Optimal allocation of quantum resources

The optimal allocation of resources is a crucial task for their efficient use in a wide range of practical applications in the fields of science and engineering. This paper investigates the optimal allocation of resources in multipartite quantum systems. In particular, we show the relevance of two optimality criteria for the application of quantum resources. Moreover, we present optimal allocation solutions for an arbitrary number of qudits in a particular resource theory. In addition, we study a third optimality criterion and demonstrate its application to scenarios involving several resource theories. Finally, we highlight the potential consequences of our results in the context of quantum networks.


Introduction
In our daily lives, we perform various activities to meet our needs, fulfill our desires or achieve our goals. These activities require the use of physical objects, which may be within our reach or require a provider. Consequently, the value of an object as a resource is determined both by our physical limitations and by the object's usefulness to perform a specific activity. Resource theories are the theoretical framework that allows such a value to be assigned to an object in a certain physical context [1,2,3]. Quantum objects are no exception Roberto Salazar: roberto.salazar@ug.edu.pl and at present resource theories are being systematically applied to demonstrate their value as resources for different operational tasks [4,5].
Once the value of a resource to perform a task has been established, the greatest challenge is to allocate such a resource to different agents so that certain optimality criteria are met. The optimal allocation of resources is a well-established research area applied in various fields, such as economics, computing, communication networks and ecology, among others [6,7,8,9,10,11,12,13]. However, to date, there has been no systematic research on the optimal allocation of quantum resources.
In this article, we investigate the optimal allocation of resources in multipartite quantum systems and illustrate their application in quantum resource theories. Moreover, we show allocations for an arbitrary number of qudits that optimize both the criteria of proportional fairness and reliability for particular quantum resource theories. Additionally, we study the criteria of equitability for resource theories with trade-offs between them and show how this applies to quantum multiresource scenarios.

Resource theories in general
The essential idea of a quantum resource theory is to study quantum information processing under a restricted set of physical operations. The permissible operations are called free and because they do not cover all physical processes that quantum mechanics allows only certain physically realiz- Figure 1: A castaway meditates on the optimal allocation of wood logs into several useful tasks such as: prepare a bonfire, build a ship, make a house or provide a fold for his flock.
able objects of a quantum system can be prepared. These accessible objects are likewise called free and any objects that is not free is called a resource. Thus a quantum resource theory identifies every physical process as being either free or prohibited, and similarly it classifies every quantum object as being either free or a resource. The theory of entanglement forms a representative example of a quantum resource theory. For two or more quantum systems, entanglement can be characterized as a resource when free operations are local quantum operations and classical communication (LOCC) and free objects are separable states.
Other important element of a resource theory is a function R that maps the objects considered by the resource theory into the non-negative real numbers. The function R must be non-increasing under the free operations i.e. a monotone and should be proportional to the advantage of using the object for some operational task.
Definition 1 (Resource theory) A quantum resource theory is defined by a triple {F, O, R} where F is the set of free objects under consideration, for instance quantum states or quantum operations, which forms a subset within the set S of all quantum objects. The set O of free operations contains functions o : S → S which preserve the set of free objects. A function R : S → [0, ∞ + [, monotone under the set of free operations and such that R(f ) = 0 for all f ∈ F. The function R is said to measure the resourcefulness of objects in the set S.
Because our research focus on multipartite objects, we assume monotones R to be well defined when applied to every subsystem. The behavior on multipartite objects of several standard measures is already well known in the literature from the study of convertibility tasks, for more details see [3].
In the following we introduce the basic definitions of the resource theory of incompatibility which recently called the attention of the community [14,15]. We will use this particular theory to illustrate the application of these definitions and show how they lead to sound, but tractable optimization problems.

