ISSN 0253-2778

CN 34-1054/N

Open AccessOpen Access JUSTC Physics 11 May 2022

Experimental investigation of quantum discord in DQC1

Cite this:
https://doi.org/10.52396/JUSTC-2021-0267
More Information
  • Author Bio:

    Tingwei Li is a postgraduate student at the University of Science and Technology of China. His research focuses on quantum information and quantum computation

    Fangzhou Jin is a teacher at Wuchang Shouyi University. He received his PhD from the University of Science and Technology of China in 2016. His research interests includes quantum control and quantum simulation of spin system

    Xing Rong is a Professor at the University of Science and Technology of China (USTC). He received a BS in 2005 and a PhD in 2011 from USTC. In 2014, he joined the Department of Modern Physics of USTC as a Professor. His research interest includes quantum control of spins in solids, quantum computation, and exploring new physics beyond the standard model with micrometer scale

  • Corresponding author: E-mail: fzjin@wsyu.edu.cn; E-mail: xrong@ustc.edu.cn
  • Received Date: 17 November 2021
  • Accepted Date: 09 February 2022
  • Available Online: 11 May 2022
  • Quantum discord has been proposed as a resource responsible for the exponential speedup in deterministic quantum computation with one pure qubit (DQC1). Investigation of the quantum discord generated in DQC1 is of significant importance from a fundamental perspective. However, in practical applications of DQC1, qubits generally interact with the environment. Thus, it is also important to investigate the discord when DQC1 is implemented in a noisy environment. We implement DQC1 on an electron spin resonance (ESR) architecture in such an environment and nonzero quantum discord is observed. Furthermore, we find that the values of discord correspond to the values of purity α and quantum Fisher information, which reflect the power of the algorithm. Our results provide further evidence for the role of discord as a resource in DQC1 and are beneficial for understanding the origin of the power of quantum algorithms.

      Quantum circuit of DQC1 algorithm and experimental results.

    Quantum discord has been proposed as a resource responsible for the exponential speedup in deterministic quantum computation with one pure qubit (DQC1). Investigation of the quantum discord generated in DQC1 is of significant importance from a fundamental perspective. However, in practical applications of DQC1, qubits generally interact with the environment. Thus, it is also important to investigate the discord when DQC1 is implemented in a noisy environment. We implement DQC1 on an electron spin resonance (ESR) architecture in such an environment and nonzero quantum discord is observed. Furthermore, we find that the values of discord correspond to the values of purity α and quantum Fisher information, which reflect the power of the algorithm. Our results provide further evidence for the role of discord as a resource in DQC1 and are beneficial for understanding the origin of the power of quantum algorithms.

    • DQC1 algorithm is implemented in a solid system in a real noisy environment.
    • Nonvanishing quantum discord is observed.
    • Quantum discord is responsible for the power of DQC1.

