Investigating the Global Properties of a Resource Theory of Contextuality

We now introduce the idea of the pre-order of objects in a resource theory. This idea is intimately connected to interconversion between objects and provides a very natural way of characterizing a given resource theory in terms of an internal structure, the structure of possible interconversions induced by the set of free operations. This idea lies at the heart of this work and we will explore it further.

In a resource theory, sometimes one is not particularly interested in the process by which a conversion occurs, but rather the important question is whether this conversion is possible or not. That is, given objects $A,B\in\mathcal{U}$, is there a transformation $A\rightarrow B$?

First, since free operations are those that can be done at no cost, it is fairly intuitive that doing nothing is a free operation, that is, for every object $A$, we have $A\rightarrow A$. Second, the possibility of freely implementing sequential composition of free operations is also reasonable. By definition, being able of getting from $A$ to $B$ and from $B$ to $C$ at no cost implies being able of getting from $A$ to $C$ at no cost. In other words, we have $A\rightarrow B,B\rightarrow C\implies A\rightarrow C$. These basic facts make of this interconversion relation a pre-order in the set of objects, meaning a binary relation that is reflexive and transitive. Following standard notation we write $A\succeq B$ whenever $A\rightarrow B$ in a resource theory, the “$\succeq$” relation thus defines a pre-order among the objects.

Now, even though this resulting ordered structure is closely related to the specific set $\mathcal{T}$ of free transformations, being actually induced by it, once this set of interconversion relations structure is given, one can “forget” the transformations that gave rise to the ordering structure and consider only questions about the induced structure itself. In this spirit, one can speak of theories of resource convertibility Coecke et al., (2016), defined exclusively by the a set of objects equipped with a pre-order and another binary relation:

Throughout this work the concept of resource interconvertibility will be the focus of our discussions, with one specific choice of free operations for a resource theory of contextuality. We thus feel free to not make further reference to the distinction between a resource theory and the associated theory of resource interconvertibility.

One of the most important aspects of a resource theory has to do with quantifying the amount of resource contained in a certain object of the theory.

Thus a monotone function gives a quantitative measure of the amount of resource available in an object. Because of their order-preserving property, these functions give us insightful information about the resource theory, as we will see in section IV.

It is worth mentioning that the pre-order structure of objects in a resource theory is more fundamental than any single resource monotone. A resource monotone captures certain aspects of the pre-order by assigning numerical values to the objects, but unless the pre-order is a total order (all its elements are comparable), it can never contain the total information available in the pre-order Amaral, (2019). In fact, even though there were early works in which one of the goals was to find what would be the correct or better resource monotone, the contemporary view is that in general there are several inequivalent monotones and there is no a priori reason to choose one over the other. The pre-order is the fundamental structure, with any particular resource monotone being a coarse-grained description of the theory.

One might be tempted to question the usefulness of worrying about resource monotones. If they provide only an incomplete description of the total information contained in the pre-order, what does one gain with their use, if anything at all? In general, as it will be in our case, the effort in constructing and investigating resource monotones does pay off and ends up being a crucial part of developing useful resource theories. The authors in Gonda and Spekkens, (2019) give an example of the usefulness of resource monotones. They introduce certain properties that they call global structures of the pre-order and use resource monotones in the characterization of such properties. Part of this work is exactly trying to answer questions of this nature for a resource theory of contextuality. Another example, is the work Duarte et al., (2018), which shows that contextuality monotones can be used to study geometrical aspects of particular sets of possible behaviors inside and outside the quantum set, as well as Amaral, (2019) and Chitambar and Gour, (2019), that present and discuss different monotones and their applicability.

Each context $\gamma\in\mathcal{C}$ represents a set of measurements in $\mathcal{M}$ that can be jointly performed. For each context $\gamma$, the set of all possible outcomes for the joint measurement of the measurements in $\gamma$ is the set $\mathcal{O}^{\gamma}$, that is, each measurement in $\gamma$ can give as result $|\mathcal{O}|$ different outputs. When we jointly perform the measurements of $\gamma$, our output is encoded in a vector $\textbf{s}\in\mathcal{O}^{\gamma}$.

