i-manager Publications

Estimation of Quantum Entropies

Alexei Kaltchenko*

Wilfrid Laurier University, Waterloo, Ontario, Canada.

Periodicity:January - June'2024
DOI : https://doi.org/10.26634/jmat.13.1.20387

Abstract

Motivated by the importance of entropy functions in quantum data compression, entanglement theory, and various quantum information-processing tasks, this study demonstrates how classical algorithms for entropy estimation can effectively contribute to the construction of quantum algorithms for universal quantum entropy estimation. Given two quantum i.i.d. sources with completely unknown density matrices, algorithms are developed for estimating quantum cross entropy and quantum relative entropy. These estimation techniques represent a quantum generalization of the classical algorithms by Lempel, Ziv, and Merhav.

Keywords

Classical and Quantum Entropy Estimation, Classical and Quantum Data Compression.

How to Cite this Article?

Kaltchenko, A. (2024). Estimation of Quantum Entropies. i-manager’s Journal on Mathematics, 13(1), 1-10. https://doi.org/10.26634/jmat.13.1.20387

References

[1]. Charya, J., Issa, I., Shende, N. V., & Wagner, A. B. (2020). Estimating quantum entropy. IEEE Journal on Selected Areas in Information Theory, 1(2), 454-468.

[2]. Bennett, C. H. (1973). Logical reversibility of computation. IBM journal of Research and Development, 17(6), 525-532.

[3]. Bernstein, E., & Vazirani, U. (1993, June). Quantum complexity theory. In Proceedings of the twenty-fifth annual ACM symposium on Theory of computing (pp. 11-20).

[4]. Braunstein, S. L., Fuchs, C. A., Gottesman, D., & Lo, H. K. (2000). A quantum analog of Huffman coding. IEEE Transactions on Information Theory, 46(4), 1644-1649.

[5]. Buhrman, H., Tromp, J., & Vitányi, P. (2001). Time and space bounds for reversible simulation. In Automata, Languages and Programming: 28th International Colloquium, ICALP 2001 Crete, Greece, July 8–12, 2001 Proceedings 28 (pp. 1017- 1027). Springer Berlin Heidelberg.

[6]. Cai, H., Kulkarni, S. R., & Verdu, S. (2002, June). Universal estimation of entropy and divergence via block sorting. In Proceedings IEEE International Symposium on Information Theory (p. 433). IEEE.

[7]. Chowdhury, A. N., Low, G. H., & Wiebe, N. (2020). A variational quantum algorithm for preparing quantum Gibbs states. arXiv preprint arXiv:2002.00055.

[8]. Gilyén, A., & Li, T. (2019). Distributional property testing in a quantum world. arXiv preprint arXiv:1902.00814.

[9]. Goldfeld, Z., Patel, D., Sreekumar, S., & Wilde, M. M. (2024). Quantum neural estimation of entropies. Physical Review A, 109(3), 032431.

[10]. Gray, R. M. (2011). Entropy and Information Theory. Springer Science & Business Media.

[11]. Gur, T., Hsieh, M. H., & Subramanian, S. (2021). Sublinear quantum algorithms for estimating von Neumann entropy. arXiv preprint arXiv:2111.11139.

[12]. Hastings, M. B., González, I., Kallin, A. B., & Melko, R. G. (2010). Measuring Renyi entanglement entropy in quantum Monte Carlo simulations. Physical Review Letters, 104(15), 157201.

[13]. Hayashi, M., & Matsumoto, K. (2002). Simple construction of quantum universal variable-length source coding. arXiv preprint quant-ph/0209124.

[14]. Jozsa, R., & Presnell, S. (2003). Universal quantum information compression and degrees of prior knowledge. In Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 459(2040), 3061- 3077.

[15]. Kaltchenko, A. (2004, June). Universal non-destructive estimation of quantum relative entropy. In International Symposium onInformation Theory, 2004. ISIT 2004. Proceedings (pp. 355-355). IEEE.

