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Balancing the minimum error rate and minimum copy consumption in quantum state discrimination

  • 1.

    CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, China

  • 2.

    CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China

Cite this:
https://doi.org/10.52396/JUSTC-2023-0155
More Information
  • Author Bio:

    Boxuan Tian was an undergraduate student of University of Scienceand Technoology of China and got this bachlor’sdegree of Science in 2024. His research interests include quantum information, quantum measurement

    Zhibo Hou is currently an Associate Professor in the School of Physics, University of Science and Technology of China. He obtained his Ph.d. of Optics in 2016. His research interests include quantum metrology, quantum information, and quantum optics

  • Corresponding author: E-mail: houzhibo@ustc.edu.cn
  • Received Date: 14 November 2023
  • Accepted Date: 31 January 2024
    Abstract

  • Abstract

    Extracting more information and saving quantum resources are two main aims for quantum measurements. However, the optimization of strategies for these two objectives varies when discriminating between quantum states $ |\psi_0\rangle$ and $|\psi_1\rangle $ through multiple measurements. In this study, we introduce a novel state discrimination model that reveals the intricate relationship between the average error rate and average copy consumption. By integrating these two crucial metrics and minimizing their weighted sum for any given weight value, our research underscores the infeasibility of simultaneously minimizing these metrics through local measurements with one-way communication. Our findings present a compelling trade-off curve, highlighting the advantages of achieving a balance between error rate and copy consumption in quantum discrimination tasks, offering valuable insights into the optimization of quantum resources while ensuring the accuracy of quantum state discrimination.

    Graphical abstract

    Balancing the minimum error rate and copy consumption in optimal quantum state discrimination.

    Abstract

    Extracting more information and saving quantum resources are two main aims for quantum measurements. However, the optimization of strategies for these two objectives varies when discriminating between quantum states $ |\psi_0\rangle$ and $|\psi_1\rangle $ through multiple measurements. In this study, we introduce a novel state discrimination model that reveals the intricate relationship between the average error rate and average copy consumption. By integrating these two crucial metrics and minimizing their weighted sum for any given weight value, our research underscores the infeasibility of simultaneously minimizing these metrics through local measurements with one-way communication. Our findings present a compelling trade-off curve, highlighting the advantages of achieving a balance between error rate and copy consumption in quantum discrimination tasks, offering valuable insights into the optimization of quantum resources while ensuring the accuracy of quantum state discrimination.

    • Public Summary

    • Introduce a state discrimination model that considers both average error rate and average copy consumption.
    • Reveals the intricate relationship between these two crucial metrics.
    • Showcase the advantages of achieving a balance between error rate and copy consumption in the discrimination task.

