As quantum computers evolves significantly [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], there arises a fundamental need for a communication network that provides unconditionally secure communication and all the network functions of the traditional internet. This network structure is the quantum Internet [15, 12, 22, 14, 11, 13]. The availability of quantum entanglement is a crucial aspect in any global-scale quantum Internet. The quantum Internet refers to a set of connected heterogeneous quantum communication networks realized by quantum nodes and channels (such as optical fibers or wireless optical quantum channels in the physical layer) [59, 60, 61, 62, 63, 16, 64]. The quantum Internet also integrates a set of classical auxiliary communication channels to transmit auxiliary classical side-information between the quantum nodes. The quantum Internet is modeled as a global-scale quantum communication network composed of quantum subnetworks and networking components. The core network of the quantum Internet is assumed to be an entangled network structure [15, 24, 25, 39, 40, 41, 42, 43, 44, 45, 46, 47, 20, 21, 27, 28]
, which is a communication network in which the quantum nodes are connected by entangled connections. An entangled connection refers to a shared entangled system (i.e., a Bell state for qubit systems to connect two quantum nodes) between the quantum nodes. In an unentangled network structure, the quantum nodes are not necessarily connected by entanglement[16, 78], and the communication between the nodes is realized in a point-to-point setting. This setting does not allow quantum communication over arbitrary distances, and an unentangled network structure can mostly be used for establishing a point-to-point quantum key distribution (QKD) [103, 70] between the quantum nodes. These short distances can be extended to longer distances by the utilization of free-space quantum channels [15, 70]. However, this solution is auxiliary, since it can be used only at some specific points of the unentangled network structure. Therefore, it does not represent an adequate and fundamental answer to the problem of long-distance quantum communication. Consequently, in an unentangled network structure, the multi-hop settings are weak for experimental, long-distance and global-scale quantum communication. On the other hand, the entangled network structure allows the parties to establish multi-hop entanglement, multi-hop QKD, high-precision sensor networks, advanced distributed computations and cryptographic functions, advanced quantum protocols, and, more importantly, the distribution of quantum entanglement over arbitrary (unlimited, in theory) distances . As an important corollary, an entangled network structure provides a strong experimental basis for realizing a global-scale quantum communication network, the quantum Internet.
In the entangled network structure of the quantum Internet, the entangled connections form entangled paths. Entanglement between a distant source and a target node is established through several intermediate repeater nodes [15, 24, 25, 26, 18]. The level of entanglement (i.e., the level of an entangled connection) is defined as the number of nodes (i.e., the hop-distance between entangled nodes) spanned by the shared entanglement, whose range is extended by the basic operation of entanglement swapping (entanglement extension). The entangled connections have several relevant attributes, the most important of which are the fidelity of entanglement and the entanglement throughput. The throughput of an entangled connection is measured as the number of entangled states per second at a given fidelity, which provides a useful metric on the basis of which further relevant metrics can be built.
Here we define measures to characterize the ratio of accessible quantum entanglement in case of complex network failures [67, 68, 65, 66] in the quantum Internet. A complex network failure models a network situation in which a set of quantum nodes and a set of entangled connections become unavailable because of an (unknown) reason. A complex failure, therefore, can cover a set of practical failure reasons: a quantum memory failure situation in which a set of nodes and connections become unavailable, quantum node and connection failure scenarios, physical-link failures or an eavesdropping activity. Specifically, a complex failure event is modeled by a network domain that is referred to as a complex failure domain. In our model, a failure domain has an abstracted center point and a given length radius [65, 66]. This domain approach allows us to describe the probability that a given node or entangled connection (i.e., a given network element) is affected by a failure in the function of the given network element’s distance from the abstracted center point of the complex failure domain.
The entanglement accessibility ratio of a given quantum network is based on the metric of the given entangled connection’s entanglement throughput. Each entangled connection is further verified by a given condition that puts a lower bound on the entanglement throughput. The entanglement accessibility ratio measures the successful accessible entanglement at a given lower bound condition for parallel complex failures in the quantum network.
We also define the cumulative probability of entanglement accessibility ratio that quantifies the cumulative probability of all complex failure events’ occurrence for which the entanglement accessibility ratio exceeds a given lower bound.
We also quantify the probability that the total entanglement accessibility ratio in the quantum network is reduced to at most a particular ratio after a complex failure. Particularly, this parameter is referred as the probabilistic reduction of entanglement accessibility ratio.