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  • [1]
    Shor P. Proceedings of the 35th Annual Symposium on the Foundations of Computer Science. Washington, D.C.: IEEE, 1994.
    [2]
    Galindo A, Martin-Delgado M A. Information and computation: Classical and quantum aspects. Reviews of Modern Physics, 2002, 74 (2): 347. doi: 10.1103/RevModPhys.74.347
    [3]
    Goettems E I, Maciel T O, Soares-Pinto D O, et al. Promoting quantum correlations in deterministic quantum computation with a one-qubit model via postselection. Physical Review A, 2021, 103 (4): 042416. doi: 10.1103/PhysRevA.103.042416
    [4]
    Göktaş O, Tham W K, Bonsma-Fisher K, et al. Benchmarking quantum processors with a single qubit. Quantum Information Processing, 2020, 19: 146. doi: 10.1007/s11128-020-02642-4
    [5]
    Zhang K, Thompson J, Zhang X, et al. Modular quantum computation in a trapped ion system. Nature Communications, 2019, 10 (1): 1–6. doi: 10.1038/s41467-018-07882-8
    [6]
    Krzyzanowska K, Copley-May M, Romain R, et al. Quantum-enhanced protocols with mixed states using cold atoms in dipole traps. Journal of Physics: Conference Series, 2017, 793 (1): 012015. doi: 10.1088/1742-6596/793/1/012015
    [7]
    Pg S, Varikuti N D, Madhok V. Exponential speedup in measuring out-of-time-ordered correlators and gate fidelity with a single bit of quantum information. Physics Letters A, 2021, 397: 127257. doi: 10.1016/j.physleta.2021.127257
    [8]
    Knill E, Laflamme R. Power of one bit of quantum information. Physical Review Letters, 1998, 81 (25): 5672. doi: 10.1103/PhysRevLett.81.5672
    [9]
    Jozsa R, Linden N. On the role of entanglement in quantum-computational speed-up. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 2003, 459 (2036): 2011–2032. doi: 10.1098/rspa.2002.1097
    [10]
    Horodecki R, Horodecki P, Horodecki M, et al. Quantum entanglement. Reviews of Modern Physics, 2009, 81 (2): 865–942. doi: 10.1103/RevModPhys.81.865
    [11]
    Datta A, Flammia S T, Caves C M. Entanglement and the power of one qubit. Physical Review A, 2005, 72 (4): 042316. doi: 10.1103/PhysRevA.72.042316
    [12]
    Boyer M, Brodutch A, Mor T. Entanglement and deterministic quantum computing with one qubit. Physical Review A, 2017, 95 (2): 022330. doi: 10.1103/PhysRevA.95.022330
    [13]
    Bennett C H, DiVincenzo D P, Fuchs C A, et al. Quantum nonlocality without entanglement. Physical Review A, 1999, 59 (2): 1070. doi: 10.1103/PhysRevA.59.1070
    [14]
    Modi K, Brodutch A, Cable H, et al. The classical-quantum boundary for correlations: Discord and related measures. Reviews of Modern Physics, 2012, 84 (4): 1655. doi: 10.1103/RevModPhys.84.1655
    [15]
    Hu M L, Hu X, Wang J, et al. Quantum coherence and geometric quantum discord. Physics Reports, 2018, 762: 1–100. doi: 10.1016/j.physrep.2018.07.004
    [16]
    Bera A, Das T, Sadhukhan D, et al. Quantum discord and its allies: A review of recent progress. Reports on Progress in Physics, 2017, 81 (2): 024001. doi: 10.1088/1361-6633/aa872f
    [17]
    Datta A, Shaji A, Caves C M. Quantum discord and the power of one qubit. Physical Review Letters, 2008, 100 (5): 050502. doi: 10.1103/PhysRevLett.100.050502
    [18]
    Madsen L S, Berni A, Lassen M, et al. Experimental investigation of the evolution of Gaussian quantum discord in an open system. Physical Review Letters, 2012, 109 (3): 030402. doi: 10.1103/PhysRevLett.109.030402
    [19]
    Radhakrishnan C, Lauriere M, Byrnes T. Multipartite generalization of quantum discord. Physical Review Letters, 2020, 124 (11): 110401. doi: 10.1103/PhysRevLett.124.110401
    [20]
    Hunt M A, Lerner I V, Yurkevich I V, et al. How to observe and quantify quantum-discord states via correlations. Physical Review A, 2019, 100 (2): 022321. doi: 10.1103/PhysRevA.100.022321
    [21]
    Faba J, Martín V, Robledo L. Two-orbital quantum discord in fermion systems. Physical Review A, 2021, 103 (3): 032426. doi: 10.1103/PhysRevA.103.032426
    [22]
    Gu M, Chrzanowski H M, Assad S M, et al. Observing the operational significance of discord consumption. Nature Physics, 2012, 8 (9): 671–675. doi: 10.1038/nphys2376