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    • References(22)
  • [1]
    Bennett C H. Quantum cryptography using any two nonorthogonal states. Physical Review Letters, 1992, 68: 3121–3124. doi: 10.1103/PhysRevLett.68.3121
    [2]
    Gisin N, Ribordy G, Tittel W, et al. Quantum cryptography. Reviews of Modern Physics, 2002, 74: 145. doi: 10.1103/RevModPhys.74.145
    [3]
    van Enk S J. Unambiguous state discrimination of coherent states with linear optics: Application to quantum cryptography. Physical Review A, 2002, 66: 042313. doi: 10.1103/PhysRevA.66.042313
    [4]
    Knill E, Laflamme R, Zurek W H. Resilient quantum computation: Error models and thresholds. Proceedings of the Royal Society of London Series A: Mathematical, Physical and Engineering Sciences, 1998, 454: 365–384. doi: 10.1098/rspa.1998.0166
    [5]
    Aharonov D, Ben-Or M. Fault tolerant quantum computation with constant error. In: Proceedings of the twenty-ninth annual ACM symposium on Theory of computing. New York: ACM, 1997 : 176–188.
    [6]
    Bennett C H, DiVincenzo D P. Quantum information and computation. Nature, 2000, 404: 247–255. doi: 10.1038/35005001
    [7]
    Helstrom C W. Quantum detection and estimation theory. Journal of Statistical Physics, 1969, 1: 231–252. doi: 10.1007/BF01007479
    [8]
    Higgins B L, Booth B M, Doherty A C, et al. Mixed state discrimination using optimal control. Physical Review Letters, 2009, 103: 220503. doi: 10.1103/PhysRevLett.103.220503
    [9]
    Calsamiglia J, de Vicente J I, Muñoz-Tapia R, et al. Local discrimination of mixed states. Physical Review Letters, 2010, 105: 080504. doi: 10.1103/PhysRevLett.105.080504
    [10]
    Higgins B L, Doherty A C, Bartlett S D, et al. Multiple-copy state discrimination: Thinking globally, acting locally. Physical Review A, 2011, 83: 052314. doi: 10.1103/PhysRevA.83.052314
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    Wiseman H M, Milburn G J. Quantum Measurement and Control. Cambridge, UK: Cambridge University Press, 2009 .
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    Acín A, Bagan E, Baig M, et al. Multiple-copy two-state discrimination with individual measurements. Physical Review A, 2005, 71: 032338. doi: 10.1103/PhysRevA.71.032338
    [13]
    Brody D, Meister B. Minimum decision cost for quantum ensembles. Physical Review Letters, 1996, 76: 1–5. doi: 10.1103/PhysRevLett.76.1
    [14]
    Slussarenko S, Weston M M, Li J G, et al. Quantum state discrimination using the minimum average number of copies. Physical Review Letters, 2017, 118: 030502. doi: 10.1103/PhysRevLett.118.030502
    [15]
    Martínez Vargas E, Hirche C, Sentís G, et al. Quantum sequential hypothesis testing. Physical Review Letters, 2021, 126: 180502. doi: 10.1103/PhysRevLett.126.180502
    [16]
    Li Y, Tan V Y F, Tomamichel M. Optimal adaptive strategies for sequential quantum hypothesis testing. Communications in Mathematical Physics, 2022, 392: 993–1027. doi: 10.1007/s00220-022-04362-5
    [17]
    Renes J M, Blume-Kohout R, Scott A J, et al. Symmetric informationally complete quantum measurements. Journal of Mathematical Physics, 2004, 45: 2171–2180. doi: 10.1063/1.1737053
    [18]
    Conlon L O, Eilenberger F, Lam P K, et al. Discriminating mixed qubit states with collective measurements. Communication Physics, 2023, 6: 337. doi: 10.1038/s42005-023-01454-z
    [19]
    Peres A, Wootters W K. Optimal detection of quantum information. Physical Review Letters, 1991, 66: 1119–1122. doi: 10.1103/PhysRevLett.66.1119
    [20]
    Xu F, Zhang X M, Xu L, et al. Experimental quantum target detection approaching the fundamental helstrom limit. Physical Review Letters, 2021, 127: 040504. doi: 10.1103/PhysRevLett.127.040504
    [21]
    Cook R L, Martin P J, Geremia J M. Optical coherent state discrimination using a closed-loop quantum measurement. Nature, 2007, 446: 774–777. doi: 10.1038/nature05655
    [22]
    Tian B, Yan W, Hou Z, et al. Minimum-consumption discrimination of quantum states via globally optimal adaptive measurements. Physical Review Letters, 2024, 132: 110801. doi: 10.1103/PhysRevLett.132.110801
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    Figure  1.  The constraints between the average copy consumption and average error rate.Here $ x=\pi/6 $, we calculate the constraint when N=5 or 10 or 15. The average score $ S $ is shown in (a) and the corresponding constraint is shown in (b).

