Unitarity

Unitarity is the requirement that time evolution in quantum mechanics be implemented by a unitary operator — one satisfying U†U = 1. The immediate consequence is that the total probability of all outcomes stays equal to one, and that the inner product between any two states is preserved as they evolve. Evolution is reversible: given the final state and the Hamiltonian, the initial state can always, in principle, be reconstructed.

A deeper consequence concerns information. Under unitary evolution a pure state remains pure forever, and the von Neumann entropy S = −Tr(ρ ln ρ) of a closed system never changes. Information defining a quantum state can be scrambled into inaccessible correlations — as when a book burns and its content hides in the photons and ash — but it is never erased. This is why physicists say quantum mechanics 'conserves information.'

Unitarity is the property that black-hole evaporation appears to violate. Hawking's 1974 calculation suggests a black hole turns a pure initial state into thermal (mixed) radiation, which would be a non-unitary, information-destroying process. The information paradox is precisely the tension between this result and the unitarity that underlies all of quantum theory; the Page curve is the quantitative statement of what a unitary evaporation must look like.

Unitarity was built into quantum mechanics from its 1920s foundations through the work of Schrödinger, Dirac, and von Neumann, who formalized states as vectors in Hilbert space and evolution as a unitary group. Its role moved to center stage in gravitational physics after Hawking's 1976 claim that black holes break it, making 'unitarity vs. gravity' one of the defining problems of twentieth- and twenty-first-century theoretical physics.