Violating the KCBS Inequality with a Toy Mechanism

Abstract

In recent years, much research has been devoted to exploring contextuality in systems that are not strictly quantum, like classical light, and many theory-independent frameworks for contextuality analysis have been developed. It has raised the debate on the meaning of contextuality outside the quantum realm, and also on whether—and, if so, when—it can be regarded as a signature of non-classicality. In this paper, we try to contribute to this debate by showing a very simple “thought experiment” or “toy mechanism” where a classical object (i.e., an object obeying the laws of classical physics) is used to generate experimental data violating the KCBS inequality. As with most thought experiments, the idea is to simplify the discussion and to shed light on issues that in real experiments, or from a purely theoretical perspective, may be cumbersome. We give special attention to the distinction between classical realism and classicality, and to the contrast between contextuality within and beyond quantum theory.

Appendix

In this appendix we provide a brief overview of the compatibility-hypergraph approach to contextuality. Our main reference is Amaral and Cunha (2018).

Definition 3

(Scenario) A scenario is a triple $\mathcal {S} \equiv (\mathcal {A},\mathcal {C},O)$ where $\mathcal {A}$, O are finite sets, whose elements represent measurements and outcomes respectively, and $\mathcal {C}$ is a collection of subsets of $\mathcal {A}$ (each one representing a maximal set of compatible measurements) satisfying the following conditions.

• (a) $\mathcal {A} = \cup \mathcal {C}$

• (b) For $C,C' \in \mathcal {C}$, $C' \subset C$ implies $C' = C$

The approach is named "hypergraph-approach" because a scenario $(\mathcal {A},\mathcal {C},O)$ can be associated with a hypergraph whose vertices are the elements of $\mathcal {A}$ and whose hyperedges are the elements of $\mathcal {C}$ (Amaral & Cunha, 2018).

The result of a joint measurement over a context C can be represented as a function $C \rightarrow O$. Therefore, the set $O^{C}$ of all functions $C \rightarrow O$ can be understood as the set of all possible outcomes of a joint measurement on C. Behaviors enable us to encode experimental data obtained from joint measurements on a measurement scenario (Amaral & Cunha, 2018).

Definition 4

(behavior) Let $\mathcal {S}$ be a scenario. A behavior on $\mathcal {S}$ is a function p which associates to each context C a probability distribution $p(\cdot | C)$ on $O^{C}$, that is to say, for each context C, $p(\cdot | C)$ is a function $O^{C} \rightarrow [0,1]$ satisfying $\sum _{s \in O^{C}} p(u | C) = 1$.

A behavior whose components coincide when restricted to intersections of contexts is said to be non-disturbing (Amaral & Cunha, 2018):

Definition 5

(Non-disturbance) A behavior p in a scenario $\mathcal {S}$ is said to be non-disturbing if, for any pair of intersecting contexts C, D, equality

$$\begin{aligned} p(\cdot | C \cap D, C) = p(\cdot | C \cap D, D) \end{aligned}$$

(7)

holds true, where, for any $E \in \mathcal {C}$ and $E' \subset E$, $p(\cdot | E',E)$ denotes the marginal of $p(\cdot | E)$ on $O^{E'}$, namely

$$\begin{aligned} \forall t \in O^{E'}: \ \ \ p(t| E', E) \doteq \sum _{\begin{array}{c} s \in O^{E} \\ s|_{E'} = t \end{array}} p(s|E). \end{aligned}$$

(8)

Classical and quantum behaviors arise when ideal measurements are performed upon classical and quantum systems respectively. They are defined as follows (Amaral & Cunha, 2018).

Definition 6

(Classical realization) Let p be a behavior in a scenario $\mathcal {S} \equiv (\mathcal {A},\mathcal {C},O)$. A probability space $\varvec{\Lambda } \equiv (\Lambda ,\Sigma ,\mu )$ is said to be a classical realization for p if we can associate each measurement A of $\mathcal {S}$ to a random variable $f_{A}: \Lambda \rightarrow O$ in $\varvec{\Lambda }$ is such a way that, for any context C, $p(\cdot |C)$ is the joint distribution of the set $\{f_{A}: A \in C\}$,which means that, for any $s \in O^{C}$,

$$\begin{aligned} p(s|C) = \mu \left( \bigcap _{A \in C} f_{A}^{-1}(\{s_{A}\}) \right) , \end{aligned}$$

(9)

where $s_{A} \equiv s(A)$.

