Complementarity of information and the emergence of the classical world

Hilbert space is a big place, exponentially larger than the arena of classical physics. The Hilbert space of macroscopic systems is dominated by states that have no classical counterparts. Yet the world observed by macroscopic observers exhibits powerful regularities that make it amenable to classical interpretations on a broad range of scales. How do we explain this? The answer, of course, is that Hilbert space is not sampled uniformly; rather, the initial state and the Hamiltonian governing evolution are both very special.

Quantum Darwinism is a framework for describing and quantifying what distinguishes quasi-classical states awash in the enormous sea of Hilbert space. Typical macroscopic observers do not directly interact with a system. Instead, they sample a (small) fragment of its environment in order to infer its state, using the environment as an information channel. Thus, when we measure the position of a chair by looking at it, our eyes do not directly interact with the chair. By opening our eyes, we merely allow them (and hence, our neurons) to become correlated with some of the photons scattered by chair (and hence, its position).

I will establish a new conservation law: The amount of quantum information about a system’s observable deposited in a fragment of the environment plus the amount of classical information yields an observable-independent total given by the quantum mutual information. In the quantum-to-classical transition, this split naturally delineates information about quantum systems accessible to observers – information that is redundantly transmitted by the environment – while showing that it is maximized for the quasi-classical pointer observable. Other observables are accessible only via correlations with the pointer observable. Further, we prove an anti-symmetry property relating accessible, classical information and quantum information. It shows that information becomes objective – accessible to many observers – only as quantum information is relegated to correlations with the global environment, and, therefore, locally inaccessible. The resulting complementarity explains why, in a quantum Universe, we perceive objective classical reality, and supports Bohr’s intuition that quantum phenomena acquire classical reality only when communicated.