Amplification, Redundancy, and Quantum Chernoff Information

Amplification was regarded, since the early days of quantum theory, as a mysterious ingredient that endows quantum microstates with macroscopic consequences, key to the “collapse of the wave packet,” and a way to avoid embarrassing problems exemplified by Schrödinger’s cat. Such a bridge between the quantum microworld and the classical world of our experience was postulated ad hoc in the Copenhagen interpretation. Quantum Darwinism views amplification as replication, in many copies, of the information about quantum states. We show that such amplification is a natural consequence of a broad class of models of decoherence, including the photon environment we use to obtain most of our information. This leads to objective reality via the presence of robust and widely accessible records of selected quantum states. The resulting redundancy (the number of copies deposited in the environment) follows from the quantum Chernoff information that quantifies the information transmitted by a typical elementary subsystem of the environment.

Figure 1

Quantum Darwinism, photons, and the emergence of objective classical reality. A quantum system, $\mathcal{S}$, initially in a nonlocal superposition, is illuminated by the environment, $\mathcal{E}$, composed of many distinct subsystems (photons) that can be lumped into fragments $\mathcal{F}$. While $\mathcal{E}$ decoheres $\mathcal{S}$, it acquires many copies of information about $\mathcal{S}$ that become available to observers who can then independently infer the state of the preferred (pointer) state of the system without perturbing $\mathcal{S}$ by direct measurements. This redundant imprinting of records is responsible for the consensus between observers that is essential for the emergence of “objective classical reality” in our quantum Universe. For the familiar photon environment, the redundancy (which quantifies amplification) can be enormous: A dust grain $1\mu \mathrm{m}$ across exposed to sunlight for just $1\mu \mathrm{s}$ will have its location (to an accuracy of $1\mu \mathrm{m}$) recorded about ${10}^8$ times in the scattered photons [5].