Type-Independent Characterization of Spacelike Separated Resources

Quantum theory describes multipartite objects of various types: quantum states, nonlocal boxes, steering assemblages, teleportages, distributed measurements, channels, and so on. Such objects describe, for example, the resources shared in quantum networks. Not all such objects are useful, however. In the context of spacelike separated parties, devices which can be simulated using local operations and shared randomness are useless, and it is of paramount importance to be able to practically distinguish useful from useless quantum resources. Accordingly, a body of literature has arisen to provide tools for witnessing and quantifying the nonclassicality of objects of each specific type. In the present Letter, we provide a framework which subsumes and generalizes all of these resources, as well as the tools for witnessing and quantifying their nonclassicality.

Figure 1

We exhaustively characterize the nonclassicality of five resources of four different types by their convertibility relations under LOSR operations (figure). We also characterize their nonclassicality using a type-independent absolute robustness monotone $M_{\mathrm{abs}}$ (table) that we introduce below. Any single monotone is only partially informative; e.g., $M_{\mathrm{abs}}$ assigns the same value to $\{{\mu }_{a|x}^{XZ}\}$ and ${\overrightarrow{P}}_{\text{Tsi}}$, even though the former is strictly more nonclassical than the latter.

Figure 2

A two-party resource $R\in \mathsf{D}(\mathcal{X}\mathcal{Y})\to \mathsf{D}(\mathcal{A}\mathcal{B})$ and its action on product input $\xi \otimes \psi$; by allowing $\mathcal{X}$, $\mathcal{Y}$, $\mathcal{A}$, $\mathcal{B}$ to be trivial, classical, or quantum, we unify all the resource types shown in Table 1.

Figure 3

Graphical proof of case 1. of Proposition 2. (a) The most general transformation from type $\mathsf{C}\mathsf{C}\to \mathsf{C}\mathsf{C}$ to type $\mathsf{I}\mathsf{I}\to \mathsf{Q}\mathsf{Q}$; without loss of generality, the side channels can be taken to be classical. One can imagine the transformation in (a) as a two-step procedure: the initial resource (b) of type $\mathsf{C}\mathsf{C}\to \mathsf{C}\mathsf{C}$ is transformed to a resource (c) of type $\mathsf{I}\mathsf{I}\to \mathsf{C}\mathsf{C}$, and then to a resource (d) of type $\mathsf{I}\mathsf{I}\to \mathsf{Q}\mathsf{Q}$. In (b)–(d), the shading indicates the portion of the figure whose type is labeled.