To describe the impacts of a given complex failure on the ratio of accessible entanglement, we define the domain entanglement accessibility ratio, which quantifies the accessible entanglement ratio after a complex failure in a particular domain in a function of the radius of the given failure domain.
We define the occurrence coefficient of an entanglement accessibility ratio (occurrence ratio) at a complex failure domain, which is measured by the ratio of the number of occurrence of a given entanglement accessibility ratio in the network after a complex failure event and the total number of occurrences of all entanglement accessibility ratios after a complex failure event.
We show that the defined measures can be extracted from the occurrence ratio, and therefore, it is enough to determine the occurrence coefficient to derive the other metrics. We propose an algorithm to determine the occurrence coefficient from the empirical quantities of the quantum network that are directly observable in the analyzed network setting. In particular, the defined entanglement accessibility measures can be derived in a purely empirical way by extracting relevant statistics from the analyzed quantum network.
The proposed protocol is not dependent from the actual physical implementation, therefore it can be applied in the heterogeneous network structure and network components of the quantum Internet (the protocol can also be applied in the quantum Internet at the utilization of magnetic field in the perturbation method [104, 105, 106] (kind of Zeeman Effect ) in the physical layer 111At a constant magnetic field perturbation, the evolution operator is diagonal. Even when the magnetic field depends only on time and not on space, the exact perturbation unitary evolution operator remains diagonal. The quantum system can be disturbed by perturbing it with electric, magnetic or electromagnetic radiation and hence, the system becomes excited and changes its state. Magnetic field-based protocol design here is more complex because a 1-D (dimensional) magnetic field will not act on a 1-D charged particle (From the Lorentz Law , , where is the charge of the particle, is the velocity, and is the magnetic field.). Note, a charged particle can also be excited in a 3-D box with a 3-D control magnetic field. Another possible extension to this problem is to consider a particle in a 3-D box perturbed by a vector electric field and a vector magnetic field
is the magnetic field.). Note, a charged particle can also be excited in a 3-D box with a 3-D control magnetic field. Another possible extension to this problem is to consider a particle in a 3-D box perturbed by a vector electric field and a vector magnetic field[109, 110]., or in electromagnetic field-based [109, 110] scenarios in the network components.).
The novel contributions of our manuscript are as follows:
We define measures to characterize the accessible quantum entanglement in case of complex network failures in the quantum Internet.
We define the terms entanglement accessibility ratio, cumulative probability of entanglement accessibility ratio, probabilistic reduction of entanglement accessibility ratio, and occurrence coefficient.
We show that the defined measures can be extracted from the occurrence ratio, and therefore, it is enough to determine the occurrence coefficient to derive the other metrics.
We propose an algorithm to determine the occurrence coefficient from the empirical quantities of the quantum network that are directly observable in the analyzed network setting of the quantum Internet.
The entanglement accessibility measures can be derived in a purely empirical way by extracting relevant statistics from the quantum Internet.
This paper is organized as follows. In Section 2, the related works are summarized. In Section 3, some preliminaries are introduced. Section 4 defines the entanglement accessibility measures. Section 5 discusses the occurrence coefficient and defines an algorithm for the empirical evaluation of the measures. In Section 6, a numerical evaluation is proposed. Finally, Section 7 concludes the paper.
2 Related Works
In this section, we review some recent results connected to the establishment of the experimental quantum Internet.
A technical roadmap on the experimental development of the quantum Internet has been provided in . The roadmap is connected to the Quantum Internet Research Group (QIRG) , which group is formulated and supported by an international researcher background and collaboration. The authors of  address some important capability milestones for the realization of a global-scale quantum Internet. The technical roadmap also addresses important future engineering problems brought up by the quantum Internet, such as the development of a standardized architectural framework for the quantum Internet, standardization and protocols of the quantum Internet, application programming interface (API) for the quantum Internet, and the definition of the application level of the quantum Internet .
In a quantum Internet scenario, entanglement purification is a procedure that takes two imperfect systems and with initial fidelity , and outputs a higher-fidelity density such that . In , the authors propose novel physical approaches to assess and optimize entanglement purification schemes. The proposed solutions provide an optimization framework of practical entanglement purification.
In , the authors defined a method for deterministic delivery of quantum entanglement on a quantum network. The results allow us to realize entanglement distribution across multiple remote quantum nodes in a quantum Internet setting.
