / 2

(b, e, h), and

S = 2

(c, f, k) for the subsystem

B

. The anisotropy parameters is

J z = 3

(a)−(c),

J z = 2

(d)−(f), and

J z = 1 / 2

(g)−(k).

Fig.2 For the two-site case of the Hamiltonian in Eq. (9) with

T = 0.5

, in subfigures (a−c)

Q A, B max

and in subfigures (d−f) the corresponding eigen energy levels vary with

J z

. The spin length is

s = 1 / 2

for the subsystem

A

and

S = 1

(a, d),

S = 3 / 2

(b, e), and

S = 2

(c, f) for subsystem

B

Fig.3 For the two-site

(1 / 2, 1)

mixed-spin chain,

D Q A → B

in (a),

Q A min

in (b), and

Q A max

in (c) vary with the anisotropy parameter

J z

at different temperature

T = 0.05, 0.2, 0.4, 0.6

Fig.4 For the two-site

(1 / 2, 1)

mixed-spin chain,

D Q A → B

in (a),

Q A min

in (b), and

Q A max

in (c) vary with different temperatures for different parameters

J z

Fig.5 The subgraphs (a, b) show the

Q A, B min

of the site

1 & 2

varies with

J z

for different total site number

n = 2, 4, 6

. In (c, d), the 6-site spin chain is considered and the results show the

Q A, B min

of the sites

1 & 2

3 & 4

, and

5 & 6

varying with

J z

. In (e, f) the 6-site spin chain is considered and the results show the

Q A, B min

of the sites

1 & 2

1 & 4

, and

1 & 6

varies with

J z

. The subgraphs (g−i) show the results of the negativity which is used to describe the entanglement of the site pairs. Here, we choose spin-

1 / 2

for the subsystem

A

and spin-

1

for the subsystem

B

at temperature

T = 0.5

Fig.6 The subgraphs (a, b) show the

Q A, B min

of the sites

1 & 2

3 & 4

, and

5 & 6

varying with

J z

. The subgraphs (c, d) show the

Q A, B min

of the sites

1 & 2

1 & 4

, and

1 & 6

varying with

J z

. Here, we choose spin-

1 / 2

for the subsystem

A

and spin-

1

for the subsystem

B

at temperature

T = 0.5

Fig. A1 In the two-site (

1 / 2

S

) mixed-spin chain,

Q A, B max

versus the parameter

J z

at different temperatures. The subgraphs (a, b, c) correspond to the case of

S = 1

, the subgraphs (d, e, f) correspond to the case of

S = 3 / 2

, and the subgraphs (g, h, i) correspond to the case of

S = 2

Fig. A2 For the two-site case of the Hamiltonian in Eq. (9) with

T = 0.5

Q A max

and

Q B max

vary with

J z

.The spin length is

s = 1

for the subsystem

A

and

S = 3 / 2

(a) and

S = 2

(b) for the subsystem

B

Fig. B1

Q A min

(a, c, e) and

Q B min

(b, d, f) versus the parameter

J z

at the temperature

T = 0.5

in the two-site (

1 / 2

3 / 2

) mixed-spin chain.

Fig. B2

Q A min

(a, c) and

Q B min

(b, d) versus the parameter

J z

at the temperature

T = 0.5

in the two-site (

1 / 2, 2

) mixed-spin chain.

