Top

Quantum Information Processing

Published in:

01-02-2024

Average Rényi entropy of a subsystem in random pure state

Authors: MuSeong Kim, Mi-Ra Hwang, Eylee Jung, DaeKil Park

Published in: Quantum Information Processing | Issue 2/2024

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

In this paper, we examine the average Rényi entropy $S_{\alpha }$ of a subsystem A when the whole composite system AB is a random pure state. We assume that the Hilbert space dimensions of A and AB are m and mn, respectively. First, we compute the average Rényi entropy analytically for $m = \alpha = 2$. We compare this analytical result with the approximate average Rényi entropy, which is shown to be very close. For general case, we compute the average of the approximate Rényi entropy ${\widetilde{S}}_{\alpha } (m,n)$ analytically. When $1 \ll n$, ${\widetilde{S}}_{\alpha } (m,n)$ reduces to $\ln m - \frac{\alpha }{2 n} (m - m^{-1})$, which is in agreement with the asymptotic expression of the average von Neumann entropy. Based on the analytic result of ${\widetilde{S}}_{\alpha } (m,n)$, we plot the $\ln m$-dependence of the Rényi information derived from ${\widetilde{S}}_{\alpha } (m,n)$. It is remarkable to note that the nearly vanishing region of the information becomes shorten with increasing $\alpha $ and eventually disappears in the limit of $\alpha \rightarrow \infty $. The physical implication of the result is briefly discussed.

previous article Quantum power iteration to efficiently obtain the dominant eigenvector from diagonalizable nonnegative matrices

next article An efficient quantum algorithm for preparation of uniform quantum superposition states

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Eq. (2.2) is called a density of the eigenvalues of the Wishart matrix.

Furthermore, F(m, n) depends on the $j^{th}$ term of some recurrence relations, where j is a function of m and n. This term is expressed with the aid of few special functions such as Lerch transcendent

$$\begin{aligned} \Phi (z,s,a) = \sum _{k=0}^{\infty } \frac{z^k}{(k + a)^s}. \end{aligned}$$

Lubkin, E.: Entropy of an $n$-system from its correlation with a $k$-reservoir. J. Math. Phys. 19, 1028 (1978)ADSCrossRef

Lloyd, S., Pagels, H.: Complexity as thermodynamic depth. Ann. Phys. 188, 186 (1988)ADSMathSciNetCrossRef

Page, D.N.: Average entropy of a subsystem. Phys. Rev. Lett. 71, 1291 (1993). arXiv:gr-qc/9305007ADSMathSciNetCrossRefPubMed

Bianchi, E., Hackl, L., Kieburg, M., Rigol, M., Vidmar, L.: Volume-law entanglement entropy of typical pure quantum states. PRX Quantum 3, 030201 (2022). arXiv:2112.06959 (quant-ph)ADSCrossRef

Foong, S.K., Kanno, S.: Proof of page’s conjecture on the average entropy of a subsystem. Phys. Rev. Lett. 72, 1148 (1994)ADSMathSciNetCrossRefPubMed

Sánchez-Ruiz, J.: Simple proof of Page’s conjecture on the average entropy of a subsystem. Phys. Rev. E 52, 5653 (1995)ADSCrossRef

Sen, S.: Average entropy of a subsystem. Phys. Rev. Lett. 77, 1 (1996). arXiv:hep-th/9601132ADSCrossRefPubMed

Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover Publications, New York (1972)

Page, D.N.: Information in black hole radiation. Phys. Rev. Lett. 71, 3743 (1993). arXiv:hep-th/9306083ADSMathSciNetCrossRefPubMed

10.

Hawking, S.W.: Breakdown of predictability in gravitational collapse. Phys. Rev. D 14, 2460 (1976)ADSMathSciNetCrossRef

11.

Preskill, J.: Do black holes destroy information? arXiv:hep-th/9209058

12.

Hawking, S.W.: Black hole explosions? Nature 248, 30 (1974)ADSCrossRef

13.