The main ingredient of our theory, for now, is what we call a behavior.

To each context $\gamma$ and output $\textbf{s}\in\mathcal{O}^{\gamma}$ the behavior gives the probability $p_{\gamma}(\textbf{s})$ of obtaining output s in a joint measurement of the elements of $\gamma$. Sometimes we also call a behavior a box, as a way of creating a mental picture, where we imagine the elements of $\mathcal{M}$ as buttons of the box, and, for each measurement, we imagine the box having $|\mathcal{O}|$ output lights that inform us the result of the measurements.

Behaviors may or may not satisfy what we call the non-disturbance condition, that we states that, given two contexts $\gamma$ and $\gamma^{\prime}$, the marginal for their intersection is well defined. If we have, for example, $\gamma=\{x,y\}$ and $\gamma^{\prime}=\{y,z\}$, the non-disturbance condition implies:

Another important idea for contextuality is the possibility of assigning a single probability distribution on the whole set $\mathcal{O}^{\mathcal{M}}$. We call this probability distribution $p_{\mathcal{M}}:\mathcal{O}^{\mathcal{M}}\rightarrow[0,1]$ a global section for the scenario. We say that $p_{\mathcal{M}}$ is a global section for a scenario $B$ if, in each context, the marginal probability distributions coincide with the ones given by $B$.

When a behavior is non-contextual, i.e. when it has a global section, all probabilities can be written as

To define the free operations of our resource theory, we begin by defining certain special operations that take behaviors (our objects) in a given scenario into behaviors, possibly in another scenario.

One of the basic operations is the operation of pre-processing a behavior. We introduce a new scenario $\Upsilon_{PRE}=(\mathcal{M}_{PRE},\mathcal{C}_{PRE},\mathcal{O}_{PRE})$, with new measurements, contexts and outputs, and a new non-contextual behavior $B_{PRE}$ associated with it. We associate each output of $B_{PRE}$ with an input of $B$, in such a way that every output configuration of $B_{PRE}$ defines a possible input configuration in $B$, that is, associated with every output $\textbf{r}\in\mathcal{O}_{PRE}$, we have a possible context $\gamma(\textbf{r})\in\mathcal{C}_{PRE}$.

With this, we define a new behavior $\mathcal{W}_{PRE}(B)$ given by

where the sum runs over all outputs r associated with the context $\beta$ in $B_{PRE}$.

Analogously, we can define the post-processing of a behavior. We again introduce $\Upsilon_{POS}=(\mathcal{M}_{POS},\mathcal{C}_{POS},\mathcal{O}_{POS})$ together with a non-contextual behavior $B_{POS}$. The same association is made between outputs $\textbf{s}\in\mathcal{O}$ and contexts $\delta(\textbf{s})\in\mathcal{M}_{POS}$. The new behavior obtained $\mathcal{W}_{POS}(B)$ is given by

We can now define the free operations we will consider in this work, the non-contextual wirings Amaral, (2019). We start with an arbitrary behavior $B$ and compose it with a pre-processing $B_{PRE}$ and a pos-processing $B_{POS}$, along with one additional possibility that the probabilities of $B_{POS}$ may also depend on the inputs and outputs of $B_{PRE}$. With this additional freedom, the probabilities of $B_{POS}$ are of the form $p_{\delta}(\textbf{t}|\beta,\textbf{r})$, but since we want to guarantee that there is no contextuality generated by the processing itself, as done in Amaral et al., (2018), we demand that

With this construction, we get the final scenario $(\mathcal{M}_{PRE},\mathcal{C}_{PRE},\mathcal{O}_{POS})$ with an associated behavior $\mathcal{W}_{NC}(B)$, given by

This particular class of operations, henceforth referred to as NCW, constitutes the free operations of our resource theory. We present here two important results derived in Amaral, (2019) about this class of operations:

For the special case of Bell scenarios, a noncontextual wirings is know as a local operation assisted by shared randomness (LOSR) Gonda and Spekkens, (2019).

