Addressing some common objections to generalized noncontextuality

When should a given operational phenomenology be deemed to admit of a classical explanation? When it can be realized in a generalized-noncontextual ontological model. The case for answering the question in this fashion has been made in many previous works and motivates research on the notion of generalized noncontextuality. Many criticisms and concerns have been raised, however, regarding the definition of this notion and of the possibility of testing it experimentally. In this work, we respond to some of the most common of these objections. One such objection is that the existence of a classical record of which laboratory procedure was actually performed in each run of an experiment implies that the operational equivalence relations that are a necessary ingredient of any proof of the failure of noncontextuality do not hold, and consequently that conclusions of nonclassicality based on these equivalences are mistaken. We explain why this concern is unfounded. Our response affords the opportunity for us to clarify certain facts about generalized noncontextuality, such as the possibility of having proofs of its failure based on a consideration of the subsystem structure of composite systems. Similarly, through our responses to each of the other objections, we elucidate some under-appreciated facts about the notion of generalized noncontextuality and experimental tests thereof.

Figure 1

The basic PM scenario.

Figure 2

The scenario with the lab notebook modeled as a physical system (denoted $X$) which is on the same footing as $S$.

Figure 3

Two experimental prepare-measure scenarios that achieve the correlations $P\left(AB\right|XY)$ of Eq. (13). (a) The causal mediary is a nonclassical GPT system, in which case the correlations are evidence of nonclassicality, and (b) the causal mediary is a classical system, in which case the correlations are not evidence of nonclassicality. The difference between experiments (a) and (b) is manifested in the operational identities that hold among the preparations of the system being transmitted. The specific GPT states and measurements in panel (a) live in the GPT known as Boxworld [11] and we follow the notation given in Eq. (8) of Ref. [21].

Figure 4

To determine if the statistics $p\left(DEF\right|ABC)$ generated by some GPT circuit are classically explainable or not, one must look at the operational identities holding among different processes that could appear in any given gate within the circuit. Thus, these operational identities can only be defined relative to the circuit structure. For instance, an operational identity between the different possibilities for the gate taking $S_1$ to $S_3,S_4$ (indexed by setting $A$ and outcome $D$) can be written as in panel (b).

Figure 5

As shown in Ref. [6], the statistics $p\left(DEF\right|ABC)$ generated by some GPT circuit are explainable within a noncontextual ontological model $\mathrm{\Gamma }$ if and only if there exists a linear map from the GPT circuit into a circuit of the same form but where all systems are classical random variables, and where all transformations are substochastic maps.