Resource theory of incompatibility
In quantum theory measurements are described by Positive Operator Value Measures (POVM). We define a POVM in dimension d and number of outcome n denoted by M : ...; m, each described by measurement operators M a|x labels each of the measurement outcomes a where a = 1, .., o such that Σ o a=1 M a|x = I for every x. This set is said to be jointly measurable (or compatible) if there exists a parent POVM G with measurement operators G λ , and conditional probability distributions p(a|x, λ), such that M a|x = Σ λ p(a|x, λ)G λ . Otherwise the set is said to be incompatible. One can introduce the resourcetheoretic framework in the case of incompatibility with jointly measurable (compatible) POVM as a free measurement set F and the set of free operations O is: any single compatible measurement operator, classical post-processing and random mixing of single compatible measurements [14].
A recent important result shows that for every resourceful measurement M there is an instance of minimal-error quantum state discrimination game for which M gives greater success probability than all free measurements [16,17,18]. It has also been shown that the relative advantage of a resourceful measurement for state discrimination is proportional to the robustness measure [19,20], which quantifies the minimal amount of noise that has to be added to a POVM to make it free. The formal expression for the general robustness of a measurement M is: (1) This is a monotone measure which means that for any measurement M and free operation φ ∈ O we have R g (φ [M]) ≤ R g (M). We use R g in our work since it provides a clear operational interpretation of measurement incompatibility and is easy to define for any subsystem: simply choose any N that acts on the same systems as M such that the mixture is some compatible POVM on the corresponding systems.

Allocation of resources
The problem of allocation of resources consist in how to distribute the use of resources into a set of tasks according to a practical criteria, which usually involves the optimization of a figure of merit. When the selected method of distribution optimizes the figure of merit in the criteria, then we have an optimal allocation of resources. A classic example of this problem can be seen in Fig. (1), where an isolated man on an island must decide the best way to use the resources he has.
In this article we restrict ourselves to study the above general problem in the case of multipartite quantum objects (a state ρ, a measurement M or a channel N ) denoted here by a common symbol σ. In a multipartite object, some subsets of the systems can be selected for different operational tasks. To describe this selection of relevant sets we shall use the notion of hypergraph H = {V H , E H }, in which hyperedges α ∈ E H connect two or more vertices v ∈ V H [21]. We define a quantum resource allocation as the distribution of a quantum multipartite resource σ over hypergraph H i.e. a list A H [σ] of values of the measure R applied to every reduction of σ to parties in an hyperedge α of H. The way in which the measure is applied for each α ∈ E H must be specified in each case, which we exemplify later in our applications of our framework. To illustrate the concept of resource allocation we present hypergraphs H 1 , H 2 and H 3 in Fig. (2) which have allocations: If the list A H satisfies an optimality criterion C then we will say that the quantum resource allocation is optimal. The relevant criterion C depends on the allocation features we wish to optimize. If a task can be carried out by parts of the entire system specified by a hypergraph H, then the performance Φ C of the allocation is given by a sum over hyperedges α ∈ E H [8,9]: Here Φ C ,α : R ≥0 → R is some monotone function of the amount of resource allocated in hyperedge α and σ α is the reduction of σ in which the complement of α has been traced out. The properties of Φ C ,α depend on how the allocation of resources at α contribute to the satisfaction of criterion C . Since the operational task has associated the resource measure R, the optimization problem consist in finding the multipartite resource σ such that A H [σ] maximizes Φ C when H, is fixed. The first optimality criterion C 1 we will consider is the optimal proportional fairness criteria. This criterion is required when it is important to allocate resources fairly among the parties. These include, for example, allocation of bandwidth in telecommunication networks, takeoff and landing slots at airports, and water resources. For this case we maximize the performance of proportional fair-ness Φ C 1 given by: Hence, each Φ C 1 ,α (·) is equal to log (·). The choice Φ C 1 ,α (·) = log (·) is useful in the case when no set of parts α has priority over others for the task performed [8]. In the case when certain priority order of the α ∈ E H exists, a different figure of merit is used to ensure equitability [8] and we will discuss it later in the article. For example, the figure of merit Φ C 1 is applied when in a communication network each demand is routed on a specified single path. From a resource point of view, the objective is to provide a resource σ α to parties in the set α to communicate with an additional party e / ∈ V H in a way as fair as possible among all α ∈ E H . It can be shown that the optimal solution A H [σ * ] satisfies: is called proportionally fair since the aggregate of proportional changes with respect to any other feasible solution is zero or negative [8].
Next, we consider the optimal reliability criterion C 2 which determines an optimal performance for a task in the presence of failures of the devices used by any one of its parts in each α ∈ E H . The appropriate performance Φ C 2 which we maximize in this case reads: where π α denotes the prior probability that only parts in α will perform the task for which σ is the resource. If every part {a 1 , . . . , a n } has a prior chance p a k to work, then: The problem of optimal reliability is a relevant problem for different engineering areas which is usually addressed by redundacy models [9]. In (4) we adapt the classical performance function to fit the scheme of resource theories (see for instance the classical definition in section 1.23 in [9]) by choosing the Φ C 2 ,α (·) to be the probability of successful performace only of parts in α rather than probability of failure, because R (σ α ) is a measure of advantage provided by σ α to the corresponding task.
The previous examples of performance and optimal reliability considered the case of a resource that should be shared in an optimal way among parties performing the same task. It should be noted, that in principle one can define different tasks for each set α ∈ E H in the two previous allocation problems. However, as we shall see on the following examples, the natural cases that involve multiple tasks are those where the availability of the resource is limited and -what is even more important -exist competition between the tasks. To describe such a situation in our notation consider for every resource σ and all α ∈ E H : with Λ α (σ) , Γ (σ) real non-negative functions of states and L α , U α some non-negative constants.
Here we use explicitly the index dependence to express the possibility that the parties in each α ∈ E H measure the resourcefulness of their σ α with a different R α . The choice of each R α is determined by the task assigned to the resource in each α. The equations (6) stand for the trade-offs between the resources assigned for tasks at different α while the (7) determine prior limitations of the resources for each task. This set of inequalities are known as knapsack constraints (KC) [8].
In the case of two or more tasks involved and nontrivial knapsack constraints a typical optimality criterion C 3 is the optimal equitability which is defined by a recursive max-min algorithm: The corresponding partial order determines the minimization over α ∈ E H in the following steps.
2. If E H = {∅} then find a solution with prior KC for H = {∅} then find a solution with prior KC for: under the additional constraint: under the additional constraints: for all α ∈ E (n−1) H and such that: for each R αm (σ αm ).