    [23]
    Dakić B, Lipp Y O, Ma X, et al. Quantum discord as resource for remote state preparation. Nature Physics, 2012, 8 (9): 666–670. doi: 10.1038/nphys2377
    [24]
    Lanyon B P, Barbieri M, Almeida M P, et al. Experimental quantum computing without entanglement. Physical Review Letters, 2008, 101 (20): 200501. doi: 10.1103/PhysRevLett.101.200501
    [25]
    Passante G, Moussa O, Trottier D A, et al. Experimental detection of nonclassical correlations in mixed-state quantum computation. Physical Review A, 2011, 84 (4): 044302. doi: 10.1103/PhysRevA.84.044302
    [26]
    Hor-Meyll M, Tasca D S, Walborn S P, et al. Deterministic quantum computation with one photonic qubit. Physical Review A, 2015, 92 (1): 012337. doi: 10.1103/PhysRevA.92.012337
    [27]
    Wang W, Han J, Yadin B, et al. Witnessing quantum resource conversion within deterministic quantum computation using one pure superconducting qubit. Physical Review Letters, 2019, 123 (22): 220501. doi: 10.1103/PhysRevLett.123.220501
    [28]
    Ferraro A, Aolita L, Cavalcanti D, et al. Almost all quantum states have nonclassical correlations. Physical Review A, 2010, 81 (5): 052318. doi: 10.1103/PhysRevA.81.052318
    [29]
    Cable H, Gu M, Modi K. Power of one bit of quantum information in quantum metrology. Physical Review A, 2016, 93 (4): 040304. doi: 10.1103/PhysRevA.93.040304
    [30]
    Dorner R, Clark S R, Heaney L, et al. Extracting quantum work statistics and fluctuation theorems by single-qubit interferometry. Physical Review Letters, 2013, 110 (23): 230601. doi: 10.1103/PhysRevLett.110.230601
    [31]
    Ryan C A, Emerson J, Poulin D, et al. Characterization of complex quantum dynamics with a scalable NMR information processor. Physical Review Letters, 2005, 95 (25): 250502. doi: 10.1103/PhysRevLett.95.250502
    [32]
    Passante G, Moussa O, Ryan C A, et al. Experimental approximation of the Jones polynomial with one quantum bit. Physical Review Letters, 2009, 103 (25): 250501. doi: 10.1103/PhysRevLett.103.250501
    [33]
    Helstrom C W. Minimum mean-squared error of estimates in quantum statistics. Physics Letters A, 1967, 25 (2): 101–102. doi: 10.1016/0375-9601(67)90366-0
    [34]
    Braunstein S L, Caves C M. Statistical distance and the geometry of quantum states. Physical Review Letters, 1994, 72 (22): 3439. doi: 10.1103/PhysRevLett.72.3439
    [35]
    Barndorff-Nielsen O E, Gill R D. Fisher information in quantum statistics. Journal of Physics A: Mathematical and General, 2000, 33 (24): 4481. doi: 10.1088/0305-4470/33/24/306
    [36]
    Weeks R A. Paramagnetic resonance of lattice defects in irradiated quartz. Journal of Applied Physics, 1956, 27 (11): 1376–1381. doi: 10.1063/1.1722267
    [37]
    Weeks R A, Nelson C M. Trapped electrons in irradiated quartz and silica: II, electron spin resonance. Journal of the American Ceramic Society, 1960, 43 (8): 399–404. doi: 10.1111/j.1151-2916.1960.tb13682.x
    [38]
    Weeks R A. Paramagnetic spectra of E2' centers in crystalline quartz. Physical Review, 1963, 130 (2): 570. doi: 10.1103/PhysRev.130.570
    [39]
    Rudra J K, Fowler W B, Feigl F J. Model for the E2' center in alpha quartz. Physical Review Letters, 1985, 55 (23): 2614. doi: 10.1103/PhysRevLett.55.2614
    [40]
    Perlson B D, Weil J A. Electron paramagnetic resonance studies of the E' centers in alpha-quartz. Canadian Journal of Physics, 2008, 86 (7): 871–881. doi: 10.1139/p08-034
    [41]
    Feng P, Wang Y, Rong X, et al. Characterization of the electronic structure of E2' defect in quartz by pulsed EPR spectroscopy. Physics Letters A, 2012, 376 (32): 2195–2199. doi: 10.1016/j.physleta.2012.05.029
    [42]
    Vandersypen L M K, Chuang I L. NMR techniques for quantum control and computation. Reviews of Modern Physics, 2005, 76 (4): 1037. doi: 10.1103/RevModPhys.76.1037
    [43]
    Braunstein S L, Caves C M, Milburn G J. Generalized uncertainty relations: Theory, examples, and Lorentz invariance. Annals of Physics, 1996, 247 (1): 135–173. doi: 10.1006/aphy.1996.0040
  • 加载中