Now we look for a special class of operations, the set of deterministic non-contextual wirings. For that, we investigate the separability of the wings of the processing operations. For example, writing the pre-processing as

we see that demanding that $p_{\beta_{i}}$ takes values in $\{0,1\}$ for each wing $\beta_{i}$ guarantees that the final form of such a probability distribution is a product of the form

For each wing of the context $\beta$, the function $f_{i}(\beta_{i})$ effectively associates a measurement $\beta_{i}$ with an output $r_{i}$, the one for which $p_{\beta_{i}}(r_{i})=1$. Hence, in a deterministic processing, each context selects a unique output string r. This allows us to formulate a definition for a deterministic non-contextual wiring operation.

Then, given a behavior B, the behavior $\mathcal{W}^{Det}(B)$ is given by

where the product $p^{Det}_{\beta}(\textbf{r})p^{Det}_{\delta(\textbf{s})}(\textbf{t})$ factorizes as

For the special case of Bell scenarios, a deterministic non-contextual wiring is know as a local deterministic operation (LDO) and a non-contextual symmetry operations is known as a local symmetry operation Gonda and Spekkens, (2019).

Just as was done for LOSR operations in Gonda and Spekkens, (2019), we can define the type of a box in terms of the cardinalities of its input and output variables. Since we are dealing with a scenario $\Upsilon=(\mathcal{M},\mathcal{C},\mathcal{O})$, the number of input and output variables are fixed by $|\mathcal{C}|$, cardinalities of input variables are fixed by $|\mathcal{M}|$, and cardinalities of output variables are fixed by $|\mathcal{O}|$. Now, as our NCW operations take boxes in $\Upsilon=(\mathcal{M},\mathcal{C},\mathcal{O})$ to boxes in $\Upsilon_{\mathcal{W}}=(\mathcal{M}_{PRE},\mathcal{C}_{PRE},\mathcal{O}_{POS})$, we define the type of an operation $\mathcal{W}$ as $[\mathcal{W}]\doteq[B]\rightarrow[\mathcal{W}(B)]$. The set of all operations of type $[B_{1}]\rightarrow[B_{2}]$ is denoted by $\underset{[B_{1}]\rightarrow[B_{2}]}{\text{{NCW}}}$. As mentioned above, some of our results will be type-specific, meaning results concerning type-preserving operations.

An important technical aspect of a resource theory is the convexity of the chosen set of free operations. Convexity is a desirable property since convex combinations represent the possibility of choosing what transformation to implement probabilistically, which should also be considered a free operation. Let us begin by considering a general free operation and write it as

where we write $p_{\beta,\delta}(\textbf{r},\textbf{t})\doteq p_{\beta}(\textbf{r})p_{\delta(\textbf{s})}(\textbf{t})$.

We ask in which circumstance can we talk about a convex sum of two such operations, say $p^{(0)}_{\beta,\delta}(r,t)$ and $p^{(1)}_{\beta,\delta}(r,t)$. A way of incorporating this notion into the formalism is by making use of the random variables already present in the operations.

Imagine we want to represent a convex combination where $p^{(0)}_{\beta,\delta}(\textbf{r},\textbf{t})$ is implemented with probability $\alpha$, and $p^{(1)}_{\beta,\delta}(\textbf{r},\textbf{t})$ with probability $1-\alpha$. What we do is to sample from a new binary probability distribution $p(\Lambda)$, in which $p(\Lambda=0)=\alpha$, $p(\Lambda=1)=1-\alpha$, such that $\Lambda=0$ results in $p^{(0)}_{\beta,\delta}$ being implemented, while $\Lambda=1$ results in $p^{(1)}_{\beta,\delta}$ being implemented.

