Hawking radiation as tunneling is non-Markovian: An information-theoretic solution to the paradox of black hole information loss
aa r X i v : . [ m a t h - ph ] N ov Hawking radiation as tunneling is non-Markovian: An information-theoreticsolution to the paradox of black hole information loss
Zeqian Chen ∗ State Key Laboratory of Resonances and Atomic and Molecular Physics,Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences,30 West District, Xiao-Hong-Shan, Wuhan 430071, China (Dated: June 22, 2018)
PACS numbers: 04.70.Dy, 03.67.-a
In an ontological description of a physical system [1], the state evolution of the system is called
Markovian if the probability distributions associated with the states of the system at all time inthe future depend only on the probability distribution associated with the present state but not onprevious ones [2]; “ non-Markovian ” is the opposite. Given a system and a set of initial conditions,classical mechanics allows us to calculate the future evolution to arbitrary precision. Any uncer-tainty we might have at a given time is caused by a lack of knowledge about the configuration.This indicates that the state (deterministic or stochastic) evolution in classical mechanics has to beMarkovian, that is, a probability distribution (or knowledge) for the actual state of a classical sys-tem at any time in the future actually only depends on the knowledge for the present state but noton probability distributions of the past ones. As is well-known, Hawking’s 1974 announcement [3]that black holes evaporate thermally has led to a much-debated paradox that the consequent loss ofinformation violates the principle of unitarity for quantum mechanics. This paradox has attracteda number of proposals, but no consensus on a solution [4]. Here, we show that Hawking radiationas tunneling [5] is non-Markovian in terms of the tunneling probability uncovered by Parikh andWilczek [6]. This implies that there are non-classical correlations among Hawking radiations. Bythe Bayesian rule in information theory [7], we compute the amount of information encoded inthose correlations and find that it is equal to the maximum information content of a black holemeasured by the Bekenstein-Hawking entropy [8]. By noting that the Bekenstein-Hawking entropyis a measure of the hole’s information capacity [9], we arrive at the conclusion that there is noinformation loss in Hawking radiation, and thereby provide an information-theoretic solution tothe paradox of black hole information loss. ∗ Electronic address: [email protected]
Recently, Zhang et al [10] have shown that there exist statistical correlations between quantaof Hawking radiation as tunneling based on the principle of energy conservation involving theaccepted emission probability spectrum from a black hole, as conjectured by Parikh and Wilczek[6]. They concluded that up to exp S BH bits of information can carried off in the correlation, whichis the maximum information context of a black hole measured by the Bekenstein-Hawking entropy S BH in the sense that exp S BH is the maximum number of bits that can be accommodated in ablack hole formed by an astrophysical collapse. As pointed out in [9], this goes a considerable waytoward resolving this long-standing “information loss paradox.”In this article, we offer a contribution to the solution of the problem by approaching the differentoptions on the table with an information-theoretic argument. For simplicity, we take the convenientunits of k = ~ = c = G = 1 . Suppose to consider Hawking radiation as tunneling for the Schwarchildblack hole. The tunneling probability for an emission at an energy E is found to be [6] P ( E ) = exp h − πE (cid:16) M − E (cid:17)i ≡ exp( △ S ) , (1)where the second equal sign expresses this result in terms of the change of the Bekenstein-Hawkingentropy for the Schwarchild black hole S BH = A/ πM with A = 4 π (2 M ) being the surfacearea of a Schwarzchild black hole with mass M and radius 2 M. Let us consider two emissions aris-ing simultaneously with energies E and E respectively. According to an information-theoretic(Bayesian) view, in which probabilities are primarily states of knowledge or evidence, the probabil-ity P ( E , E ) of such two emissions is equal to the probability of an emission at an energy E + E , that is, P ( E , E ) = P ( E + E ) = exp h − π ( E + E ) (cid:16) M −
12 ( E + E ) (cid:17)i , (2)because the black hole emits the same energy in both cases and so the information we can obtainare the same. This inference clearly can apply to the case of many emissions arising simultaneouslyand the corresponding probability formula holds true.Now consider the first three sequential emissions with energies E , E , and E in a sequence ofemissions. If the state evolution of the total system composed of a black hole and its radiations astunneling were Markovian, then the conditional probabilities P ( E | E , E ) and P ( E | E ) shouldsatisfy P ( E | E , E ) = P ( E | E ) . (3)However, by the Bayesian rule we find that [11]ln P ( E | E , E ) − ln P ( E | E ) = 8 πE E = 0 . Thus Hawking radiation as tunneling has to be non-Markovian which shows that there are non-classical correlations among