1	Giovannetti V., Lloyd S., Maccone L.. Quantum-enhanced measurements: Beating the standard quantum limit. Science, 2004, 306(5700): 1330 https://doi.org/10.1126/science.1104149
2	Braun D., Adesso G., Benatti F., Floreanini R., Marzolino U., W. Mitchell M., Pirandola S.. Quantum-enhanced measurements without entanglement. Rev. Mod. Phys., 2018, 90(3): 035006 https://doi.org/10.1103/RevModPhys.90.035006
3	Pezzè L., Smerzi A., K. Oberthaler M., Schmied R., Treutlein P.. Quantum metrology with nonclassical states of atomic ensembles. Rev. Mod. Phys., 2018, 90(3): 035005 https://doi.org/10.1103/RevModPhys.90.035005
4	P. Dowling J., P. Seshadreesan K.. Quantum optical technologies for metrology, sensing, and imaging. J. Lightwave Technol., 2015, 33(12): 2359 https://doi.org/10.1109/JLT.2014.2386795
5	Liu J., Zhang M., Chen H., Wang L., Yuan H.. Optimal scheme for quantum metrology. Adv. Quantum Technol., 2022, 5(1): 2100080 https://doi.org/10.1002/qute.202100080
6	Demkowicz-Dobrzański R., Maccone L.. Using entanglement against noise in quantum metrology. Phys. Rev. Lett., 2014, 113(25): 250801 https://doi.org/10.1103/PhysRevLett.113.250801
7	Pezzé L., Smerzi A.. Ultrasensitive two-mode interferometry with single-mode number squeezing. Phys. Rev. Lett., 2013, 110(16): 163604 https://doi.org/10.1103/PhysRevLett.110.163604
8	Schnabel R., Mavalvala N., E. McClelland D., K. Lam P.. Quantum metrology for gravitational wave astronomy. Nat. Commun., 2010, 1(1): 121 https://doi.org/10.1038/ncomms1122
9	D. Huver S., F. Wildfeuer C., P. Dowling J.. Entangled Fock states for robust quantum optical metrology, imaging, and sensing. Phys. Rev. A, 2008, 78(6): 063828 https://doi.org/10.1103/PhysRevA.78.063828
10	Ahmadi M., E. Bruschi D., Sab’ın C., Adesso G., Fuentes I.. Relativistic quantum metrology: Exploiting relativity to improve quantum measurement technologies. Sci. Rep., 2014, 4(1): 4996 https://doi.org/10.1038/srep04996
11	Sun Z., Ma J., M. Lu X., G. Wang X.. Fisher information in a quantum-critical environment. Phys. Rev. A, 2010, 82(2): 022306 https://doi.org/10.1103/PhysRevA.82.022306
12	Zhang M., M. Yu H., D. Yuan H., G. Wang X., Demkowicz-Dobrzański R., Liu J.. QuanEstimation: An open-source toolkit for quantum parameter estimation. Phys. Rev. Res., 2022, 4(4): 043057 https://doi.org/10.1103/PhysRevResearch.4.043057
13	J. Gu S.. Fidelity approach to quantum phase transitions. Int. J. Mod. Phys. B, 2010, 24(23): 4371 https://doi.org/10.1142/S0217979210056335
14	L. Wang T., N. Wu L., Yang W., R. Jin G., Lambert N., Nori F.. Quantum Fisher information as a signature of the superradiant quantum phase transition. New J. Phys., 2014, 16(6): 063039 https://doi.org/10.1088/1367-2630/16/6/063039
15	Marzolino U., Prosen T.. Fisher information approach to non-equilibrium phase transitions in a quantum XXZ spin chain with boundary noise. Phys. Rev. B, 2017, 96(10): 104402 https://doi.org/10.1103/PhysRevB.96.104402
16	Hyllus P., Laskowski W., Krischek R., Schwemmer C., Wieczorek W., Weinfurter H., Pezzé L., Smerzi A.. Fisher information and multiparticle entanglement. Phys. Rev. A, 2012, 85(2): 022321 https://doi.org/10.1103/PhysRevA.85.022321
17	Tóth G.. Multipartite entanglement and high-precision metrology. Phys. Rev. A, 2012, 85(2): 022322 https://doi.org/10.1103/PhysRevA.85.022322
18	M. Lu X., Wang X., P. Sun C.. Quantum Fisher information flow and non-Markovian processes of open systems. Phys. Rev. A, 2010, 82(4): 042103 https://doi.org/10.1103/PhysRevA.82.042103
19	Girolami D., M. Souza A., Giovannetti V., Tufarelli T., G. Filgueiras J., S. Sarthour R., O. Soares-Pinto D., S. Oliveira I., Adesso G.. Quantum discord determines the interferometric power of quantum states. Phys. Rev. Lett., 2014, 112(21): 210401 https://doi.org/10.1103/PhysRevLett.112.210401
20	S. Dhar H., N. Bera M., Adesso G.. Characterizing non-Markovianity via quantum interferometric power. Phys. Rev. A, 2015, 91(3): 032115 https://doi.org/10.1103/PhysRevA.91.032115
21	P. Chen L., N. Guo Y.. Dynamics of local quantum uncertainty and local quantum fisher information for a two-qubit system driven by classical phase noisy laser. J. Mod. Opt., 2021, 68(4): 217 https://doi.org/10.1080/09500340.2021.1887949
