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, O out θ = Q in θ . 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.One of the biggest differences between classical and quantum physics has been demonstrated using Bell's inequalities in entanglement [1]. Moreover, with the recent rapid development of quantum information technology, entanglement has been shown to have substantial importance [2]. On the other hand, interest in the so-called quantum measurement problem and quantum foundations have also resurged recently (for example, see [3] and the references therein). One of the puzzling features of quantum measurement, apart from its probabilistic nature, is that it has two quite different measurement protocols. While the single-system Copenhagen measurement protocol deals with only one system and observables, the two-system von Neumann protocol introduces the second system as a memory and uses entanglement. In this paper, using the negative entropy of entanglement [4], we will show the equivalence of these two quantum measurement schemes. The approaches can be made equivalent by considering that the second system of von Neumann's approach fills the system similar to Dirac's sea of negative energy filling up the vacuum.We then apply the equivalence of the two quantum measurements to the process of black hole evaporation. In particular, we wish to discuss two different approaches to black hole radiation, the observable-based single-system [5] and the entanglement-based two-system [6] protocols. Similar to the measurement case, these two approaches yield the same thermal radiation of a system outside the black hole. We will suggest that the black hole radiation should be the quantum measurement process. In particular, we will argue that the single-system evaporation approach should correspond to the single-system Copenhagen measurement protocol and the entanglement-based approach should correspond to the two-system von Neumann measurement scheme. Based on this, ...

Information-Probabilistic Description of the Universe

“…In our opinion, a satisfactory model of the entropy for the universe can be obtained using the ideas of information interpretation of quantum mechanics [47][48][49], in which a measurement is considered as the interaction of three systems: the quantum object, memory (measurement device) and observer. Then the total entropy of the universe can be assumed to be zero (as it is suggested by von Neumann's model) and, formally, it can be written as the sum of the information, statistical (thermodynamic) and quantum (entanglement) components [46,52],…”

“…An example is the von Neumann's multi-component measurement scheme, in which the quantum system, memory (or apparatus), and the observer himself are involved, and the entropy component for memory yields a negative value. There have been a number of notable interpretations of the negative entanglement entropy, as a potential information [44], a reversible work [45], or a memory filling the system similar to the Dirac's sea [46]. The fact that the joint quantum entropy S tot in (6) can be less than a sum of marginal entropies of subsystems, S, is one of the most fundamental differences between the classical and quantum information, and it is a key observation in quantitative realization of the principle of information-entropy conservation used in informational interpretation of quantum mechanics [47][48][49].…”

Cosmological Constant from the Entropy Balance Condition

“…In quantum mechanics a measurement is considered as the interaction of three systems: the quantum object, memory (measurement device) and observer. Then the total entropy of the ensemble of all quantum particles, which is formed by the information, statistical (thermodynamic) and quantum (entanglement) components [16,17], can be assumed to be zero [13]. In this case the universe can always remain in pure state and only allow a unitary time-evolution, as it is suggested by von Neumann's model.…”

“…Due to the presence of derivatives in (16) the cosmological constant Λ does not appears in (17) as well. So, if instead of (14), one will choice (15) and (17) as the independent system of cosmological equations, Λ obtains the role of integration constant which can be fixed from an equation of state.…”

“…So, if instead of (14), one will choice (15) and (17) as the independent system of cosmological equations, Λ obtains the role of integration constant which can be fixed from an equation of state. Indeed, combining (15) with (17) and integrating over the time we have,…”