Investigating the Global Properties of a Resource Theory of Contextuality Tiago Santos1and Barbara Amaral1 1Department of Mathematical Physics Institute of Physics

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Investigating the Global Properties of a Resource Theory of Contextuality

Tiago Santos1and Barbara Amaral1

1Department of Mathematical Physics, Institute of Physics,

University of S˜ao Paulo, R. do Mat˜ao 1371, S˜ao Paulo 05508-090, SP, Brazil

Resource theories constitute a powerful theoretical framework and a tool that captures, in an ab-

stract structure, pragmatic aspects of the most varied theories and processes. For physical theories,

while this framework deals directly with questions about the concrete possibilities of carrying out

tasks and processes, resource theories also make it possible to recast these already established theo-

ries on a new language, providing not only new perspectives on the potential of physical phenomena

as valuable resources for technological development, for example, but they also provide insights into

the very foundations of these theories. In this work, we will investigate some properties of a resource

theory for quantum contextuality, an essential characteristic of quantum phenomena that ensures

the impossibility of interpreting the results of quantum measurements as revealing properties that

are independent of the set of measurements being made. We will present the resource theory to

be studied and investigate certain global properties of this theory using tools and methods that,

although already developed and studied by the community in other resource theories, had not yet

been used to characterize resource theories of contextuality. In particular, we will use the so called

cost and yield monotones, extending the results of reference [21] to general contextuality scenarios.

arXiv:2210.03268v1 [quant-ph] 7 Oct 2022

I. INTRODUCTION

With the advent of quantum information theory, which brought to physics techniques and methods from computer

science, the laws of physics began to be probed through new kinds of questions. In particular, there arose an interest

in ﬁnding out what is possible within a theory given a set of resources and operations, that is, what the theory allows

one to actually perform [17]. In particular, concepts in foundations of quantum physics began to be investigated in the

lights of a pragmatic tradition, in which one is trying to understand and describe how and how much can a physical

system be known and controlled through human intervention [10].

One of the ways in which this pragmatic perspective has come to be formalized by the community is through

the so called resource theories. A resource theory is a framework that aims for the characterization of physical

states and processes in terms of availability, quantiﬁcation and interconversion of resourceful objects [10]. In such a

framework, a chosen property is treated as an operational resource [1] and physical phenomena are studied in order

to better leverage this speciﬁc resource. Two good examples of scientiﬁc ﬁelds that have a pragmatic ﬂavour are

thermodynamics and chemistry. Both began as endeavors to determine and better understand the ways in which

resourceful systems and materials could be transformed and used for one’s advantage. Alchemy sought to transform

basic metals into nobler ones, and one of the endeavors that marked the early days of thermodynamics was the study

of thermal non-equilibrium and its resourcefulness for extracting useful work. Even today, after so much development

in both ﬁelds, this perspective still drives much of the interest from the community [17].

Hence, the main concepts behind this kind of approach are resourceful objects and advantageous transformations

among these objects. There are many more examples of resource theories and they need not to be extremely practical

in purpose or scope. By abstracting the framework one may begin to cast many areas of science in this language and

interesting ways of understanding these ﬁelds begin to emerge. Even mathematics can be seen as a resource theory

in which the resourceful objects are mathematical propositions and the transformations are mathematical proofs,

understood as sequences of inference rules [10].

A particularly important class of resource theories are the quantum resource theories, resource theories deﬁned in

terms of quantum states, processes, protocols and concepts. Quantum resource theories are an example of how to

arrive at a particular resource theory from a theory of physics. In it we have a set of processes - state preparations,

transformations or measurements, for example - and we divide this set into costly implementable processes and freely

implementable ones. Assuming unlimited availability of elements in the free subset, one can then study the structure

that is induced on the costly set. This kind of resource theory is then speciﬁed by a chosen class of operations,

which in the case of a quantum resource theory is a restriction on the set of all quantum operations that can be

implemented. Given this restriction, some quantum states will not be accessible from some ﬁxed initial state and thus

become resourceful states which could be harnessed by some agent to reach and end not possible via the free set only

[17].