Catalog

    Figure  1.  DQC1 algorithm to estimate the normalized trace of $ {U_n}^2 $. $ I_n $ is the n-qubit identity, $ \alpha $ reflects the degree of purity of control qubit $ c $, and $ Z $ denotes the Pauli operator $ Z $. The normalized trace ${\rm{Tr}}({U_n}^2)/2^n$ can be derived from the expectation values of the output of qubit $ c $ measured on the Pauli $ X $ and $ Y $ bases.

    Figure  2.  (a) Continuous wave (${\rm{CW}}$) ESR spectrum of the $ E_2' $ center in $ \alpha $-quartz recorded at room temperature with the optical axis of the quartz oriented along the magnetic field. The split of the two peaks indicates the hyperfine coupling of the electron and nuclear spins. (b) Output of algorithm as evolution time varies for temperature $T = 80\;{\rm K}$. Temperature $ T $ and evolution time $ t $ represent $ \alpha $ and $ \theta $, respectively, as described in the text. The output value is not normalized. The area with time $ t<0 $ is related to the single-qubit gate on the control qubit. The output near $ t = 0 $ is affected by the protection pulse, and thus it differs from the actual value of the output. The decay of the output originates from the interaction with the environment during the evolution of the system, as described in the text.

    Figure  3.  Quantum discord generated for: (a)(c) $T = 20\;{\rm{K}}$ ($ \alpha = 1.16\times10^{-2} $), and (b)(d) $T = 80\;{\rm{K}}$ ($ \alpha = 2.89\times10^{-3} $). In (a) and (b), the experimental data (black square points) are practically detected, quantum discord is calculated with the original reconstructed density matrices of the output state, and the theoretical predictions (red curves) are calculated using Eq. (11) considering the interaction of the system with the environment. In (c) and (d), the experimental data (black square points) are calculated using the compensated density matrices, as described in the text, and the theoretical predictions (red curves) are calculated using Eq. (2) for perfect algorithm circuit operation. The error bars include only the estimated errors in numerically calculating quantum discord with density matrices.

    Figure  4.  (a) Experimentally detected quantum discord with varying temperature for $ \theta = 4.48\pi $ (black squares) and $ \theta = 4.27\pi $ (red dots). Discord is calculated with the original reconstructed density matrices of the output state. (b) Comparison of quantum discord calculated using the original (black squares) and compensated (red dots) density matrices for $ \theta = 4.48\pi $. The theoretical predictions (red curve) are calculated using Eq. (2) for perfect circuit operation. (c) Quantum discord calculated using compensated density matrices with varying temperature for $ \theta = 4.48\pi $ (black squares) and $ \theta = 4.27\pi $ (red dots). The theoretical predictions (black and red curves) are calculated using Eq. (2) for perfect circuit operation. (d) Same as (c), but for quantum discord with varying $ \alpha $. $ \alpha $ and the temperature is related one-to-one according to Eq. (6). Error bars only include the estimated errors for numerical calculation of the quantum discord with density matrices, and are covered by symbols of points.

    Figure  5.  Quantum Fisher information with quantum discord when implementing the DQC1 algorithm to estimate the hyperfine coupling $ A $. Theoretical predictions are calculated without considering the interaction between the system and environment. Owing to the interaction, the experimental data for quantum Fisher information and quantum discord are much smaller than the corresponding theoretical values. However, their experimentally obtained relationship is in good agreement with the theoretical prediction as shown in the figure. The error bars only include the estimated errors in numerically calculating quantum discord with density matrices.