The definition of quantum realization goes as follows (Amaral & Cunha, 2018).

Definition 7

(Quantum realization) Let p be a behavior in a scenario $\mathcal {S} \equiv (\mathcal {A},\mathcal {C},O)$. A pair $(H,\rho )$, where H is a finite-dimensional Hilbert space and $\rho$ is a density operator on H, endowed with a mapping , is said to be a quantum realization for p if the following conditions are satisfied.

• (a) O is, up to isomorphism, the set of eigenvalues of $\Theta (\mathcal {A})$, that is to say,

  $$\begin{aligned} O \cong \bigcup _{A \in \mathcal {A}} \sigma (T_{A}). \end{aligned}$$

  (10)

• (b) Each context is embedded by $\Theta$ into a commutative algebra, i.e., if $A,B \in C$ for some context C, then $T_{A}$ and $T_{B}$ commute.

• (c) For each context C, $p(\cdot | C)$ is reproduced by the Born rule, i.e., for any $s \in O^{C}$,

  $$\begin{aligned} p(s|C) = \text {Tr}\left( \rho \prod _{A \in C} P^{(A)}_{s_{A}}\right) , \end{aligned}$$

  (11)

  where $P^{(A)}_{s_{A}}$ denotes the projection $\chi _{\{s_{A}\}}(A)$ (if $s_{A} \in \sigma (T_{A})$, this is the projection associated with the subspace of H spanned by the eigenvalue $s_{A}$ of A; if $s_{A} \notin \sigma (T_{A})$, $P^{(A)}_{s_{A}} = 0$).

The definition of noncontextuality in the CH approach goes as follows (Amaral & Cunha, 2018).

Definition 8

(Noncontextuality) A behavior p in a scenario $\mathcal {S}$ is said to be non-contextual if there is a probability distribution $\overline{p}:O^{\mathcal {A}} \rightarrow [0,1]$ satisfying, for any context C,

$$\begin{aligned} p(\cdot | C) = \overline{p}_{C}, \end{aligned}$$

where $\overline{p}_{C}$ denotes the marginal of $\overline{p}$ in $O^{C}$, i.e., for any $s \in O^{C}$,

$$\begin{aligned} \overline{p}_{C}(s) \doteq \sum _{\begin{array}{c} t \in O^{\mathcal {A}} \\ t|_{C} = s \end{array} }\overline{p}(t). \end{aligned}$$

One can easily prove that non-contextuality and classicality (i.e., having a classical realization) are equivalent concepts in the compatibility-hypergraph approach (Amaral & Cunha, 2018), by which we mean that a behavior p is noncontextual (Definition 8) if and only if it has a classical realization (Definition 6).

It is easy to show that any classical behavior (i.e., any behavior satisfying Definition 6) has a quantum realization, and it is also easy to show that any quantum behavior is non-disturbing (Amaral & Cunha, 2018). Therefore, if we denote by $\mathscr {N}\mathscr {C}(\mathcal {S})$, $\mathscr {Q}(\mathcal {S})$ and $\mathscr {N}\mathscr {D}(\mathcal {S})$ the sets of noncontextual (or, equivalently, “classical”), quantum and non-disturbing behaviors, respectively, on a scenario $\mathcal {S}$, we obtain the well-known chain of inclusions

$$\begin{aligned} \mathscr {N}\mathscr {C}(\mathcal {S}) \subset \mathscr {Q}(\mathcal {S}) \subset \mathscr {N}\mathscr {D}(\mathcal {S}). \end{aligned}$$

(12)

The behavior we are interested in (Definition 1) is nondisturbing but lies outside the quantum set (Amaral & Cunha, 2018).