An example of a quantum resource theory is the resource theory of entanglement. If we restrict two or more parties

to classical communication and local quantum operations (LOCC), entangled states become resourceful. And thus

the full set of quantum states gets separated between the free set of separable states and the costly set of entangled

ones. Given access to the free set (separable states), one cannot achieve an entangled state by LOCC. Moreover,

access to entangled states allows one to perform tasks such as quantum teleportation that were not possible only via

LOCC and the free set of states [17]. There are many more examples of the use of resource theoretic framework in

quantum information theory and other areas of physics, such as in the study of asymmetry and quantum reference

frames, quantum thermodynamics, quantum coherence and superposition, non-Gaussianity and non-Markovianity

[9]. Furthermore, it has proven advantageous to recast even more foundational concepts of quantum theory, as

contextuality and Bell nonlocality, in resource theoretic frameworks.

Among the advantages of casting a quantum property in resource theory language, we can cite [9]:

•Resource theories are particularly ﬁtting for restricting our attention to operations and procedures that reﬂect

current experimental capabilities, as generally one can associate a particular resource theory to any speciﬁc

experiment by taking as free operations only those that can be performed within the limitations of the exper-

imental setup available. Thus, such theory is precisely concerned with the particular tasks that can be done

with the setup.

•Resource theories provide a means of rigorously comparing the quantity of resource present in quantum states

or channels. As by construction the amount of resource held by an object is at least equal to the amount in

another if one can transform the former into the latter by a free operation in the given theory, by studying the

interconversion relations in a theory together with the possibilities of quantiﬁcation, one is able to establish a

pre-order on the set of objects within the theory. This ordering structure oﬀers insight into the role that the

property investigated as a resource plays within the bigger theory as a whole. This particular perspective is a

great part of this work;

•Resource theory allows one to better analyze how and what fundamental processes are responsible for a certain

phenomenon. By considering the particular restrictions on the set of operations, one can point out, in a

systematic manner, what are the physical requirements for performing a speciﬁc task. Interestingly, this can

lead one to better consider resource trade-oﬀs through decomposing a certain task in terms of free operations

and resource consumption. In certain situations it might be advantageous to know if by making use of more

free objects one can lessen resource consumption.

•Because the same framework is applicable to diverse properties, by studying one property of interest within

a particular resource theory one can be actually doing much more as it might lead to identiﬁcation of struc-

tures and applications that are common to resource theories in general. As an example we note that “elegant

solutions to the problem of entanglement reversibility emerge when drawing resource-theoretic connections to

thermodynamics”.

In this work, we will investigate some properties of a resource theory for quantum contextuality, an essential char-

acteristic of quantum phenomena that ensures the impossibility of interpreting the results of quantum measurements

as revealing properties that are independent of the set of measurements being made [7]. We will present the resource

theory to be studied and investigate certain global properties of this theory using tools and methods that, although

already developed and studied by the community in other resource theories, had not yet been used to characterize

resource theories of contextuality. In particular, we will use the so called cost and yield monotones, making use of

their power in the study of resource theories for non-locality, in an attempt to extend the results of [21] to this more

general class of phenomena, contextuality.

This work is organized as follows: in section II we present the basic mathematical elements of a general resource

theory; in section III we present the resource theory of contextuality considered in this work, deﬁning the set of

objects, free objects, and free operations; in section IV we investigate the global properties of the pre-order of objects

deﬁned by the resource theory presented in section III;

II. RESOURCE THEORIES

We begin by describing the basic mathematical elements of a general resource theory [1,10,11,15]:

1. A set Uof mathematical objects that may contain the resource under consideration, together with a subset

F ⊂ U whose elements are those which are going to be considered freely available, called free objects.