In , a satellite-to-ground QKD system over 1,200 kilometres has been demonstrated. The proposed model integrated a low-Earth-orbit satellite with decoy-state QKD. The reported key rate of the protocol was above the kHz key rate over a distance up to 1200 km. The work has a relevance for an experimental quantum Internet, since the results also allow us to realize high-efficiency long-distance QKD in a global quantum Internet setting.
In , the authors demonstrated the quantum teleportation of independent single-photon qubits over 1,400 kilometres. Since an experimental realization of a global-scale quantum Internet requires the application of quantum teleportation over long-distances, the proposed results represent a fundamental of any experimental quantum Internet. In , the authors demonstrated quantum teleportation with high fidelity values between remote single-atom quantum memories.
Some other recent results connected to the development of an experimental global-scale quantum Internet are as follows. In , the authors demonstrated the Bell inequality violation using electron spins separated by 1.3 kilometres. In , the authors demonstrated modular entanglement of atomic qubits using photons and phonons. The quantum repeaters are fundamental networking elements of any experimental quantum Internet. The quantum repeaters are used in the entanglement distribution process to generate quantum entanglement between distant senders and receivers. The quantum repeaters also realize the entanglement purification and the entanglement swapping (entanglement extension) procedures. For an experimental realization of quantum repeaters based on atomic ensembles and linear optics, see .
Since quantum channels also have a fundamental role in the quantum Internet, we suggest the review paper of , and also the work of , for some specialized applications of quantum channels. For a review on some recent results of quantum computing technology, we suggest . For some recent services developed for the quantum Internet, we suggest [29, 30, 31, 32, 33, 37, 34, 35, 36, 38].
Some other related topics are as follows. The works [23, 24, 25, 26, 29, 30, 31, 16] are related to the utilization of entanglement for long-distance quantum communications and for a global-scale quantum Internet, and also to the various aspects of quantum networks in a quantum Internet setting.
For some fundamental works on quantum machine learning, see[71, 72, 85, 100], on quantum Shannon theory, see [23, 17, 58, 69, 74, 86, 101], on quantum computing see [73, 75], for schemes for reducing decoherence in quantum memory see , for quantum network coding see [80, 81, 82, 102], for transformation of multipartite pure states, see , for multistage entanglement swapping see , while for optical microcavities and photonic channels for quantum communication, see .
3.1 Entanglement Fidelity
The aim of the entanglement distribution procedure is to establish a -dimensional entangled system between the distant points and , through the intermediate quantum repeater nodes. Let , and let be the target entangled system and , subject to be generated. At a particular density generated between and , the fidelity of is evaluated as
3.2 Entangled Network Structure
Let refer to the nodes of an entangled quantum network , which consists of a transmitter node , a receiver node , and quantum repeater nodes , . Let , refer to a set of edges (an edge refers to an entangled connection in a graph representation) between the nodes of , where each identifies an -level entanglement, , between quantum nodes and of edge , respectively. Let be an actual quantum network with nodes and a set of entangled connections. An -level, , entangled connection , refers to the shared entanglement between a source node and a target node , with hop-distance
For a particular -level entangled connection with hop-distance (2), there are intermediate nodes between the quantum nodes and .
3.3 Entanglement Purification and Entanglement Throughput
Entanglement purification is a probabilistic procedure that creates a higher fidelity entangled system from two low-fidelity Bell states. The entanglement purification procedure yields a Bell state with an increased entanglement fidelity ,
Let refer to the entanglement throughput of a given entangled connection measured in the number of -dimensional entangled states established over per sec at a particular fidelity (dimension of a qubit system is ) [15, 12, 22, 23, 24, 25, 26, 29].
For any entangled connection , a condition should be satisfied, as
where is a critical lower bound on the entanglement throughput at a particular fidelity of a given , i.e., of a particular has to be at least .
4 Model Description
In this section, we define the terms and metrics for entanglement accessibility in the quantum Internet.
4.1 Failure Identifications in the Quantum Internet
Let refer to a complex failure domain that models a set of quantum nodes and a set of entangled connections in a particular network domain [65, 66], whose nodes and entangled connections are affected by a complex failure (complex – randomly affects both nodes and connections). Note, that while refers to the set of local entangled connections within the failure domain , set refers to the entangled connections of the global quantum network , therefore is a subset of ,
An complex failure event is identified by the entanglement throughput of an -th -level entangled connection as
where is a critical lower bound on the entanglement throughput.