    [1]
    Shor P. Proceedings of the 35th Annual Symposium on the Foundations of Computer Science. Washington, D.C.: IEEE, 1994.
    [2]
    Galindo A, Martin-Delgado M A. Information and computation: Classical and quantum aspects. Reviews of Modern Physics, 2002, 74 (2): 347. doi: 10.1103/RevModPhys.74.347
    [3]
    Goettems E I, Maciel T O, Soares-Pinto D O, et al. Promoting quantum correlations in deterministic quantum computation with a one-qubit model via postselection. Physical Review A, 2021, 103 (4): 042416. doi: 10.1103/PhysRevA.103.042416
    [4]
    Göktaş O, Tham W K, Bonsma-Fisher K, et al. Benchmarking quantum processors with a single qubit. Quantum Information Processing, 2020, 19: 146. doi: 10.1007/s11128-020-02642-4
    [5]
    Zhang K, Thompson J, Zhang X, et al. Modular quantum computation in a trapped ion system. Nature Communications, 2019, 10 (1): 1–6. doi: 10.1038/s41467-018-07882-8
    [6]
    Krzyzanowska K, Copley-May M, Romain R, et al. Quantum-enhanced protocols with mixed states using cold atoms in dipole traps. Journal of Physics: Conference Series, 2017, 793 (1): 012015. doi: 10.1088/1742-6596/793/1/012015
    [7]
    Pg S, Varikuti N D, Madhok V. Exponential speedup in measuring out-of-time-ordered correlators and gate fidelity with a single bit of quantum information. Physics Letters A, 2021, 397: 127257. doi: 10.1016/j.physleta.2021.127257
    [8]
    Knill E, Laflamme R. Power of one bit of quantum information. Physical Review Letters, 1998, 81 (25): 5672. doi: 10.1103/PhysRevLett.81.5672
    [9]
    Jozsa R, Linden N. On the role of entanglement in quantum-computational speed-up. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 2003, 459 (2036): 2011–2032. doi: 10.1098/rspa.2002.1097
    [10]
    Horodecki R, Horodecki P, Horodecki M, et al. Quantum entanglement. Reviews of Modern Physics, 2009, 81 (2): 865–942. doi: 10.1103/RevModPhys.81.865
    [11]
    Datta A, Flammia S T, Caves C M. Entanglement and the power of one qubit. Physical Review A, 2005, 72 (4): 042316. doi: 10.1103/PhysRevA.72.042316
    [12]
    Boyer M, Brodutch A, Mor T. Entanglement and deterministic quantum computing with one qubit. Physical Review A, 2017, 95 (2): 022330. doi: 10.1103/PhysRevA.95.022330
    [13]
    Bennett C H, DiVincenzo D P, Fuchs C A, et al. Quantum nonlocality without entanglement. Physical Review A, 1999, 59 (2): 1070. doi: 10.1103/PhysRevA.59.1070
    [14]
    Modi K, Brodutch A, Cable H, et al. The classical-quantum boundary for correlations: Discord and related measures. Reviews of Modern Physics, 2012, 84 (4): 1655. doi: 10.1103/RevModPhys.84.1655
    [15]
    Hu M L, Hu X, Wang J, et al. Quantum coherence and geometric quantum discord. Physics Reports, 2018, 762: 1–100. doi: 10.1016/j.physrep.2018.07.004
    [16]
    Bera A, Das T, Sadhukhan D, et al. Quantum discord and its allies: A review of recent progress. Reports on Progress in Physics, 2017, 81 (2): 024001. doi: 10.1088/1361-6633/aa872f
    [17]
    Datta A, Shaji A, Caves C M. Quantum discord and the power of one qubit. Physical Review Letters, 2008, 100 (5): 050502. doi: 10.1103/PhysRevLett.100.050502
    [18]
    Madsen L S, Berni A, Lassen M, et al. Experimental investigation of the evolution of Gaussian quantum discord in an open system. Physical Review Letters, 2012, 109 (3): 030402. doi: 10.1103/PhysRevLett.109.030402
    [19]
    Radhakrishnan C, Lauriere M, Byrnes T. Multipartite generalization of quantum discord. Physical Review Letters, 2020, 124 (11): 110401. doi: 10.1103/PhysRevLett.124.110401
    [20]
    Hunt M A, Lerner I V, Yurkevich I V, et al. How to observe and quantify quantum-discord states via correlations. Physical Review A, 2019, 100 (2): 022321. doi: 10.1103/PhysRevA.100.022321
    [21]
    Faba J, Martín V, Robledo L. Two-orbital quantum discord in fermion systems. Physical Review A, 2021, 103 (3): 032426. doi: 10.1103/PhysRevA.103.032426
    [22]
    Gu M, Chrzanowski H M, Assad S M, et al. Observing the operational significance of discord consumption. Nature Physics, 2012, 8 (9): 671–675. doi: 10.1038/nphys2376
