The concept of conservation of information in quantum mechanics can be somewhat subtle and is still an area of active research and discussion. While the expansion of the observable universe may lead to an increase in the amount of information available locally, it is important to distinguish between different notions of information in the context of quantum mechanics.
In quantum mechanics, information is typically associated with the state of a quantum system, which is described by a wave function or a density matrix. The evolution of a closed quantum system is governed by the unitary evolution prescribed by Schrödinger's equation, which is reversible and conserves information. In other words, the total amount of information within a closed quantum system remains constant over time.
However, when we consider the entanglement of quantum systems, the picture becomes more nuanced. Entanglement is a form of correlation between quantum systems where the state of one system becomes intrinsically linked to the state of another, even when they are spatially separated. When two systems become entangled, the information about their individual states is no longer separable or localized within each system alone.
In the context of an expanding universe, the increase in the amount of information can be attributed to the proliferation of entangled states due to the expansion of space itself. As the universe expands, distant regions that were previously causally disconnected can come into contact, leading to the formation of new entangled states. This can result in an apparent increase in the amount of information locally accessible.
It is important to note that the conservation of information in quantum mechanics refers to the total amount of information within a closed system. When we consider the entire universe as a closed quantum system, the conservation of information holds. However, when we focus on localized regions or subsystems within the universe, the increase in information due to entanglement can occur.
The interplay between the expansion of the universe, entanglement, and information conservation is a complex topic that is still being explored. It involves understanding the role of entanglement entropy, the holographic principle, and the dynamics of quantum fields in curved spacetime. Ongoing research in quantum gravity and quantum information theory aims to shed more light on these fundamental aspects of information conservation in the context of an expanding universe.