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.