In the center of , for all entangled connections of the set of ,
and therefore, the probability that an event occurs at for all elements of is
As the distance from the center of increases, the complex failure probability decreases, e.g.,
Let be the center of domain , and let be the radius of defined as in terms of the hop-distance of an abstracted shortest entangled path in , as
where is the nearest affected quantum node to , is the farthest affected quantum node from , while is an abstracted shortest entangled path between and , with a hop-distance , as
where , are intermediate quantum nodes between and on the entangled path .
where identifies the level of the entangled connection .
4.2 Entanglement Accessibility Ratio
Let set refer to those entangled connections of for which the condition (see (4)) holds after a complex failure . Let
be a random variable that quantifies the ratio of total entanglement throughput in a complex failure event at a given(see (4)). This quantity is referred as the entanglement accessibility ratio (EAR) after a complex failure and identified by the ratio of total entanglement throughput after a complex failure of and the total entanglement throughput without a failure event  at a given lower bound condition (4) as
where is the number of connections in the set of , and is the cardinality of connection set after a failure occurs in .
4.3 Cumulative Probability of Entanglement Accessibility Ratio
Let be a critical lower bound on the entanglement accessibility ratio of (see (15)) at a given condition and a complex failure . A cumulative probability of all complex failure events’ occurrence for which the yielding ratio at a given is at least (see (15)),
is referred to as the cumulative probability of entanglement accessibility ratio (CP-EAR) , defined as
is the cumulative distribution function ofat a condition .
The probability density function (PDF) of ratio
after a complex failure is therefore
4.4 Probabilistic Reduction of Entanglement Accessibility Ratio
Assume that the cumulative distribution function of at a condition is given as
Using (20), the probabilistic reduction of entanglement accessibility ratio (PR-EAR) at a given ratio , condition , and probability is defined as
As follows, the PR-EAR parameter in (21) quantifies the probability that the total entanglement accessibility ratio is reduced to at most ratio after a complex failure .
4.5 Domain-Dependent Entanglement Accessibility Ratio
The domain-dependent entanglement accessibility ratio (DD-EAR) quantifies the accessible entanglement ratio after a complex failure in a particular domain in a function of the radius of as
where is the PDF of ratio at an -radius length complex failure domain , defined as
A complex network failure situation of a quantum repeater network with failure domain is illustrated in Fig. 1. A complex failure is associated with domain , . In the center of the , for all connections , and . As the distance from the center of increases, the failure probability decreases, e.g., . The condition holds for , where is a critical lower bound on an -th -level entangled connection , for the established entangled connections of ,
5 Evaluation of Entanglement Accessibility
In this section, first, we define a coefficient that describes the occurrence of a given entanglement accessibility ratio after a multiple complex failure scenario. Then we propose an empirical method to determine the occurrence coefficient from the observable quantities of a particular quantum network of the quantum Internet.
5.1 Occurrence Coefficient
Let refer to the occurrence coefficient of a particular entanglement accessibility ratio at a complex failure domain in , defined as
where is the number of occurrence of a given entanglement accessibility ratio in after a failure , while quantifies the total number of occurrences of all accessible ratios in after a failure .
Extending (24) to all the complex failure domains yields
where quantifies the occurrence of ratio via (24) for an -th domain .
For an -domain setting with domains , can be derived from the function as
while at is
At a particular failure domain radius of a given , let
where as shown in (19).
For all domains , (28) extends to
where quantifies the occurrence of ratio for an -th domain via a particular radius using (28). Then in a scenario is expressed as
To find this quantity at a given network empirically, we propose an algorithm as follows.
5.2 Empirical Evaluation of Occurrence Coefficient
We propose an algorithm, , for the empirical determination of the O-EAR coefficient (see (24)) at a complex failure domain scenario and then the evaluation of (see (25)) by the extended analysis of all domains . Some preliminary definitions are as follows.
To describe the topology of , let be the node-to-node incidence matrix of , and let refer to a temporal incidence matrix for the iteration steps of the algorithm.
Each -level entangled connection is characterized by a particular entanglement throughput rate , which are used to determine the total accessible entanglement at a connection set at no failure as
Then let and be the source and target quantum nodes of a demand associated to user , , where is the number of users. Then let be the total required entanglement by a demand as
f refers to the connection set of .
