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Quantum information distance based on classical descriptions

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Abstract

Classical information distance between two individual finite objects is the length of the shortest program that makes objects convert into each other. This is based on classical Kolmogorov complexity which is an accepted theory of absolute information in bits contained in an individual finite object. Nowadays, quantum Kolmogorov complexity is developed as the absolute information in bits or qubits contained in an individual pure quantum state. In this paper, we give a definition of quantum information distance between two individual pure quantum states based on Vitányi’s definition of quantum Kolmogorov complexity. This extends classical information distance to the quantum domain retaining classical descriptions. We show that our definition is robust, that is, the quantum information distance between two pure quantum states does not depend on the underlying quantum Turing machine.

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References

  • Adleman LM, Demarrais J, Huang MDA (1997) Quantum computability. SIAM J Comput 26(5):1524–1540

    Article  MathSciNet  Google Scholar 

  • Benatti F, Kruger T, Müller M., Siegmund-Schultze R, Szkora A (2006) Entropy and quantum Kolmogorov complexity: a quantum Brudno’s theorem. Commun Math Phys 265:437

    Article  MathSciNet  Google Scholar 

  • Bennett CH, Gacs P, Li M, Vitányi PMB, Zurek W (1998) Information distance. IEEE Trans Inf Theory 44:1407–1423

    Article  MathSciNet  Google Scholar 

  • Bernstein E, Vazirani U (1997) Quantum complexity theory. SIAM J Comput 26(5):1411–1473

    Article  MathSciNet  Google Scholar 

  • Berthiaume A, Van Dam W, Laplante S (2001) Quantum Kolmogorov complexity. J Comput, System Sci 63:201–221

    Article  MathSciNet  Google Scholar 

  • Chaitin G (1969) On the length of programs for computing finite binary sequences: statistical considerations. J ACM 16:145–159

    Article  MathSciNet  Google Scholar 

  • Cilibrasi RL, Vitányi PMB (2005) Clustering by compression. IEEE Trans Inf Theory 51:1523–1545

    Article  MathSciNet  Google Scholar 

  • Cilibrasi RL, Vitányi PMB (2007) The Google similarity distance. IEEE Trans Knowl Data Eng 19:370–383

    Article  Google Scholar 

  • Dai S (2021) Quantum information distance. Int J Quantum Inform 19(6):2150031

    Article  MathSciNet  Google Scholar 

  • Deutsch D (1985) Quantum theory, the Church-Turing principle and the universal quantum computer. Proc R Soc London Series A Math Phys Sci 400:97–117

    MathSciNet  MATH  Google Scholar 

  • Gács P (2001) Quantum algorithmic entropy. J Phys A: Math Gen 34:6859–6880

    Article  MathSciNet  Google Scholar 

  • Hu B, Bi L, Dai S (2017) Information distances versus entropy metric. Entropy 19(6):260

    Article  Google Scholar 

  • Kolmogorov A (1965) Three approaches to the quantitative definition of information. Probl Inf Transm 1(1):1–7

    Google Scholar 

  • Li M (2007) Information distance and its applications. Int J Found Comput Sci 18:669–681

    Article  MathSciNet  Google Scholar 

  • Li M, Badger J, Chen X, Kwong S, Kearney P, Zhang H (2001) An information-based sequence distance and its application to whole mito chondrial genome phylogeny. Bioinformatics 17:149–154

    Article  Google Scholar 

  • Li M, Chen X, Li X, Ma B, Vitányi PMB (2004) The similarity metric. IEEE Trans Inf Theory 50:3250–3264

    Article  MathSciNet  Google Scholar 

  • Li M, Vitányi PMB (2019) An introduction to Kolmogorov complexity and its applications, 4th edn. Springer, New York

    Book  Google Scholar 

  • Miyadera T (2011) Quantum Kolmogorov complexity and bounded quantum memory. Phys Rev A 83:042316

    Article  Google Scholar 

  • Miyadera T, Imai H (2009) Quantum Kolmogorov complexity and quantum key distribution. Phys Rev A 79:012324

    Article  Google Scholar 

  • Mora C, Briegel H, Kraus B (2006) Quantum Kolmogorov complexity and its applications. Int J Quantum Inform 5:12

    MATH  Google Scholar 

  • Müller M (2008) Strongly universal quantum turing machines and invariance of Kolmogorov complexity. IEEE Trans Inform Theory 54:763

    Article  MathSciNet  Google Scholar 

  • Nishimura H, Ozawa M (2002) Computational complexity of uniform quantum circuit families and quantum Turing machines. Theor Comput Sci 276:147–181

    Article  MathSciNet  Google Scholar 

  • Rogers C, Vedral V (2008) The second quantized quantum turing machine and Kolmogorov complexity. Mod Rev Phys Lett B 22(2):1203–1210

    Article  MathSciNet  Google Scholar 

  • Solomonoff R (1964) A formal theory of inductive inference, part I. Inf Control 7(1):1–22

    Article  MathSciNet  Google Scholar 

  • Terwijn SA, Torenvliet L, Vitányi PMB (2011) Nonapproximability of the normalized information distance. J Comput Syst Sci 77(4):738–742

    Article  MathSciNet  Google Scholar 

  • Vitányi PMB (2001) Quantum Kolmogorov complexity based on classical descriptions. IEEE Trans Infor Theory 47:2464–2479

    Article  MathSciNet  Google Scholar 

  • Zhang X, Hao Y, Zhu XY, Li M (2008) New information distance measure and its application in question answering system. J Comput Sci Technol 23:557–572

    Article  MathSciNet  Google Scholar 

Download references

Funding

This project was supported by the National Science Foundation of China under Grant No. 62006168 and Zhejiang Provincial Natural Science Foundation of China under Grant No. LQ21A010001.

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  1. School of Electronics and Information Engineering, Taizhou University, Taizhou, 318000, China

    Songsong Dai

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  1. Songsong Dai
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Correspondence to Songsong Dai.

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Dai, S. Quantum information distance based on classical descriptions. Quantum Mach. Intell. 4, 7 (2022). https://doi.org/10.1007/s42484-022-00064-2

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