What exactly does Bekenstein bound?

Patrick Hayden; Jinzhao Wang

doi:10.22331/q-2025-03-20-1664

What exactly does Bekenstein bound?

Patrick Hayden and Jinzhao Wang

Stanford Institute for Theoretical Physics, Stanford University, Stanford, CA 94305

Published:	2025-03-20, volume 9, page 1664
Editor:	Fabio Anza
Eprint:	arXiv:2309.07436v3
Doi:	https://doi.org/10.22331/q-2025-03-20-1664
Citation:	Quantum 9, 1664 (2025).

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Abstract

The Bekenstein bound posits a maximum entropy for matter with finite energy confined to a spatial region. It is often interpreted as a fundamental limit on the information that can be stored by physical objects. In this work, we test this interpretation by asking whether the Bekenstein bound imposes constraints on a channel's communication capacity, a context in which information can be given a mathematically rigorous and operationally meaningful definition. We study specifically the $\textit{Unruh channel}$ that describes a stationary Alice exciting different species of free scalar fields to send information to an accelerating Bob, who is confined to a Rindler wedge and exposed to the noise of Unruh radiation. We show that the classical and quantum capacities of the Unruh channel obey the Bekenstein bound that pertains to the decoder Bob. In contrast, even at high temperatures, the Unruh channel can transmit a significant number of $\textit{zero-bits}$, which are quantum communication resources that can be used for quantum identification and many other primitive protocols. Therefore, unlike classical bits and qubits, zero-bits and their associated information processing capability are generally not constrained by the Bekenstein bound. However, we further show that when both the encoder and the decoder are restricted, the Bekenstein bound does constrain the channel capacities, including the zero-bit capacity.

► BibTeX data

@article{Hayden2025whatexactlydoes,
  doi = {10.22331/q-2025-03-20-1664},
  url = {https://doi.org/10.22331/q-2025-03-20-1664},
  title = {What exactly does {B}ekenstein bound?},
  author = {Hayden, Patrick and Wang, Jinzhao},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1664},
  month = mar,
  year = {2025}
}

► References

[1] J. D. Bekenstein, Universal upper bound on the entropy-to-energy ratio for bounded systems, Physical Review D 23 (1981) 287.
https://doi.org/10.1103/PhysRevD.23.287

[2] J. D. Bekenstein, Generalized second law of thermodynamics in black-hole physics, Physical Review D 9 (1974) 3292.
https://doi.org/10.1103/PhysRevD.9.3292

[3] R. Landauer, Irreversibility and heat generation in the computing process, IBM Journal of Research and Development 5 (1961) 183.
https://doi.org/10.1147/rd.53.0183

[4] C. H. Bennett, Notes on Landauer's principle, reversible computation, and Maxwell's demon, Studies In History and Philosophy of Science Part B: Studies In History and Philosophy of Modern Physics 34 (2003) 501.
https://doi.org/10.1016/S1355-2198(03)00039-X

[5] R. Bousso, Black hole entropy and the bekenstein bound, Jacob Bekenstein: The Conservative Revolutionary (2020) 139.
https://doi.org/10.1142/9789811203961_0008

[6] D. N. Page, Comment on a universal upper bound on the entropy-to-energy ratio for bounded systems, Physical Review D 26 (1982) 947.
https://doi.org/10.1103/PhysRevD.26.947

[7] J. D. Bekenstein, How does the entropy/information bound work?, Foundations of Physics 35 (2005) 1805.
https://doi.org/10.1007/s10701-005-7350-7

[8] H. Casini, Relative entropy and the bekenstein bound, Classical and Quantum Gravity 25 (2008) 205021.
https://doi.org/10.1088/0264-9381/25/20/205021

[9] B. Schumacher, Quantum coding, Physical Review A 51 (1995) 2738.
https://doi.org/10.1103/PhysRevA.51.2738

[10] D. D. Blanco and H. Casini, Localization of negative energy and the bekenstein bound, Physical Review Letters 111 (2013) 221601.
https://doi.org/10.1103/PhysRevLett.111.221601

[11] R. Longo and F. Xu, Comment on the bekenstein bound, Journal of Geometry and Physics 130 (2018) 113.
https://doi.org/10.1016/j.geomphys.2018.03.010

[12] J. D. Bekenstein, Black holes and information theory, Contemporary Physics 45 (2004) 31.
https://doi.org/10.1080/00107510310001632512