For a given demand , let
quantify the total required entanglement of demand with connection set along entangled connections traversed by respective paths in .
identify a set of demands with both end nodes and not affected by a complex failure .
Assuming that a complex failure with a domain occurs in , the total accessible entanglement after a complex failure is
where is the connection set of after the failure.
The center of a domain and the corresponding radius length of
are modeled as uniformly distributed random continuous variables.
At a given upper bound on the entanglement throughput of , the remaining accessible entanglement throughput is defined as
where refers to a current rate.
The algorithm aims to determine the empirical estimation of the occurrence function .
The algorithm for a multiple complex failure scenario is given in Algorithm 1.
A brief description of the method is as follows. In steps 1 and 2, some initializations are performed for further calculations. Steps 3 to 5 derive the ratio of accessible entanglement at a given failure domain . The steps aim to determine the ratio of total accessible entanglement in a given complex domain failure scenario. For each demand that has unaffected end nodes, a path searching is performed to find the shortest alternate path for all demands to serve requirement of a given . If an alternate path exists but the entangled connections of the path are not able to serve the required entanglement , then a new shortest path is determined. The calculations are performed for all demands that are present with a nonzero required entanglement in the network. In step 6, the iteration is extended for the evaluation of all failure domains .
5.3.1 Step 1
In step 1, a temporal incidence matrix is initialized by , and the value of the total accessible entanglement via set after a complex failure is set to zero, , where is defined in (35). To identify the set of quantum nodes affected by , for all nodes their corresponding probability is determined via (14) in a function of distance node from center of . Then to distinguish the unusable connections after has occurred for all connections for which condition does not hold (see (4)), set the corresponding elements of to 0.
5.3.2 Step 2
In step 2, for all entangled connections of , the amount of the utilizable throughput rate is set to a maximum of the given entangled connection , , where , the upper bound on the throughput of an entangled connection , and are given by (36). Initialize a set of demands with both end nodes and not affected by as given by (34). The quantity of (see (33)), which describes the required total entanglement by demand with connection set along entangled connections traversed by respective paths in , is set to the amount of the total entanglement required for , (see (32)). As a final substep, determine the shortest path for by using the temporarily incidence matrix as characterized in step 1.
5.3.3 Step 3
In step 3, some computations are performed for the demands of set , whose demands are not affected by the failure. The value of the total accessible entanglement via connection set of a given demand after a complex failure , (see (31)), is set to the minimal amount of utilizable throughput rate of , thus
From step 2, it follows that will be equal to the maximal entanglement rate of that entangled connection, which yields the min-max optimization
After this substep, the relation of and is verified, and the next steps are selected based on it. If the value of the required total entanglement of demand along entangled connections traversed by respective paths in does not exceed , the value of the total accessible entanglement of demand after a complex failure , then value of total accessible entanglement via connection set after a complex failure is increased by . Conversely, if exceeds , then is increased by . As the value of is determined, depending on the relation of and , the value of the required total entanglement is either decreased by or by . This substep therefore yields if , but results in if . Depending on the relation of and , a final computation is also performed in this step. For each entangled connection traversed by the shortest path , the amount of remaining utilizable entanglement throughput is decreased as if , and if holds.
5.3.4 Step 4
In step 4, a set of demands is determined via condition . It follows that some demanded entanglement cannot be served fully; thus, in this step, the entanglement assigned to the demands should be increased as much as possible. These demands are still associated with a nonzero required entanglement ratio in the network, and therefore, these queries should be processed. This step focuses on the service of these demands via the corresponding calculations that are similar to the calculations of step 3. The value is increased by a given , which is a given ratio of the maximum of the total accessible entanglement throughput of the entangled connections of the next shortest path . Then the value of is decreased by ratio .
5.3.5 Step 5
In step 5, all demands are served until there is no nonzero required entanglement present in the network. All demands are served if for all . The serving process of demands also stops if there is no next shortest path in the network; therefore, holds. Finally, the empirical estimation of the ratio of accessible entanglement after a failure is determined as (see (37)). The estimation of (see (24)) uses the empirical value of after a complex failure via connection set , and also the empirical value of the via connection set . Using the resulting estimate in (37), can be determined via the estimation in (38).
5.3.6 Step 6
Finally, step 6 extends the results for all the failure events occurring in to determine (see (25)).
5.4 Computational Complexity
The computational complexity of algorithm