2. A set Tof transformations between objects that can be freely constructed or implemented, that is, without

consuming any resource, called free transformations. The notation A→B, in which A, B ∈ F, denotes that

there is a free transformation F∈ T such that F(A) = Band will be used when the speciﬁc transformation

is not important, but only it’s existence. In terms of deﬁning the free transformations, if the free objects are

ﬁxed, a transformation Fis a free transformation when, for every free object A, the resulting object B=F(A)

is also a free object.

3. The possibility of combining objects and transformations through binary relations among them. If Aand B

are objects of the theory, the composite object regarding both is denoted by A⊗B. In a similar manner, if

we have two transformations Fand G, we consider the composite transformation F⊗Gas performing the two

transformations in parallel, so that if F(A) = Band G(C) = D, then (F⊗G)(A⊗B) = C⊗D.

Thus we come to a deﬁnition of a resource theory, in terms of the elements described above, as follows:

Deﬁnition 1. Aresource theory is deﬁned by the tuple (U,F,T,⊗), in which Uconsists of the set of objects to

which the theory refers, F ⊂ U is the set of free objects of the theory, Tis a set of free transformations acting on the

objects and a binary operation ⊗that allows parallel combinations of objects and operations.

For the mathematically oriented reader, we would like to mention that, as [10] discusses, this formalization can

be summed up by stating that objects and free transformations in a resource theory are, respectively, objects and

morphisms in a symmetric monoidal category. In fact, the author of that work also states that “the diﬀerence between

a resource theory and a symmetric monoidal category is not a mathematical one, but rather one of interpretational

nature”, that “a particular symmetric monoidal category is called a resource theory whenever one wants to think of

its objects as resourceful and its morphisms as transformations or conversions between these resourceful objects”. We

refer the reader to references as [10,14].

A. The pre-order of objects

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 ∈ U, is there

a transformation A→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→A. Second, the possibility of freely implementing sequential

composition of free operations is also reasonable. By deﬁnition, being able of getting from Ato Band from Bto Cat

no cost implies being able of getting from Ato Cat no cost. In other words, we have A→B, B →C=⇒A→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Bwhenever A→Bin a resource theory,

the “” relation thus deﬁnes a pre-order among the objects.

Now, even though this resulting ordered structure is closely related to the speciﬁc set Tof 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 [10], deﬁned exclusively by the a set of objects equipped with a

pre-order and another binary relation:

Deﬁnition 2. Given a resource theory R= (U,F,T,⊗), the theory of resource convertibility associated with R

is the tuple ˜

R= (U,F,⊗,), in which is the pre-order relation induced on the objects by the set of free operations.

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.

Deﬁnition 3. We redeﬁne a resource theory to be the reduced tuple R= (U,F,T)of aforementioned elements

together with the induced theory of resource convertibility redeﬁned as ˜

R= (U,F,).

B. Monotones

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.

Deﬁnition 4. Let (U,F,)be a resource theory. We deﬁne a resource monotone as a function deﬁned on the set

of objects, that preserves the the pre-order structure, that is, for all A, B ∈ U,

M:U → ¯

Rsuch that BA=⇒M(B)≤M(A),(1)

in which ¯

Rmeans the set of extended real numbers R∪ {−∞,∞}.

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 [1]. 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 [16] give an

example of the usefulness of resource monotones. They introduce certain properties that they call global structures

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InvestigatingtheGlobalPropertiesofaResourceTheoryofContextualityTiagoSantos1andBarbaraAmaral11DepartmentofMathematicalPhysics,InstituteofPhysics,UniversityofS~aoPaulo,R.doMat~ao1371,S~aoPaulo05508-090,SP,BrazilResourcetheoriesconstituteapowerfultheoreticalframeworkandatoolthatcaptures,inanab-stracts...

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Investigating the Global Properties of a Resource Theory of Contextuality Tiago Santos1and Barbara Amaral1 1Department of Mathematical Physics Institute of Physics

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