state. A pure state is assigned a von Neumann entropy of 0 whereas a mixed state has a finite entropy. The unitary evolution of a state according to Schrödinger's equation preserves the entropy. Therefore Hawking's argument suggests that the process of black-hole evaporation cannot be described within the framework of unitary evolution. Although this paradox is often phrased in terms of quantum mechanics, the evolution from a pure state to a mixed state is also inconsistent with Liouville's theorem in classical physics. (see e.g.). In equations, Hawking showed that if one denotes the creation and annihilation operators at a frequency for a quantum field propagating in the black-hole background by and then the expectation value of the product of these operators in the state formed by the collapse of a black hole would satisfy where where k is the Boltzmann constant and T is the temperature of the black hole. (See, for example, section 2.2 of.) There are two important aspects of the formula above. The first is that the form of the radiation depends only on a single parameter, the temperature, even though the initial state of the black hole cannot be characterized by one parameter. Second, the formula implies that the black hole radiates mass at a rate given by where a is constant related to fundamental constants, including the Stefan–Boltzmann constant and certain properties of the black hole spacetime called its greybody factors. The temperature of the black hole is in turn dependent on the mass, charge and angular momentum of the black hole. For a Schwarzschild black hole the temperature is given by This means that if the black hole starts out with an initial mass , it evaporates completely in a time proportional to . The important aspect of the formulas above is they suggest that the final gas of radiation formed through this process depends only on the black hole's temperature and is independent of other details of the initial state. This leads to the following paradox. Consider two distinct initial states that collapse to form a Schwarzschild black hole of the same mass. Even though the states were distinct to start with, since the mass (and, hence, the temperature) of the black holes is the same, they will emit the same Hawking radiation. Once the black holes evaporate completely, in both cases, one will be left with a featureless gas of radiation. This