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{{Short description|Paradox concerning information loss in black-hole evaporation}}
{{Quantum book backlink|Advanced and frontier topics}}

The '''quantum black hole information paradox''' is a conflict between [[Physics:Quantum mechanics|quantum mechanics]], [[Physics:Quantum field theory|quantum field theory]], and the gravitational description of black holes. In quantum mechanics, the complete state of a closed system is expected to evolve unitarily, so information about an earlier state is in principle preserved. In semiclassical black-hole physics, however, Hawking radiation appears thermal and seems to depend only on a few macroscopic properties of the black hole.

The paradox arises when a black hole forms from matter in a pure quantum state and then evaporates completely through Hawking radiation. If the final radiation contains only thermal information about the black hole's mass, charge, and angular momentum, then many different initial states could lead to the same final state. This would mean that information has been permanently lost, contradicting ordinary quantum-mechanical unitarity.<ref name="hawkbreak">{{cite journal |doi=10.1103/PhysRevD.14.2460 |title=Breakdown of predictability in gravitational collapse |year=1976 |last1=Hawking |first1=S. W. |journal=Physical Review D |volume=14 |issue=10 |pages=2460–2473 |bibcode=1976PhRvD..14.2460H}}</ref>

The paradox is one of the central problems in [[Physics:Quantum gravity problem|quantum gravity]]. It links black-hole thermodynamics, Hawking radiation, entropy, holography, the Page curve, and the microscopic structure of spacetime. Many physicists now believe that information is preserved during black-hole evaporation, but there is still debate about the precise mechanism by which Hawking's original semiclassical calculation must be corrected.<ref name="entrhawk">{{cite journal |last1=Almheiri |first1=Ahmed |last2=Hartman |first2=Thomas |last3=Maldacena |first3=Juan |last4=Shaghoulian |first4=Edgar |last5=Tajdini |first5=Amirhossein |title=The entropy of Hawking radiation |journal=Reviews of Modern Physics |date=21 July 2021 |volume=93 |issue=3 |pages=035002 |doi=10.1103/RevModPhys.93.035002 |arxiv=2006.06872 |bibcode=2021RvMP...93c5002A |s2cid=219635921 }}</ref><ref name="lessons">{{cite journal |last1=Raju |first1=Suvrat |title=Lessons from the information paradox |journal=Physics Reports |date=January 2022 |volume=943 |pages=1–80 |doi=10.1016/j.physrep.2021.10.001 |arxiv=2012.05770 |bibcode=2022PhR...943....1R |s2cid=228083488 }}</ref>

<div style="float:right; border:1px solid #e0d890; background:#fff8cc; padding:6px; margin:0 0 1em 1em; width:360px;">
[[File:Quantum_black_hole_information_paradox_yellow.png|340px]]
<div style="font-size:90%;">The information paradox asks whether information that falls into a black hole is lost, hidden, or encoded in subtle correlations in Hawking radiation.</div>
</div>

== Basic idea ==

A black hole is classically described by a small number of external parameters, such as mass, electric charge, and angular momentum. The no-hair idea suggests that many details of the matter that formed the black hole are not visible from outside the horizon.

Hawking's calculation showed that black holes should emit radiation with a thermal spectrum.<ref name="Hawking">{{cite journal |last=Hawking |first=Stephen |title=Particle Creation by Black Holes |journal=Communications in Mathematical Physics |volume=43 |issue=3 |pages=199–220 |date=1 August 1975 |doi=10.1007/BF02345020 |bibcode=1975CMaPh..43..199H |s2cid=55539246 |url=https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-43/issue-3/Particle-creation-by-black-holes/cmp/1103899181.pdf }}</ref> If the radiation is exactly thermal, then it carries no detailed memory of the matter that collapsed. When the black hole has fully evaporated, only radiation remains. The question is whether that radiation contains the original information.

In quantum mechanics, the wave function of a closed system evolves by a unitary operator. This means that the state at one time determines the state at any other time:

<math display="block">|\Psi(t_1)\rangle = U(t_1,t_2)|\Psi(t_2)\rangle.</math>

If black-hole evaporation turns a pure state into a mixed thermal state, then the process is not unitary. That is the core of the paradox.

