General Method for Classicality Certification in the Prepare and Measure Scenario

Preparing and measuring physical systems are the operational building blocks of any physical experiment, and to describe them is the first purpose of any physical theory. Remarkably, even when only uncharacterized preparation and measurement devices are present, it is sometimes possible to distinguish between the behaviors of quantum and classical systems from only observational data. Certifying the physical origin of measurement statistics in the prepare and measure scenario is of primal importance for developing quantum networks, distributing quantum keys, and certifying randomness, to mention a few applications, but, surprisingly, no general methods to do so are known. We progress on this problem by crafting a general, sufficient condition to certify that a given set of preparations can only generate classical statistics, for any number of generalized measurements. As an application, we employ the method to demonstrate nonclassicality activation in the prepare and measure scenario, also considering its application in random access codes. Following that, we adapt our method to certify, again through a sufficient condition, whether a given set of measurements can never give rise to nonclassical behaviors, irrespective of what preparations they may act upon. This, in turn, allows us to find a large set of incompatible measurements that cannot be used to demonstrate nonclassicality, thus showing incompatibility is not sufficient for nonclassicality in the prepare and measure scenario.

Popular Summary

Quantum systems can display many advantages over their classical counterparts for information processing tasks. However, in general, one needs to have a high level of control over the systems to assess their nonclassicality, since not all quantum resources are suitable for all tasks. On a more fundamental side, it is desirable to identify those quantum resources that are unsuitable for the task at hand, that is, those that would lead to no advantage whatsoever. We pursue this goal for a simple, yet very important, experimental scenario. We consider a generic experiment where a party prepares a system, classical or quantum, in one of several possible states, and sends it to a second party, that performs one of several possible measurements. This prepare and measure scenario is paradigmatic and is underneath many quantum information primitives, such as quantum key distribution, random access codes, and superdense coding.

Within the prepare and measure scenario, we provide a method to certify that a given set of quantum states can only generate classical statistics. Our method can also be extended to test sets of measurements. We find a large set of incompatible measurements that cannot be used to demonstrate nonclassicality; therefore, showing incompatibility is not sufficient for nonclassicality. The techniques we present can be employed on many advanced applications that are based on prepare and measure schemes.

Figure 1

Operational and causal representations of a PAM scenario. On the left, the black-box shows a preparation, conditioned on input $x$, being sent to a measurement device with measurement choice $y$ and output $b$. On the right, the DAG turns explicit the unobservable variables $a$ and $\lambda$ in the causal structure.

Figure 2

Representation of the method for $d=2$. Each vertex on the Bloch sphere represents a measurement operator. To every ${\mathrm{\Pi }}_{0|y}$ we associate a corresponding antipodal ${\mathrm{\Pi }}_{1|y}$. Measurements inside the set enclosed by $\mathbf{conv}(\mathcal{M})$ can be simulated by mixing these extremal ones. In particular, any measurement $\{{\mathrm{\Pi }}_{b|u}^{\eta }{\}}_b$ in a ball with radius $\eta$ inscribed in the polytope is simulatable in such manner.

Figure 3

Application of program (6a) to $\mathcal{S}(\theta ,\varphi )=\{{\rho }_{\mathit{x}},{\rho }_{\mathit{z}},{\rho }_{\mathit{r}(\theta ,\varphi )}\}$. Levels are the maximum visibility $\alpha$ such that preparation set $\alpha \mathcal{S}(\theta ,\varphi )$ has a classical model. (a) For the $|\mathcal{Y}|=6$ icosahedron measurements, $\eta \approx 0.79$, and program (6a) can be applied directly. (b) A rhombicuboctahedron corresponds to $|\mathcal{Y}|=12$ PMs and $\eta \approx 0.86$. As the number of deterministic strategies scales exponentially, the computation is only possible by iteratively optimizing over subsets of deterministic strategies.

Figure 4

Preparations $\mathcal{S}(\alpha ,\theta )=\{{\rho }_{{\mathit{r}}_{\mathbf{1}}},{\rho }_{{\mathit{r}}_{\mathbf{2}}},{\rho }_{{\mathit{r}}_{\mathbf{3}}},{\rho }_{{\mathit{r}}_{\mathbf{4}}}\}$ for $\alpha =0.8$. At $\theta =0$ all preparations are at $\alpha \mathit{y}$. For $\theta =\pi /2$, $\mathcal{S}=\{-\alpha \mathit{x},\alpha \mathit{x},-\alpha \mathit{z},\alpha \mathit{z}\}$, presenting the largest violation of inequality (7).

Figure 5

Nonclassicality activation for the preparations in Fig. 4 and measurements arranged in a rhombicuboctahedron with corresponding $\eta \approx 0.86$. As $S\propto \alpha$, every preparation set above the $S=4$ curve is nonclassical. On the other hand, the blue scatter shows the maximum visibility such that any triad of states in the preparation set are classical. The shaded region thence stands for preparations that exhibit nonclassicality activation by increasing the number of preparations.

Figure 6

Mirror-symmetric measurements used to show the existence of incompatible measurements that do not exhibit nonclassical statistics for any preparation set.

Figure 7

Incompatibility is insufficient for nonclassicality in the PAM scenario. For each $\theta$ (see Fig. 6), any $\chi$ above the incompatibility robustness curve stands for an incompatible measurement set, and any $\alpha$ below the measurement classicality lower bound represents measurements that certifiedly do not generate nonclassical statistics, regardless what preparations they act upon. Thenceforth, the shaded region contains incompatible albeit classical measurements.