Emergence of Noncontextuality under Quantum Darwinism

Quantum Darwinism proposes that the proliferation of redundant information plays a major role in the emergence of objectivity out of the quantum world. Is this kind of objectivity necessarily classical? We show that if one takes Spekkens’s notion of noncontextuality as the notion of classicality and the approach of Brandão, Piani, and Horodecki to quantum Darwinism, the answer to the above question is “‘yes,” if the environment encodes the proliferated information sufficiently well. Moreover, we propose a threshold on this encoding, above which one can unambiguously say that classical objectivity has emerged under quantum Darwinism.

Popular Summary

It is widely known that quantum theory possesses very counterintuitive properties. Moreover, quantum technologies aim to obtain advantages from such nonclassical features. From this point of view, contextuality is one vital resource. Indeed, contextuality is a generalization of nonlocality and it underpins advantages in crucial tasks. Furthermore, experimental tests of contextuality have also been performed, always confirming the contradiction with the classical predictions. On the one hand, such nonclassical manifestations bring excitement to the quantum world but, on the other, they pose a problem: how to reconcile the quantum description of natural phenomena with the classical world we experience every day?

In our work, we tackle this problem from the viewpoint of contextuality and show that noncontextuality emerges under quantum Darwinism. In a Darwinist process, some information about a quantum system is redundantly stored in portions of its environment, allowing independent observers to discover such information—and to agree on what they see. In other words, quantum Darwinism leads to the emergence of (a notion of) objectivity. We then show that the information that they agree upon admits noncontextual modeling, thus deeming contextual descriptions unnecessary. Remarkably, our methods may also serve quantum-state-discrimination research. We show that a high success probability on such a task leads to information on the geometrical disposition of the states being discriminated against.

In summary, our work may find application in different areas of quantum information. Moreover, it contributes to the study of the quantum-to-classical transition, reconciling two essential notions of classicality: noncontextuality and objectivity.

Figure 1

Different approaches to open systems. (a) The decoherence paradigm. The focus is on the central system $A$. The environment is treated as a monolithic system, usually to be traced out after the interaction. (b) The environment split into fractions. The environment is more accurately described as a composition of several subsystems. (c) Environment-as-a-witness dynamics and the quantum Darwinism process. Under this paradigm, the environment is promoted to an information channel, spreading redundant information about system $A$ to independent observers. In the example of the figure, the number of portions in $S_t$ is $t=3$ and $B_{S_t}=\{B_1,B_2,B_3\}$. Theorem 4 provides us with a sufficient condition for such redundant information to be noncontextual, i.e., classical.

Figure 2

The scenario from the perspective of Bob $B_j$: preparation $P\in \mathcal{P}$ of the central system is followed by a fixed transformation (i.e., interaction with the environment, always prepared in the same way), leading to an effective preparation $P^{\prime }=T_j(P)$. The nontraced-out portion of the environment is given by the labels $S_t$ (here, $t=3$). For Bob, $[P^{\prime }]\leftrightarrow {\mathrm{\Phi }}^{B_j}({\rho }^A)$ and $[b|M]\leftrightarrow F_b$.

Figure 3

An example of Caratheodory’s theorem: an example of an affinely dependent set with four states ($k_{\max }=4$). Their affine hull $E$ forms a plane; thus $\dim (E)=2$ and $\mathrm{ConvHull}[\{{\sigma }_k{\}}_k]$ is a quadrilateral contained in $E$. An interior point, ${\sigma }_T$, can be written as a convex combination of $2+1=3$ vertices (as shown by the red vertices and the purple dashed triangle). Thus, we can write ${\sigma }_T=\sum _k{q}_k{\sigma }_k$ with $q_4=0$.