== Hawking radiation ==

Hawking radiation is produced by quantum fields in the curved spacetime around a black hole. In Hawking's semiclassical calculation, gravity is treated classically while quantum fields are allowed to fluctuate on the black-hole background.

For a Schwarzschild black hole, the Hawking temperature is

<math display="block">
T = {\hbar c^3 \over 8 \pi k G M}.
</math>

This formula shows that smaller black holes are hotter. As a black hole radiates, it loses mass and becomes hotter, so the evaporation accelerates. The lifetime of a Schwarzschild black hole is proportional to the cube of its initial mass.

The problem is not merely that a black hole radiates. The problem is that the radiation appears to be thermal and therefore insensitive to the detailed quantum state of the matter that formed the black hole.

== Pure states and mixed states ==

In quantum mechanics, a pure state contains maximal information about a system's quantum state. A mixed state represents statistical uncertainty or incomplete information. Unitary quantum evolution preserves the purity of a closed system.

Hawking's original argument suggested that a black hole could form from matter in a pure state and evaporate into a mixed state of thermal radiation. Such a pure-to-mixed transition would violate unitarity.

This can also be described using von Neumann entropy. A pure state has zero von Neumann entropy, while a mixed state has nonzero entropy. If black-hole evaporation is unitary, then the fine-grained entropy of the total radiation should eventually return to zero. If Hawking's original calculation is taken literally, the entropy instead continues to increase.

== Page curve ==

Don Page argued that if black-hole evaporation is unitary, the entropy of the Hawking radiation should follow a characteristic pattern. It should initially increase as the black hole becomes entangled with its radiation, then reach a maximum, and finally decrease back to zero when the black hole has evaporated completely.<ref name="Page1993">{{cite journal |volume=71 |last=Page |first=Don N. |title=Information in Black Hole Radiation |journal=Physical Review Letters |pages=3743–3746 |date=6 December 1993 |issue=23 |doi=10.1103/PhysRevLett.71.3743 |arxiv=hep-th/9306083 |bibcode=1993PhRvL..71.3743P |pmid=10055062 |s2cid=9363821 }}</ref>

This expected entropy pattern is called the '''Page curve'''. Deriving the Page curve from a theory of gravity is often treated as a major sign that the information paradox has been resolved in that framework.

Recent work using replica wormholes and quantum extremal surfaces has reproduced Page-curve behavior in specific models of black-hole evaporation.<ref name="Penington2020">{{cite journal |last1=Penington |first1=Geoffrey |title=Entanglement wedge reconstruction and the information paradox |journal=Journal of High Energy Physics |date=September 2020 |volume=2020 |issue=9 |pages=2 |doi=10.1007/JHEP09(2020)002 |arxiv=1905.08255 |bibcode=2020JHEP...09..002P |s2cid=160009640 }}</ref><ref name="AEMM2019">{{cite journal |last1=Almheiri |first1=Ahmed |last2=Engelhardt |first2=Netta |last3=Marolf |first3=Donald |last4=Maxfield |first4=Henry |title=The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole |journal=Journal of High Energy Physics |date=December 2019 |volume=2019 |issue=12 |pages=63 |doi=10.1007/JHEP12(2019)063 |arxiv=1905.08762 |bibcode=2019JHEP...12..063A |s2cid=160009599 }}</ref><ref name="HREntropy">{{cite journal |arxiv=1911.12333 |title=Replica Wormholes and the Entropy of Hawking Radiation |first1=A. |last1=Almheiri |first2=T. |last2=Hartman |first3=J. |last3=Maldacena |first4=E. |last4=Shaghoulian |first5=A. |last5=Tajdini |year=2020 |doi=10.1007/JHEP05(2020)013 |volume=2020 |journal=Journal of High Energy Physics |issue=5 |s2cid=208310010}}</ref>

== Holography ==

The holographic principle suggests that the information inside a gravitational region may be encoded on a lower-dimensional boundary. In the context of black holes, holography offers a way for the physics outside the black hole to contain information that appears classically hidden behind the horizon.

