obeys two principles that are relevant for the paradox. These are the principles of quantum determinism, which means that given a present wave function, its future changes are uniquely determined by the evolution operator and also the principle of 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" is used to refer to 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. 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 to be irreversible if one keeps track only of coarse-grained features of the system and does not keep track of its microscopic details, as is usually done in thermodynamics. However, 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 irreversible 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 emitted by isolated black holes. This calculation utilized the framework of general relativity and quantum field theory. The calculation of Hawking radiation is performed at the black-hole horizon; for a large enough black hole the curvature at the horizon is small and therefore both these theories should be valid. Hawking relied