    [1]
    Bennett C H. Quantum cryptography using any two nonorthogonal states. Physical Review Letters, 1992, 68: 3121–3124. doi: 10.1103/PhysRevLett.68.3121
    [2]
    Gisin N, Ribordy G, Tittel W, et al. Quantum cryptography. Reviews of Modern Physics, 2002, 74: 145. doi: 10.1103/RevModPhys.74.145
    [3]
    van Enk S J. Unambiguous state discrimination of coherent states with linear optics: Application to quantum cryptography. Physical Review A, 2002, 66: 042313. doi: 10.1103/PhysRevA.66.042313
    [4]
    Knill E, Laflamme R, Zurek W H. Resilient quantum computation: Error models and thresholds. Proceedings of the Royal Society of London Series A: Mathematical, Physical and Engineering Sciences, 1998, 454: 365–384. doi: 10.1098/rspa.1998.0166
    [5]
    Aharonov D, Ben-Or M. Fault tolerant quantum computation with constant error. In: Proceedings of the twenty-ninth annual ACM symposium on Theory of computing. New York: ACM, 1997 : 176–188.
    [6]
    Bennett C H, DiVincenzo D P. Quantum information and computation. Nature, 2000, 404: 247–255. doi: 10.1038/35005001
    [7]
    Helstrom C W. Quantum detection and estimation theory. Journal of Statistical Physics, 1969, 1: 231–252. doi: 10.1007/BF01007479
    [8]
    Higgins B L, Booth B M, Doherty A C, et al. Mixed state discrimination using optimal control. Physical Review Letters, 2009, 103: 220503. doi: 10.1103/PhysRevLett.103.220503
    [9]
    Calsamiglia J, de Vicente J I, Muñoz-Tapia R, et al. Local discrimination of mixed states. Physical Review Letters, 2010, 105: 080504. doi: 10.1103/PhysRevLett.105.080504
    [10]
    Higgins B L, Doherty A C, Bartlett S D, et al. Multiple-copy state discrimination: Thinking globally, acting locally. Physical Review A, 2011, 83: 052314. doi: 10.1103/PhysRevA.83.052314
    [11]
    Wiseman H M, Milburn G J. Quantum Measurement and Control. Cambridge, UK: Cambridge University Press, 2009 .
    [12]
    Acín A, Bagan E, Baig M, et al. Multiple-copy two-state discrimination with individual measurements. Physical Review A, 2005, 71: 032338. doi: 10.1103/PhysRevA.71.032338
    [13]
    Brody D, Meister B. Minimum decision cost for quantum ensembles. Physical Review Letters, 1996, 76: 1–5. doi: 10.1103/PhysRevLett.76.1
    [14]
    Slussarenko S, Weston M M, Li J G, et al. Quantum state discrimination using the minimum average number of copies. Physical Review Letters, 2017, 118: 030502. doi: 10.1103/PhysRevLett.118.030502
    [15]
    Martínez Vargas E, Hirche C, Sentís G, et al. Quantum sequential hypothesis testing. Physical Review Letters, 2021, 126: 180502. doi: 10.1103/PhysRevLett.126.180502
    [16]
    Li Y, Tan V Y F, Tomamichel M. Optimal adaptive strategies for sequential quantum hypothesis testing. Communications in Mathematical Physics, 2022, 392: 993–1027. doi: 10.1007/s00220-022-04362-5
    [17]
    Renes J M, Blume-Kohout R, Scott A J, et al. Symmetric informationally complete quantum measurements. Journal of Mathematical Physics, 2004, 45: 2171–2180. doi: 10.1063/1.1737053
    [18]
    Conlon L O, Eilenberger F, Lam P K, et al. Discriminating mixed qubit states with collective measurements. Communication Physics, 2023, 6: 337. doi: 10.1038/s42005-023-01454-z
    [19]
    Peres A, Wootters W K. Optimal detection of quantum information. Physical Review Letters, 1991, 66: 1119–1122. doi: 10.1103/PhysRevLett.66.1119
    [20]
    Xu F, Zhang X M, Xu L, et al. Experimental quantum target detection approaching the fundamental helstrom limit. Physical Review Letters, 2021, 127: 040504. doi: 10.1103/PhysRevLett.127.040504
    [21]
    Cook R L, Martin P J, Geremia J M. Optical coherent state discrimination using a closed-loop quantum measurement. Nature, 2007, 446: 774–777. doi: 10.1038/nature05655
    [22]
    Tian B, Yan W, Hou Z, et al. Minimum-consumption discrimination of quantum states via globally optimal adaptive measurements. Physical Review Letters, 2024, 132: 110801. doi: 10.1103/PhysRevLett.132.110801
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