When
In the previous recursive algorithm minimization of R α (σ α ) over α ∈ E H considers R α i (σ α i ) ≤ R α j σ α j iff U α i ≤ U α j because in this case allocating resources in α j is potentially more advantageous than in α i . We also note that the above algorithm will halt in finite number steps because the number of α ∈ E H is finite. Solutions to optimal equitability are known to be non-unique, but nevertheless they provide a set of solutions considered safe for the usual practical applications [8]. If this is not satisfactory, a unique solution can be ensured by additional requirements such as Pareto optimality [8] of the final order

Results
Before presenting a concrete example of resource allocation let us recall the notion of unbiased bases. Two orthogonal measurements with labels j, j ∈ {0, 1} are unbiased iff they satisfy the following condition: where D is the total dimension of the system [22,23]. Two unbiased bases consisting of product states only were introduced in [24], for which case we write |u x a = u x a 1 ⊗. . . ⊗ u x a N with each u x a k a state from two unbiased bases. Therefore, for N systems of dimension d two product unbiased measurements |u , onto the unbiased bases. Then we arrive at the following theorem: Theorem 1 Consider an optimal allocation with arbitrary hypergraph H of a given measurement M over a number N of qudits of dimension d and with optimality criterion C . If the criterion C requires maximization of performance function Φ C , with allocation A H [M] = {R g (M α )} α∈E H defined by the general robustness R g , then the measurement M composed of two product unbiased bases is a feasible optimal solution in the resource theory of incompatibility.
Proof: First, we need to specify in which way R g applies to every possible measurement M α indexed by the hyperedge α of the hypergraph. We use equation (1) specified to the measurement M α , where the mixture (M α + sN α )/(1 + s) belongs to the set F α of compatible measurements only on the composed system α ∈ E H and N α is some measurement also on the systems in α. This α-wise application of R g is meaningful since it should measure the advantage of M α over any compatible measurement of parties in α ∈ E H . Now, we remark the recent proof in [15] that unbiased bases are optimal incompatible measurements under the general robustness monotone R g in any dimension d. Since, we can always find two product unbiased bases to define M, then because product unbiased bases are nested measurements the evaluation of R g for each α ∈ E H yields the maximal value, hence for any monotone function or Φ C 2 -the M measurement achieves the maximal value q.e.d.
As an illustration, the maximal value Φ max Another example is the maximal value Φ max Since the resources in this particular resource theory are advantageous for state discrimination, our result shows that the device which implements the optimal measurement M also has optimal performance for such a task in the presence of failures of any subsystem. The scenario represented by H 2 is the simplest case in which "at most a number K of devices may fail", which defines a problem of allocation relevant to the performance of different engineering tasks [9].
Our next result involves nonlocality and contextuality, which are well known resources for communication and randomness amplification tasks [27,28,29,31,30]. A monotone for these state resource theories is [31,30]: where I is the Bell correlation function, Φ is the set of free operations (see Appendix A) and B c is the classical bound. We will use the resource theories of nonlocality and contextuality to exemplify a resource scenario in which the optimal equitability criteria C 3 is useful. In this scenario the allocation of the resource state ρ is defined by an hypergraph H 3 : The objective of party b is to perform a task that improves with the contextuality of his local state ρ b = Tr a (ρ), while parties a, b should perform a task that requires nonlocality of their bipartite state ρ a,b = ρ. For our application of C 3 is important the existence of a fundamental trade-off between both resources [26] : Here I s (ρ) stands for a cyclic correlation, where the average of observable The correlation I s (ρ) witnesses nonlocality or contextuality, depending on the kind of constraints satisfied by the {B 1 , ..., B s } observables [26]. Then, if we replace