Somewhere in the universe: Where is the information stored when histories decohere?

In the context of the decoherent histories approach to quantum theory, we investigate the idea that decoherence is connected with the storage of information about the decohering system somewhere in the universe. The known connection between decoherence and the existence of records is extended from the case of pure initial states to mixed states, where it is shown that records may still exist but are necessarily imperfect. We formulate an information-theoretic conjecture about decoherence due to an environment; the number of bits required to describe a set of decoherent histories is approximately equal to the number of bits of information thrown away to the environment in the coarse-graining process. This idea is verified in a simple model consisting of a particle coupled to an environment that can store only one bit of information. We explore the decoherence and information storage in the quantum Brownian motion model, in which a particle trajectory is decohered as a result of coupling to an environment of harmonic oscillators in a thermal state. It is shown that the variables that the environment naturally measures and stores information about are nonlocal functions of time, which are essentially the Fourier components of the function $x\left(t\right)\mathrm{}$ (describing the particle trajectory). In particular, the records storing the information about the Fourier modes are the positions and momenta of the environmental oscillators at the final time. We show that it is possible to achieve decoherence even if there is only one oscillator in the environment. The information count of the histories and records in the environment adds up according to our conjecture. These results give quantitative content to the idea that decoherence is related to “information lost.” Some implications of these ideas for quantum cosmology are discussed.