Hawking, S.W.: Particle creation by black holes. Commun. Math. Phys. 43, 199 (1975)ADSMathSciNetCrossRef

14.

Hayden, P., Preskill, J.: Black holes as mirrors: quantum information in random subsystems. J. High Energy Phys. 09, 120 (2007). arXiv:0708.4025 (hep-th)ADSMathSciNetCrossRef

15.

Yoshida, B., Kitaev, A.: Efficient decoding for the Hayden–Preskill protocol. arXiv:1710.03363 (hep-th)

16.

Alonso-Serrano, A., Visser, M.: Multipartite analysis of average-subsystem entropies. Phys. Rev. A 96, 052302 (2017)ADSCrossRef

17.

Hwang, J., Lee, D.S., Nho, D., Oh, J., Park, H., Yeom, D., Zoe, H.: Page curves for tripartite systems. Class. Quant. Grav. 34, 145004 (2017). arXiv:1608.03391 (hep-th)ADSMathSciNetCrossRef

18.

Shapourian, H., Liu, S., Kudler-Flam, J., Vishwanath, A.: Entanglement negativity spectrum of random mixed states: a diagrammatic approach. PRX Quantum 2, 030347 (2021). arXiv:2011.01277 (cond-mat)ADSCrossRef

19.

Liu, C., Chen, X., Balents, L.: Quantum Entanglement of the Sachdev–Ye–Kitaev Models. Phys. Rev. B 97, 245126 (2018). arXiv:1709.06259 (cond-mat)ADSCrossRef

20.

Yu, X., Gong, Z., Cirac, J.I.: Free-fermion page curve: canonical typicality and dynamical emergence. arXiv:2209.08871 (quant-ph)

21.

Ruggiero, P., Turkeshi, X.: Quantum information spreading in random spin chains. Phys. Rev. B 106, 134205 (2022). arXiv:2206.02934 (cond-mat)ADSCrossRef

22.

Iosue, J.T., Ehrenberg, A., Hangleiter, D., Deshpande, A., Gorshkov, A.V.: Page curves and typical entanglement in linear optics. arXiv:2209.06838 (quant-ph)

23.

Bianchi, E., Hackl, L., Kieburg, M.: The Page curve for fermionic Gaussian states. Phys. Rev. B 103, 241118 (2021). arXiv:2103.05416 (quant-ph)ADSCrossRef

24.

Bhattacharjee, B., Nandy, P., Pathak, T.: Eigenstate capacity and Page curve in fermionic Gaussian states. Phys. Rev. B 104, 214306 (2021). arXiv:2109.00557 (quant-ph)ADSCrossRef

25.

Yang, Z., Chamon, C., Hamma, A., Mucciolo, E.R.: Two-component structure in the entanglement spectrum of highly excited states. Phys. Rev. Lett. 115, 267206 (2015). arXiv:1506.01714 (cond-mat)ADSCrossRefPubMed

26.

Vidmar, L., Rigol, M.: Entanglement entropy of eigenstates of quantum chaotic Hamiltonians. Phys. Rev. Lett. 119, 220603 (2017). arXiv:1708.08453 (cond-mat)ADSCrossRefPubMed

27.

Nakagawa, Y.O., Watanabe, M., Fujita, H., Sugiura, S.: Universality in volume-law entanglement of scrambled pure quantum states. Nat. Commun. 9, 1635 (2018)ADSCrossRefPubMedPubMedCentral

28.

Kaneko, K., Iyoda, E., Sagawa, T.: Characterizing complexity of many-body quantum dynamics by higher-order eigenstate thermalization. Phys. Rev. A 101, 042126 (2020). arXiv:1911.10755 (cond-mat)ADSMathSciNetCrossRef

29.

Lu, T., Grover, T.: Renyi entropy of chaotic eigenstates. Phys. Rev. E 99, 032111 (2019). arXiv:1709.08784 (cond-mat)ADSMathSciNetCrossRefPubMed

30.