appropriately I s and B s c in (11) to define a monotone M s (ρ) the inequality (12) implies the resource relation: with B s Q is the quantum Tsirelson bound (i.e. quantumly saturable) for I s (ρ) and sgn(·) the sign function. From an allocation of resources perspective the inequality (14) defines a knapsack constraint like (6). The individual bounds The optimal equitable solutions in this case are simply: A nontrivial scenario for optimal equitability arises in the case of monogamy activation, for example in the following relationship [26]: Here, I 5 (ρ b ) is a contextual cycle correlation with B k observables as in (13), while I B A (ρ a,b ) , I B C (ρ b,c ) stand for I 3322 inequalities [26]: (16) and analogously for I B C (ρ b,c ) by replacing each A k by C k . Now, we study the allocation scenario defined by H 3 , and with operational tasks associated with I B A (ρ a,b ) and I 5 (ρ b ). In this case we assume ρ b,c to be resourceless, but with a fixed value I B C (ρ b,c ) = λ < 4. Then, we can state an equitability problem defined by the constraints: The monotones M B A (ρ a,b ) and M 5 (ρ b ) are defined analogously as M s (ρ) in (11), and B γ Q , B 5 Q are the quantum bounds for I 3322 and I 5 (ρ) respectively. In this case the solution is not trivial and reads: with µ = 4−λ (for the proof see Appendix A, section A.2). As mentioned before nonlocal and contextual resources are useful for randomness amplification [27,28], Therefore, an application of the solutions to the problems presented is an equitable assignment of security into quantum networks. In consequence, the above examples show that optimal equitable allocation of resources are relevant for concrete tasks in quantum information.

Discussion
In quantum information, a variety of approaches are used to determine the practical value of quantum systems, such as cryptography, communication capacity, computational complexity and thermodynamics. The common factor among these approaches is the search for quantum resources to benefit agents in the development of specific tasks. This is why, observed as a whole, each of these can be seen as a part of an economy that manages such resources, i.e. an economy of quantum resources. This article applies three different criteria of the optimal allocation of quantum resources and demonstrate the existence of solutions to the corresponding optimization problems. It would be interesting to consider scenarios where exist an interplay between quantum processing machines (computers) and a quantum communication network (see [32,33,34]). Indeed our example of the optimal equitability criteria with nonlocality and local contextuality may be seen as a starting point for analysis of network scenarios. Indeed nonlocality may be a selftesting benchmark for entanglement while contextuality may be related to coherence. Speaking very roughly, the first may be related to communication between the nodes, while the second -to quantum information processing at the nodes. If we have limited fault tolerant resources for protecting both local and nonlocal quantumness, then the optimal equitability may help for finding a balance between local computation and delegation of tasks especially if is desired the latter to be secure (see [38]). Independently the very nonlocality at single computing node may be one day an important resource itself (see [39]). We leave these and related topics for further research.
In addition, the scheme we present opens the possibility of finding applications for multi-party systems that nest resources symmetrically [35]. In this sense, our results suggest that resource nesting may be potentially a tool for distributing resources according to certain criteria relevant to the corresponding tasks.
In conclusion it seems natural to expect that the field of quantum resources will need the tools provided for allocation of resources in near future. It may be helpful at the level of designing quantum information processing protocols (or -may be -even some practical experiments with limited quantum coherent effects). On the other hand it seems that the methods and tools for allocation of resources can assist the theoretical analysis of trade-offs between different resources in quantum information. One can even expect that as such they can contribute to development of some form of economy of quantum resources as a field of research in itself.