[13] J. D. Bekenstein, Energy cost of information transfer, Physical Review Letters 46 (1981) 623.
https://doi.org/10.1103/PhysRevLett.46.623

[14] J. D. Bekenstein, Entropy content and information flow in systems with limited energy, Physical Review D 30 (1984) 1669.
https://doi.org/10.1103/PhysRevD.30.1669

[15] J. D. Bekenstein and M. Schiffer, Quantum limitations on the storage and transmission of information, International Journal of Modern Physics C 1 (1990) 355.
https://doi.org/10.1142/S0129183190000225

[16] S. Hod, Universal bound on dynamical relaxation times and black-hole quasinormal ringing, Physical Review D 75 (2007) 064013.
https://doi.org/10.1103/PhysRevD.75.064013

[17] G. Carullo, D. Laghi, J. Veitch and W. Del Pozzo, Bekenstein-Hod universal bound on information emission rate is obeyed by LIGO-Virgo binary black hole remnants, Physical Review Letters 126 (2021) 161102.
https://doi.org/10.1103/PhysRevLett.126.161102

[18] L. Buoninfante, G. G. Luciano, L. Petruzziello and F. Scardigli, Bekenstein bound and uncertainty relations, Physics Letters B 824 (2022) 136818.
https://doi.org/10.1016/j.physletb.2021.136818

[19] R. Bousso, Universal limit on communication, Physical Review Letters 119 (2017) 140501.
https://doi.org/10.1103/PhysRevLett.119.140501

[20] S. A. Fulling, Nonuniqueness of canonical field quantization in riemannian space-time, Physical Review D 7 (1973) 2850.
https://doi.org/10.1103/PhysRevD.7.2850

[21] P. C. Davies, Scalar production in schwarzschild and rindler metrics, Journal of Physics A: Mathematical and General 8 (1975) 609.
https://doi.org/10.1088/0305-4470/8/4/022

[22] W. G. Unruh, Notes on black-hole evaporation, Physical Review D 14 (1976) 870.
https://doi.org/10.1103/PhysRevD.14.870

[23] P. Hayden and A. Winter, Weak decoupling duality and quantum identification, IEEE Transactions on Information Theory 58 (2012) 4914.
https://doi.org/10.1109/TIT.2012.2191695

[24] P. Hayden and G. Penington, Approximate quantum error correction revisited: introducing the alpha-bit, Communications in Mathematical Physics 374 (2020) 369.
https://doi.org/10.1007/s00220-019-03696-9

[25] A. S. Holevo, Bounds for the quantity of information transmitted by a quantum communication channel, Problemy Peredachi Informatsii 9 (1973) 3.
https://doi.org/10.1007/BF01032583

[26] M. J. Donald, Further results on the relative entropy, Mathematical Proceedings of the Cambridge Philosophical Society 101 (1987) 363.
https://doi.org/10.1017/S0305004100066058

[27] L. Wang and R. Renner, One-shot classical-quantum capacity and hypothesis testing, Physical Review Letters 108 (2012) 200501.
https://doi.org/10.1103/PhysRevLett.108.200501

[28] D. Marolf, D. Minic and S. F. Ross, Notes on spacetime thermodynamics and the observer dependence of entropy, Physical Review D 69 (2004) 064006.
https://doi.org/10.1103/PhysRevD.69.064006

[29] D. Marolf, A few words on entropy, thermodynamics, and horizons, General Relativity and Gravitation (2005) 83.
https://doi.org/10.1142/9789812700988_0005

[30] K. Brádler, P. Hayden and P. Panangaden, Quantum communication in rindler spacetime, Communications in Mathematical Physics 312 (2012) 361.
https://doi.org/10.1007/s00220-012-1530-z

[31] K. Brádler, P. Hayden and P. Panangaden, Private information via the unruh effect, Journal of High Energy Physics 2009 (2009) 074.
https://doi.org/10.1088/1126-6708/2009/08/074

[32] K. Brádler, N. Dutil, P. Hayden and A. Muhammad, Conjugate degradability and the quantum capacity of cloning channels, Journal of Mathematical Physics 51 (2010) 072201.
https://doi.org/10.1063/1.3456083

[33] P. M. Alsing and G. J. Milburn, Teleportation with a uniformly accelerated partner, Physical Review Letters 91 (2003) 180404.
https://doi.org/10.1103/PhysRevLett.91.180404