Formally, we want to implement

where the extra superscript $(\Lambda)$ in $p_{\beta,\delta}^{(\Lambda)}(r,t)$ denotes the dependence of the operation on the initial sampling over $p(\Lambda)$ through the explicit dependence of $p(\lambda|\Lambda)$ on the variable $\Lambda$. Now, defining $\tilde{p}(\lambda)\doteq\sum_{\Lambda}p(\lambda|\Lambda)p(\Lambda)$ allows us to write

which is the standard form of a non-contextual wiring operation. That is, general convex combinations can be naturally incorporated in the formalism, which is surely desirable. When the set of free operations defining a resource theory is convex, we say that the resource theory is a convex resource theory. An thus we have just derived an important technical result:

Now, recalling the discussion on deterministic operations, the convexity of our set of operations gives us another powerful result. Notice that in a general operation with respective contexts $\beta,\delta$, etc, we have that for each particular measurement in each context $\beta_{i},\delta_{i},...$, we can write

so that

in which $\vec{\Lambda}\doteq(\Lambda_{1},...,\Lambda_{|\beta|})$ and $P(\vec{\Lambda})\doteq\sum_{\lambda}\prod_{i}p(\Lambda_{i}|\lambda)p(\lambda)$. That is, any operation $p_{\beta,\delta}(r,t)$ is a convex combination of products of deterministic operations, and we arrive at a generalization of Fine’s theorem Fine, (1982) and Wolfe et al., (2020):

In other words, $Y_{(S|f)}(a)$ gives the value of $f$ for the most resourceful object in $S$ that can be freely obtained from $a$. On the other hand, $C_{(S|f)}(a)$ gives the value of $f$ for the least resourceful object in $S$ from which one can freely obtain $a$.

One of the goals of a resource theory description is the complete characterization of the pre-order of objects, i.e., actually knowing which of the four possible interconversion relations holds for each pair of objects. The result obtained above about the decomposition of non-contextual free operations into combinations of extremal deterministic ones proves to be actually useful in this characterization.

Let $\mathcal{P}^{NCW}_{[B_{2}]}(B_{1})$ denote the set of behaviors of type $[B_{2}]$ that can be obtained by general non-contextual wirings from $B_{1}$, and $\mathcal{V}^{Det}_{[B_{2}]}(B_{1})$ denote the set of behaviors of type $[B_{2}]$ obtained from $B_{1}$ through deterministic non-contextual wirings. The finite cardinality of the set $\mathcal{V}^{Det}_{[B_{2}]}(B_{1})$ and the existence of a polytope of free operations can be nicely expressed in the following result:

This statement is equivalent to

which is a very important result, since it actually allows one to check if $B_{1}\rightarrow B_{2}$ through an efficient algorithm. To check if $B_{1}\rightarrow B_{2}$ holds, one has only to determine all the deterministic operations that take behaviors of type $[B_{1}]$ to behaviors of type $[B_{2}]$ (which are finite in number), compute the image of $B_{1}$ under these deterministic operations and then determine whether $B_{2}$ can be obtained through a convex combination of these images of $B_{1}$. The answer to last step can be decided with the use of linear programming (Wolfe et al.,, 2020).

Now, although the result just obtained is indeed useful, reducing greatly the number of operations needed to know if $B_{1}\rightarrow B_{2}$, in order to fully characterize the pre-order of objects through this method, one would need to apply the linear program to every pair of objects, which is certainly not practical. Another alternative to characterize the pre-order is by the use of monotones.

Our goal now is the characterization of the resource theory of contextuality defined by non-contextual wirings in terms of these global properties.

The monotone construction we will use is partially based on the notion of maximally violating behaviors Amaral et al., (2018), behaviors that are analogous to the PR-boxes for the CHSH Bell scenario Popescu and Rohrlich, (1994). Because of this, we will first focus on a specific type of contextuality scenario, the $n$-cycle Araújo et al., (2013), since for this class of behaviors, both non-contextuality inequalities and their quantum violations are known.

The $n$-cycle scenario consists of $n$ dichotomic measurements $X_{i}$ and a set of contexts of the form $\{X_{i},X_{i+1}\}$ modulo $n$. There is a lot to be said about this kind of scenario, but for now let us focus on necessary considerations to define the monotones that will be used henceforth.