Negative Entropy and Black Hole Information

Abstract

Based on negative entropy in entanglement, it is shown that a single-system Copenhagen measurement protocol is equivalent to the two-system von Neumann scheme with the memory filling up the system with negative information similar to the Dirac sea of negative energy. After equating the two quantum measurement protocols, we then apply this equivalence to the black hole radiation. That is, the black hole evaporation corresponds to the quantum measurement process and the two evaporation approaches, the observable-based single-system and the two-system entanglement-based protocols, can be made equivalent using quantum memory. In particular, the measurement choice θ with the memory state inside the horizon in the entanglement-based scheme is shown to correspond to the observable of the measurement choice θ outside the horizon in the single-system protocol, that is, $\mathcal{O}_{\theta}^{\mathrm{out}} = Q_{\theta}^{\mathrm{in}}$. This indicates that the black hole as quantum memory is filling up with negative information outside the horizon, and its entropy corresponds to the logarithm of a number of equally probable measurement choices. This shows that the black hole radiation is no different than ordinary quantum theory.