[34] D. E. Bruschi, J. Louko, E. Martín-Martínez, A. Dragan and I. Fuentes, Unruh effect in quantum information beyond the single-mode approximation, Physical Review A 82 (2010) 042332.
https://doi.org/10.1103/PhysRevA.82.042332

[35] E. Martín-Martínez, D. Hosler and M. Montero, Fundamental limitations to information transfer in accelerated frames, Physical Review A 86 (2012) 062307.
https://doi.org/10.1103/PhysRevA.86.062307

[36] M. M. Wilde, Quantum Information Theory. Cambridge University Press, 2013, 10.1017/CBO9781139525343.
https://doi.org/10.1017/CBO9781139525343

[37] A. S. Holevo, The capacity of the quantum channel with general signal states, IEEE Transactions on Information Theory 44 (1998) 269.
https://doi.org/10.1109/18.651321

[38] B. Schumacher, Sending entanglement through noisy quantum channels, Physical Review A 54 (1996) 2614.
https://doi.org/10.1103/PhysRevA.54.2614

[39] B. Schumacher and M. D. Westmoreland, Sending classical information via noisy quantum channels, Physical Review A 56 (1997) 131.
https://doi.org/10.1103/PhysRevA.56.131

[40] B. Schumacher and M. D. Westmoreland, Optimal signal ensembles, Physical Review A 63 (2001) 022308.
https://doi.org/10.1103/PhysRevA.63.022308

[41] B. Schumacher and M. D. Westmoreland, Relative entropy in quantum information theory, Contemporary Mathematics 305 (2002) 265.
https://doi.org/10.1090/conm/305/05225

[42] S. Lloyd, Capacity of the noisy quantum channel, Physical Review A 55 (1997) 1613.
https://doi.org/10.1103/PhysRevA.55.1613

[43] P. W. Shor, The quantum channel capacity and coherent information, in Lecture Notes, MSRI Workshop on Quantum Computation, 2002, DOI.
https://doi.org/10.1090/msri/2002/quantum

[44] I. Devetak, The private classical capacity and quantum capacity of a quantum channel, IEEE Transactions on Information Theory 51 (2005) 44.
https://doi.org/10.1109/TIT.2004.839515

[45] C. H. Bennett, P. W. Shor, J. A. Smolin and A. V. Thapliyal, Entanglement-assisted classical capacity of noisy quantum channels, Physical Review Letters 83 (1999) 3081.
https://doi.org/10.1103/PhysRevLett.83.3081

[46] C. H. Bennett, P. W. Shor, J. A. Smolin and A. V. Thapliyal, Entanglement-assisted capacity of a quantum channel and the reverse shannon theorem, IEEE Transactions on Information Theory 48 (2002) 2637.
https://doi.org/10.1109/TIT.2002.802612

[47] I. Devetak and P. W. Shor, The capacity of a quantum channel for simultaneous transmission of classical and quantum information, Communications in Mathematical Physics 256 (2005) 287.
https://doi.org/10.1007/s00220-005-1317-6

[48] K. Brádler, An infinite sequence of additive channels: the classical capacity of cloning channels, IEEE Transactions on Information Theory 57 (2011) 5497.
https://doi.org/10.1109/TIT.2011.2158470

[49] M. Horodecki, P. W. Shor and M. B. Ruskai, Entanglement breaking channels, Reviews in Mathematical Physics 15 (2003) 629.
https://doi.org/10.1142/S0129055X03001709

[50] M. B. Hastings, Superadditivity of communication capacity using entangled inputs, Nature Physics 5 (2009) 255.
https://doi.org/10.1038/nphys1224

[51] C. King, K. Matsumoto, M. Nathanson and M. B. Ruskai, Properties of conjugate channels with applications to additivity and multiplicativity, arXiv preprint quant-ph/0509126 (2005).
https://doi.org/10.48550/arXiv.quant-ph/0509126
arXiv:quant-ph/0509126

[52] R. Schützhold and W. G. Unruh, Comment on ``teleportation with a uniformly accelerated partner", arXiv preprint quant-ph/0506028 (2005).
https://doi.org/10.48550/arXiv.quant-ph/0506028
arXiv:quant-ph/0506028

[53] T. G. Downes, I. Fuentes and T. C. Ralph, Entangling moving cavities in noninertial frames, Physical Review Letters 106 (2011) 210502.
https://doi.org/10.1103/PhysRevLett.106.210502