The AdS/CFT correspondence provides a concrete example of holography. It relates certain gravitational theories in anti-de Sitter spacetime to ordinary quantum field theories without gravity on the boundary. Since the boundary theory is unitary, many physicists take AdS/CFT as evidence that black-hole evaporation should also be unitary in the gravitational description.

This view has strongly influenced modern thinking about the information paradox. It suggests that Hawking radiation is not exactly featureless thermal radiation, but contains subtle correlations that encode the black hole's initial state.<ref name="lessons" />

== Replica wormholes and islands ==

Recent developments have shown how the Page curve can emerge from gravitational path integrals in certain simplified models. The key ingredients include replica wormholes and the island formula.

In these calculations, the entropy of Hawking radiation receives contributions from additional saddle points in the gravitational path integral. These contributions can make part of the black-hole interior, called an island, count as part of the radiation system for entropy calculations. This prevents the radiation entropy from increasing forever and reproduces the Page curve.<ref name="entrhawk" /><ref name="Penington2020" /><ref name="AEMM2019" />

The island and replica-wormhole results are widely regarded as major progress. However, there is still debate over how directly these model calculations apply to realistic black holes in our universe.

== Fuzzballs and firewalls ==

The fuzzball proposal, developed in string theory, suggests that individual black-hole microstates do not have a conventional empty interior. Instead, the black hole is replaced by a complicated horizon-scale quantum structure. In this view, information is stored in the detailed microstate structure and can affect the outgoing radiation.<ref name="MathurPedag">{{cite journal |last1=Mathur |first1=Samir D. |title=The information paradox: a pedagogical introduction |journal=Classical and Quantum Gravity |date=21 November 2009 |volume=26 |issue=22 |pages=224001 |doi=10.1088/0264-9381/26/22/224001 |arxiv=0909.1038 |bibcode=2009CQGra..26v4001M |s2cid=18878424 }}</ref><ref>{{cite journal |last1=Skenderis |first1=Kostas |last2=Taylor |first2=Marika |title=The fuzzball proposal for black holes |journal=Physics Reports |date=October 2008 |volume=467 |issue=4–5 |pages=117–171 |doi=10.1016/j.physrep.2008.08.001 |arxiv=0804.0552 |bibcode=2008PhR...467..117S |s2cid=118403957 }}</ref>

The firewall proposal is another radical possibility. It suggests that an observer falling through the horizon might encounter high-energy quantum effects rather than empty space. Firewalls were proposed in response to tensions between unitarity, semiclassical geometry, and the equivalence principle.

Both fuzzballs and firewalls modify the conventional picture of a smooth black-hole horizon. They show how deeply the information paradox challenges the usual separation between quantum theory and classical spacetime geometry.

== Remnants and final-stage release ==

Another possible resolution is that information remains inside the black hole until the final stages of evaporation. When the black hole becomes Planck-sized, quantum-gravity effects could release the information or leave behind a stable remnant.

This idea has the advantage that large black holes can remain close to the semiclassical picture for most of their lifetime. However, remnant scenarios face difficulties. A tiny remnant may need to store an arbitrarily large amount of information, and this can create problems for effective field theory and entropy bounds.<ref name="Giddings1995">{{cite conference |first=Steven B. |last=Giddings |title=The black hole information paradox |arxiv=hep-th/9508151 |book-title=Particles, Strings and Cosmology |year=1995 |conference=Johns Hopkins Workshop on Current Problems in Particle Theory 19 and the PASCOS Interdisciplinary Symposium 5 |bibcode=1995hep.th....8151G }}</ref><ref name="Preskill1992">{{cite conference |first=John |last=Preskill |title=Do Black Holes Destroy Information? |conference=International Symposium on Black Holes, Membranes, Wormholes, and Superstrings |arxiv=hep-th/9209058 |year=1992 |bibcode=1993bhmw.conf...22P }}</ref>

