Strong Quantum Darwinism and Strong Independence are Equivalent to Spectrum Broadcast Structure

How the objective everyday world emerges from the underlying quantum behavior of its microscopic constituents is an open question at the heart of the foundations of quantum mechanics. Quantum Darwinism and spectrum broadcast structure are two different frameworks providing key insight into this question. Recent works, however, indicate these two frameworks can lead to conflicting predictions on the objectivity of the state of a system interacting with an environment. Here, we provide a resolution to this issue by defining strong quantum Darwinism and proving that it is equivalent to spectrum broadcast structure when combined with strong independence of the subenvironments. We further show that strong quantum Darwinism is sufficient and necessary to signal state objectivity without the requirement of strong independence. Our Letter unveils the deep connection between strong quantum Darwinism and spectrum broadcast structure, thereby making fundamental progress toward understanding and solving the emergence of classicality from the quantum world. Together they provide us a sharper understanding of the transition in terms of state structure, geometry, and quantum and classical information.

Figure 1

Quantum Darwinism and spectrum broadcast structure. (a) Quantum Darwinism recognizes that the environment is made up of different fragments, for example, $\mathcal{E}={\mathcal{E}}_1\otimes {\mathcal{E}}_2\otimes \cdots \otimes {\mathcal{E}}_5$. Different observers access the properties of the system by measuring different environment fragments. (b) A spectrum broadcast structure state can be viewed as the existence of different branches, where there are different, distinguishable states ${\{{\rho }_i^{{\mathcal{E}}_k}\}}_i$ given different conditional system states.

Figure 2

Accessible classical information given by the Holevo information $\chi ({\mathcal{S}}^{\mathrm{\Pi }}:\mathcal{F})$ about the system stored in different fractions of the environment. (Green dashed) For pure states picked out at random (through a Haar measure), the amount of accessible classical information and quantum discord are typically roughly the same. (Red solid) For a state that satisfies strong quantum Darwinism, such as the reduced GHZ state, $\chi ({\mathcal{S}}^{\mathrm{\Pi }}:\mathcal{F})$ will be approximately equal to the system entropy $H(\mathcal{S})$ for small fractions of the environment, suggesting the existence of multiple copies of the information and hence redundancy [cf. analogous Fig. 2 in Zurek [4] ].