[54] P. M. Alsing and I. Fuentes, Observer-dependent entanglement, Classical and Quantum Gravity 29 (2012) 224001.
https://doi.org/10.1088/0264-9381/29/22/224001

[55] A. Harrow, Coherent communication of classical messages, Physical Review Letters 92 (2004) 097902.
https://doi.org/10.1103/PhysRevLett.92.097902

[56] D. Sutter, V. B. Scholz, A. Winter and R. Renner, Approximate degradable quantum channels, IEEE Transactions on Information Theory 63 (2017) 7832.
https://doi.org/10.1109/TIT.2017.2757478

[57] F. Leditzky, E. Kaur, N. Datta and M. M. Wilde, Approaches for approximate additivity of the holevo information of quantum channels, Physical Review A 97 (2018) 012332.
https://doi.org/10.1103/PhysRevA.97.012332

[58] I. Devetak and J. Yard, Exact cost of redistributing multipartite quantum states, Physical Review Letters 100 (2008) 230501.
https://doi.org/10.1103/PhysRevLett.100.230501

[59] A. S. Holevo, M. Sohma and O. Hirota, Capacity of quantum gaussian channels, Physical Review A 59 (1999) 1820.
https://doi.org/10.1103/PhysRevA.59.1820

[60] A. S. Holevo and R. F. Werner, Evaluating capacities of bosonic gaussian channels, Physical Review A 63 (2001) 032312.
https://doi.org/10.1103/PhysRevA.63.032312

[61] V. Giovannetti, S. Guha, S. Lloyd, L. Maccone, J. H. Shapiro and H. P. Yuen, Classical capacity of the lossy bosonic channel: The exact solution, Physical Review Letters 92 (2004) 027902.
https://doi.org/10.1103/PhysRevLett.92.027902

[62] F. Caruso, V. Giovannetti and A. S. Holevo, One-mode bosonic gaussian channels: a full weak-degradability classification, New Journal of Physics 8 (2006) 310.
https://doi.org/10.1088/1367-2630/8/12/310

[63] R. D. Sorkin, Impossible measurements on quantum fields, Directions in General Relativity: Proceedings of the 1993 International Symposium, Maryland 2 (1993) 293.
https://doi.org/10.1017/CBO9780511524653.024

[64] D. Beckman, D. Gottesman, A. Kitaev and J. Preskill, Measurability of wilson loop operators, Physical Review D 65 (2002) 065022.
https://doi.org/10.1103/PhysRevD.65.065022

[65] L. Borsten, I. Jubb and G. Kells, Impossible measurements revisited, Physical Review D 104 (2021) 025012.
https://doi.org/10.1103/PhysRevD.104.025012

[66] H. Bostelmann, C. J. Fewster and M. H. Ruep, Impossible measurements require impossible apparatus, Physical Review D 103 (2021) 025017.
https://doi.org/10.1103/PhysRevD.103.025017

[67] J. Polo-Gómez, L. J. Garay and E. Martín-Martínez, A detector-based measurement theory for quantum field theory, Physical Review D 105 (2022) 065003.
https://doi.org/10.1103/PhysRevD.105.065003

[68] C. J. Fewster and R. Verch, Quantum fields and local measurements, Communications in Mathematical Physics 378 (2020) 851.
https://doi.org/10.1007/s00220-020-03800-6

[69] G. Smith and J. A. Smolin, Extensive nonadditivity of privacy, Physical Review Letters 103 (2009) 120503.
https://doi.org/10.1103/PhysRevLett.103.120503

[70] O. Fawzi, P. Hayden and P. Sen, From low-distortion norm embeddings to explicit uncertainty relations and efficient information locking, Journal of the ACM (JACM) 60 (2013) 1.
https://doi.org/10.1145/2535928

[71] P. Hayden and G. Penington, Learning the alpha-bits of black holes, Journal of High Energy Physics 2019 (2019) 1.
https://doi.org/10.1007/JHEP12(2019)007

[72] D. N. Page, Defining entropy bounds, Journal of High Energy Physics 2008 (2008) 007.
https://doi.org/10.1088/1126-6708/2008/10/007

[73] L. B. Levitin, Optimal quantum measurements for two pure and mixed states, Quantum Communications and Measurement (1995) 439.
https://doi.org/10.1007/978-1-4615-2345-1_46

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