The black hole information paradox is an unsolved problem in physics and a paradox that appears when the predictions of quantum mechanics and general relativity are combined. The theory of general relativity predicts the existence of black holes that are regions of spacetime from which nothing—not even light—can escape. In the 1970s, Stephen Hawking applied the semiclassical approach of quantum field theory in curved spacetime to such systems and found that an isolated black hole would emit a form of radiation (now called Hawking radiation in his honor). He also argued that the detailed form of the radiation would be independent of the initial state of the black hole, and depend only on its mass, electric charge and angular momentum.

The information paradox appears when one considers a process in which a black hole is formed through a physical process and then evaporates away entirely through Hawking radiation. Hawking's calculation suggests that the final state of radiation would retain information only about the total mass, electric charge and angular momentum of the initial state. Since many different states can have the same mass, charge and angular momentum, this suggests that many initial physical states could evolve into the same final state. Therefore, information about the details of the initial state would be permanently lost; however, this violates a core precept of both classical and quantum physics: that, in principle only, the state of a system at one point in time should determine its state at any other time. Specifically, in quantum mechanics the state of the system is encoded by its wave function. The evolution of the wave function is determined by a unitary operator, and unitarity implies that the wave function at any instant of time can be used to determine the wave function either in the past or the future. In 1993, Don Page argued that if a black hole starts in a pure quantum state and evaporates completely by a unitary process, the von Neumann entropy of the Hawking radiation initially increases and then decreases back to zero when the black hole has disappeared. This is called the Page curve.

It is now generally believed that information is preserved in black-hole evaporation. For many researchers, deriving the Page curve is synonymous with solving the black hole information puzzle. But views differ as to precisely how Hawking's original semiclassical calculation should be corrected. In recent years, several extensions of the original paradox have been explored. Taken together, these puzzles about black hole evaporation have implications for how gravity and quantum mechanics must be combined. The information paradox remains an active field of research in quantum gravity.

In quantum mechanics, the evolution of the state is governed by the Schrödinger equation. The Schrödinger equation obeys two principles that are relevant to the paradox—quantum determinism, which means that given a present wave function, its future changes are uniquely determined by the evolution operator, and reversibility, which refers to the fact that the evolution operator has an inverse, meaning that the past wave functions are similarly unique. The combination of the two means that information must always be preserved. In this context "information" means all the details of the state, and the statement that information must be preserved means that details corresponding to an earlier time can always be reconstructed at a later time.

Mathematically, the Schrödinger equation implies that the wavefunction at a time t₁ can be related to the wavefunction at a time t₂ by means of a unitary operator. $${\displaystyle |\Psi (t_{1})\rangle =U(t_{1},t_{2})|\Psi (t_{2})\rangle .}$$ Since the unitary operator is bijective, the wavefunction at t₂ can be obtained from the wavefunction at t₁ and vice versa.

The reversibility of time evolution described above applies only at the microscopic level, since the wavefunction provides a complete description of the state. It should not be conflated with thermodynamic irreversibility. A process may appear irreversible if one keeps track only of the system's coarse-grained features and not of its microscopic details, as is usually done in thermodynamics. But at the microscopic level, the principles of quantum mechanics imply that every process is completely reversible.

Starting in the mid-1970s, Stephen Hawking and Jacob Bekenstein put forward theoretical arguments that suggested that black-hole evaporation loses information, and is therefore inconsistent with unitarity. Crucially, these arguments were meant to apply at the microscopic level and suggested that black-hole evaporation is not only thermodynamically but microscopically irreversible. This contradicts the principle of unitarity described above and leads to the information paradox. Since the paradox suggested that quantum mechanics would be violated by black-hole formation and evaporation, Hawking framed the paradox in terms of the "breakdown of predictability in gravitational collapse".

The arguments for microscopic irreversibility were backed by Hawking's calculation of the spectrum of radiation that isolated black holes emit. This calculation utilized the framework of general relativity and quantum field theory. The calculation of Hawking radiation is performed at the black hole horizon and does not account for the backreaction of spacetime geometry; for a large enough black hole the curvature at the horizon is small and therefore both these theories should be valid. Hawking relied on the no-hair theorem to arrive at the conclusion that radiation emitted by black holes would depend only on a few macroscopic parameters, such as the black hole's mass, charge, and spin, but not on the details of the initial state that led to the formation of the black hole. In addition, the argument for information loss relied on the causal structure of the black hole spacetime, which suggests that information in the interior should not affect any observation in the exterior, including observations performed on the radiation the black hole emits. If so, the region of spacetime outside the black hole would lose information about the state of the interior after black-hole evaporation, leading to the loss of information.