[16]. Kaltchenko, A., & Yang, E. H. (2003). Universal compression of ergodic quantum sources. arXiv preprint quant- ph/0302174.

[17]. Kaltchenko, A., Yang, E. H., & Timofeeva, N. (2007, September). Entropy estimators with almost sure convergence and an o (n-1) variance. In 2007 IEEE Information Theory Workshop (pp. 644-649). IEEE.

[18]. Kontopoulou, E. M., Dexter, G. P., Szpankowski, W., Grama, A., & Drineas, P. (2020). Randomized linear algebra approaches to estimate the von neumann entropy of density matrices. IEEE Transactions on Information Theory, 66(8), 5003- 5021.

[19]. Langford, J. (2002). Generic quantum block compression. Physical Review A, 65(5), 052312.

[20]. Li, T., & Wu, X. (2018). Quantum query complexity of entropy estimation. IEEE Transactions on Information Theory, 65(5), 2899-2921.

[21]. Luongo, A., & Shao, C. (2020). Quantum algorithms for spectral sums. arXiv preprint arXiv:2011.06475.

[22]. Montanaro, A., & de Wolf, R. (2013). A survey of quantum property testing. arXiv preprint arXiv:1310.2035.

[23]. Moore, C., & Nilsson, M. (2001). Parallel quantum computation and quantum codes. SIAM Journal on Computing, 31(3), 799-815.

[24]. Schumacher, B., & Westmoreland, M. D. (2001). Indeterminate-length quantum coding. Physical Review A, 64(4), 042304.

[25]. Schumacher, B., & Westmoreland, M. D. (2002). Relative entropy in quantum information theory. Contemporary Mathematics, 305, 265-290.

[26]. Shangnan, Z. (2021). Quantum data compression and quantum cross entropy. arXiv preprint arXiv:2106.13823.

[27]. Shields, P. C. (1998). The interactions between ergodic theory and information theory. IEEE Transactions on Information Theory, 44(6), 2079-2093.

[28]. Srikumar, M., Hill, C. D., & Hollenberg, L. C. (2021). Clustering and enhanced classification using a hybrid quantum autoencoder. Quantum Science and Technology, 7(1), 015020.

[29]. Subramanian, S., & Hsieh, M. H. (2021). Quantum algorithm for estimating α-Renyi entropies of quantum states. Physical Review A, 104(2), 022428.

[30]. Vedral, V. (2002). The role of relative entropy in quantum information theory. Reviews of Modern Physics, 74(1), 197.

[31]. Verdú, S. (2019). Empirical estimation of information measures: A literature guide. Entropy, 21(8), 720.

[32]. Wang, Y., Zhao, B., & Wang, X. (2023). Quantum algorithms for estimating quantum entropies. Physical Review Applied, 19(4), 044041.

[33]. Yano, H., & Yamamoto, N. (2023). Quantum information criteria for model selection in quantum state estimation. Journal of Physics A: Mathematical and Theoretical, 56(40), 405301.

[34]. Yirka, J., & Subaşı, Y. (2021). Qubit-efficient entanglement spectroscopy using qubit resets. Quantum, 5, 535.

[35]. Ziv, J., & Lempel, A. (1977). A universal algorithm for sequential data compression. IEEE Transactions on Information Theory, 23(3), 337-343.

[36]. Ziv, J., & Lempel, A. (1978). Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory, 24(5), 530-536.

[37]. Ziv, J., & Merhav, N. (1993). A measure of relative entropy between individual sequences with application to universal classification. IEEE Transactions on Information Theory, 39(4), 1270-1279.

If you have access to this article please login to view the article or kindly login to purchase the article

Single Article

	North Americas,UK, Middle East,Europe		India	Rest of world
	USD	EUR	INR	USD-ROW
Online	15	15

ADD TO CART

Options for accessing this content:

If you would like institutional access to this content, please recommend the title to your librarian.
If you already have i-manager's user account: Login above and proceed to purchase the article.
New Users: Please register, then proceed to purchase the article.