    [23]
    Dakić B, Lipp Y O, Ma X, et al. Quantum discord as resource for remote state preparation. Nature Physics, 2012, 8 (9): 666–670. doi: 10.1038/nphys2377
    [24]
    Lanyon B P, Barbieri M, Almeida M P, et al. Experimental quantum computing without entanglement. Physical Review Letters, 2008, 101 (20): 200501. doi: 10.1103/PhysRevLett.101.200501
    [25]
    Passante G, Moussa O, Trottier D A, et al. Experimental detection of nonclassical correlations in mixed-state quantum computation. Physical Review A, 2011, 84 (4): 044302. doi: 10.1103/PhysRevA.84.044302
    [26]
    Hor-Meyll M, Tasca D S, Walborn S P, et al. Deterministic quantum computation with one photonic qubit. Physical Review A, 2015, 92 (1): 012337. doi: 10.1103/PhysRevA.92.012337
    [27]
    Wang W, Han J, Yadin B, et al. Witnessing quantum resource conversion within deterministic quantum computation using one pure superconducting qubit. Physical Review Letters, 2019, 123 (22): 220501. doi: 10.1103/PhysRevLett.123.220501
    [28]
    Ferraro A, Aolita L, Cavalcanti D, et al. Almost all quantum states have nonclassical correlations. Physical Review A, 2010, 81 (5): 052318. doi: 10.1103/PhysRevA.81.052318
    [29]
    Cable H, Gu M, Modi K. Power of one bit of quantum information in quantum metrology. Physical Review A, 2016, 93 (4): 040304. doi: 10.1103/PhysRevA.93.040304
    [30]
    Dorner R, Clark S R, Heaney L, et al. Extracting quantum work statistics and fluctuation theorems by single-qubit interferometry. Physical Review Letters, 2013, 110 (23): 230601. doi: 10.1103/PhysRevLett.110.230601
    [31]
    Ryan C A, Emerson J, Poulin D, et al. Characterization of complex quantum dynamics with a scalable NMR information processor. Physical Review Letters, 2005, 95 (25): 250502. doi: 10.1103/PhysRevLett.95.250502
    [32]
    Passante G, Moussa O, Ryan C A, et al. Experimental approximation of the Jones polynomial with one quantum bit. Physical Review Letters, 2009, 103 (25): 250501. doi: 10.1103/PhysRevLett.103.250501
    [33]
    Helstrom C W. Minimum mean-squared error of estimates in quantum statistics. Physics Letters A, 1967, 25 (2): 101–102. doi: 10.1016/0375-9601(67)90366-0
    [34]
    Braunstein S L, Caves C M. Statistical distance and the geometry of quantum states. Physical Review Letters, 1994, 72 (22): 3439. doi: 10.1103/PhysRevLett.72.3439
    [35]
    Barndorff-Nielsen O E, Gill R D. Fisher information in quantum statistics. Journal of Physics A: Mathematical and General, 2000, 33 (24): 4481. doi: 10.1088/0305-4470/33/24/306
    [36]
    Weeks R A. Paramagnetic resonance of lattice defects in irradiated quartz. Journal of Applied Physics, 1956, 27 (11): 1376–1381. doi: 10.1063/1.1722267
    [37]
    Weeks R A, Nelson C M. Trapped electrons in irradiated quartz and silica: II, electron spin resonance. Journal of the American Ceramic Society, 1960, 43 (8): 399–404. doi: 10.1111/j.1151-2916.1960.tb13682.x
    [38]
    Weeks R A. Paramagnetic spectra of E2' centers in crystalline quartz. Physical Review, 1963, 130 (2): 570. doi: 10.1103/PhysRev.130.570
    [39]
    Rudra J K, Fowler W B, Feigl F J. Model for the E2' center in alpha quartz. Physical Review Letters, 1985, 55 (23): 2614. doi: 10.1103/PhysRevLett.55.2614
    [40]
    Perlson B D, Weil J A. Electron paramagnetic resonance studies of the E' centers in alpha-quartz. Canadian Journal of Physics, 2008, 86 (7): 871–881. doi: 10.1139/p08-034
    [41]
    Feng P, Wang Y, Rong X, et al. Characterization of the electronic structure of E2' defect in quartz by pulsed EPR spectroscopy. Physics Letters A, 2012, 376 (32): 2195–2199. doi: 10.1016/j.physleta.2012.05.029
    [42]
    Vandersypen L M K, Chuang I L. NMR techniques for quantum control and computation. Reviews of Modern Physics, 2005, 76 (4): 1037. doi: 10.1103/RevModPhys.76.1037
    [43]
    Braunstein S L, Caves C M, Milburn G J. Generalized uncertainty relations: Theory, examples, and Lorentz invariance. Annals of Physics, 1996, 247 (1): 135–173. doi: 10.1006/aphy.1996.0040

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