A.1 Resource theories of nonlocality and contextuality
The resource theories of nonlocality and contextuality usually consider as objects boxes with at least two integer inputs x 1 , x 2 and two integer outputs a 1 , a 2 , which are characterized by the conditional probabilities P (a 1 , a 2 | x 1 , x 2 ) called behaviours [30,31]. In our research we consider only quantum resources, hence every behaviour can be written as: For a bipartite state ρ 1,2 and measurements M (1) a 2 |x 2 for systems 1, 2. Moreover, in each case we consider fixed measurements, such that in our study the resource theories of nonlocality and contextuality can as well be considered resource theories of quantum states. Because of this some examples of free operations for nonlocality and contextuality can be local operations as well mixing of states. If in both cases the set Φ is defined as the closure of free operations under composition, the measure M s (ρ) defined in (11) is a monotone: due to the definition of supremum.

A.2 Non-trivial solutions of allocation in the monogamy activation scenario
Here we provide in more detail the possible equitable solutions to the problem with knapsack constraints (17). First, let's define the auxiliary variables µ = 4 − λ, ν 1 = B I 3322 Q − 4 and ν 2 = B 5 Q − 3. In Appendix C we show that ν 1 ≥ 1 and from reference [36] v 2 = 0.9442, hence ν 1 > ν 2 . From the above and the partial order imposed by the equitability criteria C 3 we have: which means that in this scenario the advantage provided by nonlocality of ρ {a,b} is potentially greater than the noncontextuality of ρ {b} , therefore has priority. Additionally we know (See ref [26,36,37]) that all values for I 5 ρ {b} , I B A ρ {a,b} , λ satisfying the trade-off (15) and Tsirelson bounds to be achievable for some ρ a,b,c , then the supreme in each step of the optimal equitability criterion can be replaced by a maximization. Later, because the optimization is over a convex set -of M 5 ρ {b} , M B A (ρ a,b )the solution must lie at a boundary. A solution in the boundary M 5 ρ {b} = ν 2 will violate the requirements (17) or (21) because µ ≤ 2ν 2 , on the other hand a solution with M B A ρ {a,b} = ν 1 , implies M 5 ρ {b} = 0 which is actually a minimum. Then, to find a maximum for M 5 ρ {b} under the constraints, we search at the boundary: Replacing (22) in (21) we obtain: in terms of binary operators {A 1 , A 2 , A 3 , B 1 , B 4 , B 6 } with outcomes 0 and 1, However, in this article we use binary operators with outcomes −1 and 1, Then, in this appendix we will show how to transform B v Q into an operator B γ Q in terms of binary operators with outcomes −1 and 1. First, note that: for all i ∈ {1, 2, 3} and j ∈ {1, 4, 6} respectively. In consequence: Second, we apply (27) After some algebraic simplifications we obtain: If now we use observables B 1 = −B 1 , B 4 = −B 4 and A 3 = −A 3 which are just re-labeling of outcomes for the observables B 1 , B 4 , A 3 we have the following relation: Now, in the article we used the alternative form of I 3322 in terms of binary operators with outputs +1, −1 whose corresponding Bell operator is: By proper identification of terms in (30) and (30), we obtain the operator identity: Additionally, from the definition of Tsirelson bounds B v Q and B γ Q we have: sup but in reference [37] is shown that a lower bound for B v Q is 0.25 and an upper bound is 0.25085..., then identity (30) and Tsirelson bound definition implies: 1 ≤ B γ Q − 4 ≤ 1.0034 (33) which are the bounds used in section A.2 of the Appendix.