Łydżba, P., Rigol, M., Vidmar, L.: Eigenstate entanglement entropy in random quadratic Hamiltonians. Phys. Rev. Lett. 125, 180604 (2020). arXiv:2006.11302 (cond-mat)MathSciNetCrossRefPubMed

31.

Liu, H., Vardhan, S.: A dynamical mechanism for the Page curve from quantum chaos. J. High Energy Phys. 2021, 88 (2021). arXiv:2002.05734 (hep-th)MathSciNetCrossRef

32.

Sinha, S., Ray, S., Sinha, S.: Fingerprint of chaos and quantum scars in kicked Dicke model: an out-of-time-order correlator study. arXiv:2101.05155 (cond-mat)

33.

de Miranda, J.T., Micklitz, T.: Subsystem trace-distances of random states. arXiv:2210.03213 (quant-ph)

34.

Oliveira, R., Dahlsten, O.C.O., Plenio, M.B.: Generic entanglement can be generated efficiently. Phys. Rev. Lett. 98, 130502 (2007). arXiv:quant-ph/0605126ADSCrossRefPubMed

35.

Plato, A.D.K., Dahlsten, O.C., Plenio, M.B.: Random circuits by measurements on weighted graph states. Phys. Rev. A 78, 042332 (2008). arXiv:0806.3058 (quant-ph)ADSCrossRef

36.

Bera, A., Roy, S.S.: Growth of genuine multipartite entanglement in random unitary circuits. Phys. Rev. A 102, 062431 (2020). arXiv:2003.12546 (quant-ph)ADSMathSciNetCrossRef

37.

Hayden, P., Winter, A.: Counterexamples to the maximal $p$-norm multiplicativity conjecture for all $p > 1$. Commun. Math. Phys. 284, 263 (2008). arXiv:0807.4753 (quant-ph)ADSCrossRef

38.

Brandao, F.G.S.L., Horodecki, M.: On Hastings’ counterexamples to the minimum output entropy additivity conjecture. Open. Syst. Inf. Dyn. 17, 31 (2010). arXiv:0907.3210 (quant-ph)MathSciNetCrossRef

39.

Fukuda, M., King, C.: Entanglement of random subspaces via the Hastings bound. J. Math. Phys. 51, 042201 (2010). arXiv:0907.5446 (quant-ph)ADSMathSciNetCrossRef

40.

Prudnikov, A.P., Brychkov, Y.A., Marichev, O.I.: Integrals and Series. Gordon and Breach Science Publishers, New York (1986)

41.

Gradshteyn, I.S., Ryzbik, I.M.: Table of Integrals, Series, and Products. Academic Press, San Diego (2000)

42.

Bianchi, E., Donà, P.: Typical entanglement entropy in the presence of a center: Page curve and its variance. Phys. Rev. D 100, 105010 (2019). arXiv:1904.08370 (hep-th)ADSCrossRef

43.

Susskind, L., Thorlacius, L.: Gedanken experiments involving black holes. Phys. Rev. D 49, 966 (1994). arXiv:hep-th/9308100ADSMathSciNetCrossRef

44.

Sekino, Y., Susskind, L.: Fast scramblers. J. High Energy Phys. 0810, 065 (2008). arXiv:0808.2096 (hep-th)ADSCrossRef

Title: Average Rényi entropy of a subsystem in random pure state
Authors: MuSeong Kim
Mi-Ra Hwang
Eylee Jung
DaeKil Park
Publication date: 01-02-2024
Publisher: Springer US
Published in: Quantum Information Processing / Issue 2/2024
Print ISSN: 1570-0755
Electronic ISSN: 1573-1332
DOI: https://doi.org/10.1007/s11128-023-04249-x

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"

Other articles of this Issue 2/2024

Primitivity for random quantum channels

End-to-end entanglement establishment with lower latency in quantum networks

Secure quantum signature scheme without entangled state

A rational hierarchical (t,n)-threshold quantum secret sharing scheme

Withstanding detector attacks in continuous-variable quantum key distribution via mean-restricted unary linear regression

Multi-party semiquantum key distribution with multi-qubit GHZ states