Macroscopic observables (16 pages)

Alberto Barchielli, L. Lanz, G. Lupieri

Rathindra Nath Sen

In a certain sense, object and apparatus were essentially on the same footing in von Neumann's measurement theory. They were systems with k and l degrees of freedom respectively (page 146). There were no constraints on the numbers k and l , which could, for example, be of the same order of magnitude. Quite possibly, it was this lack of differentiation between object and apparatus that led Wigner to assert that ‘the state of the apparatus has no classical description’ (page 137), a state of affairs that produced the infinite von Neumann chain which only ended in the observer's consciousness. In this and the following chapter we shall break with von Neumann and assume that k is small, i.e., of the order of unity, and that l is large, i.e., within a few orders of magnitude of Avogadro's number. The room for manoeuvre that this provides will allow us to break with Wigner and explore systems with states that do have classical descriptions. It will not surprise the informed reader that the room for manoeuvre created by our assumption will be filled, very substantially, by von Neumann's own work. This chapter is divided into three sections. Section 9.1 is devoted to a theorem of von Neumann on observables that commute with each other. This prepares the way to our treatment of macroscopic observables, which is based on the commuting approximations to P and Q devised by von Neumann; these results, together with their antecedents, are presented in Section 9.2. In Section 9.3, an attempt is made to resolve some of Wigner's doubts.

Stephen D. H. Hsu

We discuss the black hole information problem, including the recent claim that unitarity requires a horizon firewall, emphasizing the role of decoherence and macroscopic superpositions. We consider the formation and evaporation of a large black hole as a quantum amplitude, and note that during intermediate stages (e.g., after the Page time), the amplitude is a superposition of macroscopically distinct (and decohered) spacetimes, with the black hole itself in different positions on different branches. Small but semiclassical observers (who are themselves part of the quantum amplitude) that fall into the hole on one branch will miss it entirely on other branches and instead reach future infinity. This observation can reconcile the subjective experience of an infalling observer with unitarity. We also discuss implications for the nice slice formulation of the information problem, and to complementarity.

Pan Li, Yi Ling

The Page curve plotted by the typical random state approximation is not applicable to a system with conserved quantities, such as the evaporation process of a charged black hole during which the electric charge does not radiate out with a uniform rate macroscopically. In this context the symmetry-resolved entanglement entropy may play a significant role in describing the entanglement structure of such a system. We attempt to impose constraints on microscopic quantum states to match with the macroscopic phenomenon of the charge radiation during black hole evaporation. Specifically, we consider a simple qubit system with conserved spin/charge serving as a toy model for the evaporation of charged black holes. We propose refined rules for selecting a random state with conserved quantities to simulate the distribution of charges during the different stages of evaporation, and obtain refined Page curves that exhibit distinct features in contrast to the original Page curve. We find the refined Page curve may have a different Page time and exhibit asymmetric behavior on both sides of the Page time. Such refined Page curves may provide more realistic description for the entanglement between the charged black hole and radiation during the process of evaporation.