== Soft hair ==

Stephen Hawking, Malcolm Perry, and Andrew Strominger proposed that black holes may carry soft hair: low-energy gravitational or electromagnetic degrees of freedom associated with asymptotic symmetries.<ref name="HawkingSoftHair">{{Cite journal |title=Soft Hair on Black Holes |arxiv=1601.00921 |date=2016-01-05 |first1=Stephen W. |last1=Hawking |first2=Malcolm J. |last2=Perry |first3=Andrew |last3=Strominger |doi=10.1103/PhysRevLett.116.231301 |pmid=27341223 |volume=116 |issue=23 |pages=231301 |journal=Physical Review Letters |bibcode=2016PhRvL.116w1301H |s2cid=16198886 }}</ref>

Soft hair may store some information about the black hole's history. Whether it is sufficient to solve the full information paradox remains debated.

== Information loss as a real effect ==

A minority view is that information may genuinely be lost in black-hole evaporation. This would require modifying ordinary quantum mechanics or accepting non-unitary time evolution in gravitational processes.

Such proposals are controversial because unitarity is deeply built into quantum theory. They also raise questions about energy conservation, locality, and the foundations of quantum measurement.<ref name="Giddings1995" /><ref name="Preskill1992" />

== Relation to quantum gravity ==

The information paradox is not just a puzzle about black holes. It is a clue about the structure of quantum gravity. It suggests that at least one familiar principle must be revised or reinterpreted: locality, smooth horizons, semiclassical spacetime, or the way information is encoded in gravitational systems.

Modern resolutions often point toward some form of holography or nonlocal encoding of information. This does not necessarily mean that information can be transmitted faster than light, but it does suggest that gravitational theories may not factor into independent inside and outside degrees of freedom in the same way as ordinary quantum field theories on fixed backgrounds.

== Status ==

The black hole information paradox remains an active field of research. Many researchers believe that information is preserved and that Hawking's original conclusion of information loss was incomplete. The Page curve, holography, replica wormholes, and island calculations provide strong evidence in this direction in certain models.

However, there is still no universally accepted, experimentally confirmed microscopic account of black-hole evaporation in realistic quantum gravity. The paradox therefore remains one of the most important open problems at the intersection of black holes, thermodynamics, quantum information, and spacetime.

== See also ==

{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

=References=
{{reflist|3}}

== Further reading ==

* {{Cite book |last1=Susskind |first1=Leonard |title=The Black Hole War: My Battle with Stephen Hawking to Make the World Safe for Quantum Mechanics |publisher=Little, Brown |date=2008-07-07 |isbn=9780316032698}}
* {{Cite book |last1=Susskind |first1=Leonard |last2=Lindesay |first2=James |title=Black Holes, Information and the String Theory Revolution |publisher=World Scientific |year=2005 |isbn=978-981-256-083-4}}
* {{cite journal |last1=Mathur |first1=Samir D. |title=The information paradox: a pedagogical introduction |journal=Classical and Quantum Gravity |date=21 November 2009 |volume=26 |issue=22 |pages=224001 |doi=10.1088/0264-9381/26/22/224001 |arxiv=0909.1038 |bibcode=2009CQGra..26v4001M |s2cid=18878424 }}
* {{cite journal |last1=Raju |first1=Suvrat |title=Lessons from the information paradox |journal=Physics Reports |date=January 2022 |volume=943 |pages=1–80 |doi=10.1016/j.physrep.2021.10.001 |arxiv=2012.05770 |bibcode=2022PhR...943....1R |s2cid=228083488 }}
* {{cite journal |last1=Almheiri |first1=Ahmed |last2=Hartman |first2=Thomas |last3=Maldacena |first3=Juan |last4=Shaghoulian |first4=Edgar |last5=Tajdini |first5=Amirhossein |title=The entropy of Hawking radiation |journal=Reviews of Modern Physics |date=21 July 2021 |volume=93 |issue=3 |pages=035002 |doi=10.1103/RevModPhys.93.035002 |arxiv=2006.06872 |bibcode=2021RvMP...93c5002A |s2cid=219635921 }}

{{Author|Harold Foppele}}

{{Sourceattribution|Quantum Black hole information paradox|1}}

Physics:Quantum black hole information paradox - Revision history

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