Quantum Darwinism in an Everyday Environment: Huge Redundancy in Scattered Photons C. Jess Riedel 1 and Wojciech H. Zurek 1 $^{1}$Theory Division, LANL, Los Alamos, New Mexico 87545, USA (Dated: November 5, 2018) We study quantum Darwinism—the redundant recording of information about the preferred states of a decohering system by its environment—for an object illuminated by a blackbody. In the cases of point-source and isotropic illumination, we calculate the quantum mutual information between the object and its photon environment. We demonstrate that this realistic model exhibits fast and extensive proliferation of information about the object into the environment and results in redundancies orders of magnitude larger than the exactly soluble models considered to date. PACS numbers: 03.65.Ta, 03.65.Yz, 42.50.Ar The theory of decoherence [1] is the foundation of the modern understanding of the quantum-classical transi- tion. However, the standard analysis largely ignores the environment by tracing over it when, in fact, the envi- ronment plays a crucial role in how real observers find out about the world. Classically, properties of systems are objective in that they can be independently mea- sured and agreed upon by arbitrarily many observers without disturbing the system itself. This can arise in a purely quantum universe when many copies of infor- mation about a system’s properties are imprinted onto the environment. Under the condition of effective deco- herence, this information can only describe the pointer states of the system, not superpositions thereof [2]. In this sense, pointer states are distinguished not only for forming the stable basis in which the density matrix of the system diagonalizes but also for being redundantly copied into the environment. Quantum Darwinism [3] is a framework for analyzing the flow of information about these “fittest” states, helping to elucidate the origin of classical objectivity. Quantum Darwinism has been investigated for a spin- 1 2 particle monitored by a pure [4] and mixed [5, 6] bath of spins and a harmonic oscillator monitored by a pure bath of oscillators [7, 8]. These studies support the intuition that redundant proliferation of pointer-state information should be common in decohering systems, but the models are abstract and limited in size by the feasibility of nu- merical calculations. Therefore, they do not allow one to estimate the redundancies expected in physically realis- tic systems. In this Letter we shed light on this question by considering an object illuminated by a black body. When an object in a macroscopic superposition is ex- posed to radiation, scattering photons will quickly reduce its pure, nonlocal state to a mixture of localized alterna- tives via collisional decoherence [9]. (See also [10–12] for refinements and corrections.) The fantastic rate of col- lisional decoherence has been confirmed experimentally [13, 14]. Here we investigate this process as an example of quantum Darwinism. Following Joos and Zeh [9], we consider a dielectric sphere of radius a and relative permittivity ϵ initially in a superposition, so that | ψ ( ⃗ x ) | 2 ≈ [ δ ( ⃗ x − ⃗ x$_{1}$ )+ δ ( ⃗ x − ⃗ x$_{2}$ )] / 2 for some Δ x = | ⃗ x$_{1}$ − ⃗ x$_{2}$ | (see Fig. 1). We ignore the self- Hamiltonian of the object so that its effective Hilbert space S is spanned by the two position eigenstates. The sphere is illuminated from a single direction by radiation from a point-source black body at a temperature T . Observers typically access a small part of the environ- ment (in this case, the photons that enter one’s eye), so we will estimate how much information about the object is available in a subset of the environmental photons. We assume our environment consists of a large but fixed num- ber N of photons: E = ⊗ N n =1 E$_{i}$ , where E$_{i}$ is the Hilbert space of a single photon in a box of volume V . We then define F$_{f}$ = ⊗ fN n =1 E$_{i}$ to be the fragment corresponding to some fraction f of the environment composed of f N photons. Since the photons have identical initial condi- tions and interactions, the choice of photons with which to construct the fragment is unimportant. To get our fi- !  FIG. 1. A dielecric sphere of radius a and permittivity ϵ is ini- tially in a superposition with separation Δ x = | x$_{1}$ − x$_{2}$ | . The object is subjected to plane-wave radiation with thermally distributed wavelength λ and propagating in a direction ˆ n that makes an angle θ with the vector ⃗ Δ x . ! $_{"}$x a # $ x$_{1}$ x$_{2}$ ˆ  n  arXiv:1001.3419v3 [quant-ph] 9 Jul 2010