22	B. A. Mohamed A., Eleuch H.. Dynamics of two magnons coupled to an open microwave cavity: Local quantum Fisher- and local skew-information coherence. Eur. Phys. J. Plus, 2022, 137(7): 853 https://doi.org/10.1140/epjp/s13360-022-03042-6
23	Slaoui A.Bakmou L.Daoud M.Ahl Laamara R., A comparative study of local quantum Fisher information and local quantum uncertainty in Heisenberg XY model, Phys. Lett. A 383(19), 2241 (2019)
24	Habiballah N., Khedif Y., Daoud M.. Local quantum uncertainty in XYZ Heisenberg spin models with Dzyaloshinski‒Moriya interaction. Eur. Phys. J. D, 2018, 72(9): 154 https://doi.org/10.1140/epjd/e2018-90255-y
25	Ozaydin F., A. Altintas A.. Parameter estimation with Dzyaloshinski‒Moriya interaction under external magnetic fields. Opt. Quantum Electron., 2020, 52(2): 70 https://doi.org/10.1007/s11082-019-2185-1
26	Ozaydin F., A. Altintas A.. Quantum metrology: Surpassing the shot-noise limit with Dzyaloshinskii‒Moriya interaction. Sci. Rep., 2015, 5(1): 16360 https://doi.org/10.1038/srep16360
27	Haseli S.. Local quantum Fisher information and local quantum uncertainty in two-qubit Heisenberg XYZ chain with Dzyaloshinskii‒Moriya interactions. Laser Phys., 2020, 30(10): 105203 https://doi.org/10.1088/1555-6611/abac65
28	V. Fedorova A., A. Yurischev M.. Behavior of quantum discord, local quantum uncertainty, and local quantum Fisher information in two-spin-1/2 Heisenberg chain with DM and KSEA interactions. Quantum Inform. Process., 2022, 21(3): 92 https://doi.org/10.1007/s11128-022-03427-7
29	Liu J., X. Jing X., Zhong W., G. Wang X.. Quantum Fisher information for density matrices with arbitrary ranks. Commum. Theor. Phys., 2014, 61(1): 45 https://doi.org/10.1088/0253-6102/61/1/08
30	A. Nielsen M.L. Chuang I., Quantum Computation and Quantum Information, Cambridge: Cambridge University Press, 2000
31	Ollivier H., H. Zurek W.. Quantum discord: A measure of the quantumness of correlations. Phys. Rev. Lett., 2001, 88(1): 017901 https://doi.org/10.1103/PhysRevLett.88.017901
32	Groisman B., Popescu S., Winter A.. Quantum, classical, and total amount of correlations in a quantum state. Phys. Rev. A, 2005, 72(3): 032317 https://doi.org/10.1103/PhysRevA.72.032317
33	Dakić B., Vedral V., Brukner Č.. Necessary and sufficient condition for nonzero quantum discord. Phys. Rev. Lett., 2010, 105(19): 190502 https://doi.org/10.1103/PhysRevLett.105.190502
34	M. Lu X.J. Xi Z.Sun Z.G. Wang X., Geometric measure of quantum discord under decoherence, Quantum Inf. Comput. 10(11–12), 0994 (2010)
35	Werlang T., Souza S., F. Fanchini F., J. Villas Boas C.. Robustness of quantum discord to sudden death. Phys. Rev. A, 2009, 80(2): 024103 https://doi.org/10.1103/PhysRevA.80.024103
36	Datta A., Shaji A., M. Caves C.. Quantum discord and the power of one qubit. Phys. Rev. Lett., 2008, 100(5): 050502 https://doi.org/10.1103/PhysRevLett.100.050502
37	L. Luo S., S. Fu S.. Geometric measure of quantum discord. Phys. Rev. A, 2010, 82(3): 034302 https://doi.org/10.1103/PhysRevA.82.034302
38	Henderson L., Vedral V.. Classical, quantum and total correlations. J. Phys. Math. Gen., 2001, 34(35): 6899 https://doi.org/10.1088/0305-4470/34/35/315
39	X. Chen Y., Yin Z.. Thermal quantum discord in anisotropic Heisenberg XXZ model with Dzyaloshinskii‒Moriya interaction. Commum. Theor. Phys., 2010, 54(1): 60 https://doi.org/10.1088/0253-6102/54/1/12
40	Chen Q., Zhang C., Yu S., X. Yi X., H. Oh C.. Quantum discord of two-qubit X states. Phys. Rev. A, 2011, 84(4): 042313 https://doi.org/10.1103/PhysRevA.84.042313
41	Ali M., R. P. Rau A., Alber G.. Quantum discord for two-qubit X states. Phys. Rev. A, 2010, 81(4): 042105 https://doi.org/10.1103/PhysRevA.81.042105
42	L. Braunstein S., M. Caves C.. Statistical distance and the geometry of quantum states. Phys. Rev. Lett., 1994, 72(22): 3439 https://doi.org/10.1103/PhysRevLett.72.3439
43	Vidal G., F. Werner R.. Computable measure of entanglement. Phys. Rev. A, 2002, 65(3): 032314 https://doi.org/10.1103/PhysRevA.65.032314
44	Sun Z., G. Wang X., Z. Hu A., Q. Li Y.. Entanglement properties in mixed-spin Heisenberg systems. Physica A, 2006, 370(2): 483 https://doi.org/10.1016/j.physa.2006.03.020

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