these radiations [12].Next we compute the information content hidden in those correlations of sequentially tunneledparticles, that is, the entropy of all tunneled particles. Let us consider a radiation process ofsequential emissions with energies E , E , . . . , E n so that M = P ni =1 E i . According to the Chainrule for entropy in information theory [7], the total entropy H ( E , E , . . . , E n ) of this radiationprocess which eventually exhausts the black hole is H ( E , E , . . . , E n ) = n X i =1 H ( E i | E i − , . . . , E ) (4)where H ( E i | E i − , . . . , E ) = − ln P ( E i | E i − , . . . , E ) with H ( E | E ) = − ln P ( E ) . Again, by theBayesian rule we find that [13] H ( E , E , . . . , E n ) = 4 πM , (5)which is exactly the same as the Bekenstein-Hawking entropy of the Schwarzchild black hole withmass M and radius 2 M. In conclusion, we have shown that the information flow associated with the state evolution of thetotal system composed of a black hole and its radiations as tunneling is non-Markovian and thusthere are non-classical correlations among these radiations. We further find that the entropy of thesystem is conserved if information hidden in those correlations of sequentially tunneled particles isincluded. Our analysis relies on an information-theoretic argument within which no conservationlaw (such as the principle of energy conversation) is involved. The crucial feature of our approach isthat it is operational, in the sense that we only refer to directly information-theoretic objects, butdo not assume anything about the underlying evolution of black holes (such as the energy changeof the black hole after an emission). This thereby provides a complete solution to the paradox ofblack hole information loss from an information-theoretic viewpoint [14]. Finally, we remark thatthere is a close relationship between non-Markovian and unitary evolutions [15]. However, it is notour purpose to enter into this debate here.The author is grateful to Qing-yu Cai for helpful discussions on this topic. This work wassupported in part by the NSFC under Grant No. 11171338 and National Basic Research Programof China under Grant No. 2012CB922102. [1] R. W. Spekkens, Contextuality for preparations, transformations, and unsharp measurements,
Phys.Rev. A (2005), 052108(17). [2] A. Montina, Exponetial complexity and ontological theories of quantum mechanics, Phys. Rev. A (2008), 022104(10).[3] S. W. Hawking, Black hole explosions ? Nature (London) (1974), 30-31.[4] S. Hossenfelder and L. Smolin, Conservative solutions to the black hole information problem,
Phys.Rev. D (2010), 064009(13).[5] L. M. Krauss and F. Wilczek, Self-interaction correction to black hole radiance, Nuclear Physics
B433 (1995), 403-420; Effect of self-interaction correction on charged black hole radiance,
B437 (1995),231-242;[6] M. K. Parikh and F. Wilczek, Hawking radiation as tunneling,
Phys. Rev. Lett. (2000), 5042-5045.[7] T. M. Cover and J. A. Thomas, Elements of Information Theory (John Wiley & Sons, Inc., New York,1991).[8] J. D. Bekenstein, Black holes and entropy, Phys. Rev. D (1973), 2333-2346 ; S. W. Hawking, Particlecreation by black holes, Comm. Math. Phys. (1975), 199-220.[9] W. Israel and Z. Yun, Band-aid for information loss from black holes, Phys. Rev. D (2010),124036(10).[10] B. Zhang, Q. Cai, L. You, and M. Zhan, Hidden messenger revealed in Hawking radiation: A resolutionto the paradox of black hole information loss, Phys. Lett. B (2009), 98-101.[11] Indeed, by the Bayesian formula we have the conditional probability P ( E | E ) = P ( E , E ) /P ( E ) = P ( E + E ) /P ( E ) where the second equality is obtained by (2). Similarly, we have P ( E | E , E ) = P ( E + E + E ) /P ( E + E ) . [12] If one only considers two emissions with energies E and E , by the Bayesian formula one can findthe conditional probabilty P ( E | E ) = P ( E + E ) /P ( E ) = P ( E ) and conclude that there existstatistical correlations between two emissions. However, in this case one cannot infer whether or notthese correlations are non-classical. This also applies to the case of the argument presented in [10].[13] By the Bayesian formula, H ( E i | E i − , . . . , E ) = − ln P ( E i | E i − , . . . , E ) = − ln P ( E , . . . , E i ) +ln P ( E , . . . , E i − ) . Hence, H ( E , E , . . . , E n ) = − ln P ( E , . . . , E n ) = − ln P ( E + . . . + E n ) = 4 πM . [14] Our analysis can be extended to charged black holes, Kerr black holes, and Kerr-Neumann black holesin a similar way as done in B. Zhang, Q. Cai, M. Zhan, and L. You, Entropy is conserved in Hawkingradiation as tunneling: A revisit of the black hole information loss paradox, Ann. Phys. (2011),350-363.[15] D. T. Gillespie, Why quantum mechanics cannot be formulated as a Markov process,
Phys. Rev. A (1994), 1607-1612; L. Hardy et al , Comment on “Why quantum mechanics cannot be formulated as aMarkov process,” Phys. Rev. A (1997), 3301-3303; D. T. Gillespie, Reply to “Comment on ‘Whyquantum mechanics cannot be formulated as a Markov process’ ,” Phys. Rev. A (1997), 3304-3306;G. A. Skorobogatov and S. I. Svertilov, Quantum mechanics can be formulated as a non-Markovianstochastic process, Phys. Rev. A58