In the quantum Internet, the quantum nodes share quantum entanglement among one other, which provides an entangled ground-base network structure for the various quantum networking protocols [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. In a quantum Internet scenario, the aim of the quantum repeater elements is to extend the range of entanglement through several steps [20, 21, 22, 23, 24, 28, 27, 30, 31, 32, 33, 34, 35, 36]. The available entanglement at the end points has several critical parameters, most importantly the fidelity of the established entanglement (fidelity of entanglement [25, 26]) and the probability of the existence of a given entangled connection [1, 8]. In an experimental setting, these critical parameters are time-varying since the noise of local quantum memories that store the shared entanglement in the quantum nodes evolves over time, and the probability of entangled connections (shared entanglement between a node pair) also changes dynamically [1, 5, 6, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 37].
In the quantum Internet, several entangled paths (paths formulated by several entangled connections) could exist between a given source-target quantum node pair [1, 5, 6, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64]. This fact allows us to introduce a method that utilizes this multipath property to change these critical parameters via the number of entangled paths associated with a given end-to-end node pair: the available fidelity of entanglement and the probability of an entangled connection. The model utilizes the reliability (probability) of the entangled connections and the entanglement fidelity coefficient as primary metrics. The decomposition is motivated by the fact that a maximization of the entanglement throughput (number of transmitted Bell states per a given time unit at a particular fidelity) parameter requires also the maximization of the connection probability, and the entanglement fidelity.
In this work, we define a method for entanglement access control in entangled quantum networks. The entanglement differentiation is achieved via a controlled variability of entanglement fidelity and entangled connection probability between source and target quantum nodes in a quantum repeater network. The proposed approach allows us to define different priority levels of entanglement access for the legal users of the quantum network with respect to the number of available paths. The number of available paths injects an additional degree of freedom to the quantum network, allowing for the selection of the entanglement fidelity and connection reliability for the end nodes. In a straightforward application of our method, the high-priority demands are associated with high fidelity and high connection probability in the end nodes of the user, while the lower-priority users get lower fidelity and lower connection probability in their end nodes. To achieve the differentiation, we define the appropriate cost and path cost functions and the criteria regarding the entanglement fidelity and connection probability for the quantum nodes and entangled connections of the entangled path. The entanglement differentiation utilizes a different number of paths between the source and target nodes allowing a distinction to be made between single-path and multipath scenarios. In a single-path setting, only one entangled path exists between source and target nodes, and therefore the fidelity of entanglement and the probability of existence of the entangled connections between the end nodes are determined only by the nodes of the given entangled path. In a multipath scenario, more than one path exists from a source to a target.
In our model, a given criterion regarding the entanglement fidelity of the local nodes has to be satisfied for all node pairs on the path referred to as integrated connection probability and fidelity criterion for all entangled connections of the entangled path. The integrated criterion allows us to reach a given entanglement fidelity and a given connection probability between the end nodes of the quantum network.
Our solution utilizes time-varying parameters since the cost functions deal with the evolution of the entanglement fidelity parameter and the connection probabilities, which evolve over time. We define the entanglement access control algorithm for an arbitrary topology quantum repeater network. We also reveal the computational complexity of the method.
The novel contributions of our manuscript are as follows:
We define a method to achieve controlled entanglement accessibility in the quantum Internet.
The algorithm defines entangled paths between the source and target nodes in function of a particular path cost function.
The path cost is determined by an integrated criterion on the entanglement fidelity and the probability of entangled connection.
The proposed scheme has moderate complexity, providing an efficient entanglement accessibility differentiation, allowing for the construction of different priority levels of entanglement accessibility for users.
The results can be straightforwardly applied in the entangled quantum networks of the quantum Internet.
This paper is organized as follows. In Section 2, the basic components of the model are summarized. In Section 3, the entanglement accessibility methods are discussed. Section 4 proposes the integrated criterion related to entanglement fidelity. Section 5 defines the entanglement access control algorithm. Finally, Section 6 concludes the results.
2 System Model
2.1 Entangled Network
The quantum Internet setting is modeled as follows . Let refer to the nodes of an entangled quantum network , with a transmitter quantum node , a receiver quantum node , and quantum repeater nodes , . Let , , refer to a set of edges between the nodes of , where each identifies an -level entangled connection, , between quantum nodes and of edge , respectively. The entanglement levels of the entangled connections in the entangled quantum network structure are defined as follows.
2.1.1 Entanglement Levels in the Quantum Internet
In a quantum Internet setting, an entangled quantum network consists of single-hop and multi-hop entangled connections, such that the single-hop entangled nodes111The -level entangled nodes refer to quantum nodes and connected by an entangled connection . are directly connected through an -level entanglement, while the multi-hop entangled nodes communicate through -level entanglement. Focusing on the doubling architecture [1, 5, 6] in the entanglement distribution procedure, the number of spanned nodes is doubled in each level of entanglement swapping (entanglement swapping is applied in an intermediate node to create a longer distance entanglement ). Therefore, the hop distance in for the -level entangled connection between is denoted by 
with intermediate quantum nodes between and . Therefore, refers to a direct entangled connection between two quantum nodes and without intermediate quantum repeaters, while identifies a multilevel entanglement.
2.1.2 Entanglement Fidelity
be the target Bell state subject to be created at the end of the entanglement distribution procedure between a particular source node and receiver node . The entanglement fidelity at an actually created noisy quantum system between and is
where is a value between and , for a perfect Bell state and for an imperfect state. The entanglement fidelity represents the accuracy of our information about a quantum state [1, 5, 6]. The fidelity in (3) measures the amount of overlap between the target state (2) and the density matrix that represents our system. In the entanglement distribution procedure, the usage of the entanglement fidelity metric rather than other correlation measure functions (concurrence, negativity, quantum discord, quantum coherent information, etc.)  is motivated by the fact that the fidelity of entanglement is an improvable parameter in a practical setting. The improvement of the fidelity is realizable by the so-called entanglement purification process . The entanglement purification takes imperfect entangled states and outputs a higher-fidelity entangled system. Without loss of generality, in an experimental quantum Internet setting, an aim is to reach over long distances [1, 5, 6].
2.1.3 Practical Implementation
An experimental quantum network refers to a set of source users (quantum nodes), destination users (quantum nodes), several intermediate quantum repeaters between them, and to a set of physical node-to-node connections between the physical nodes (the physical attributes of the level connections identify the physical attributes of the physical links between the neighboring nodes). A quantum node is a quantum device with internal quantum memory , and with the capability of performing local operations (such as the internal processes connected to entanglement purification, entanglement swapping, error correction, etc.)[28, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64]. In a practical setting, the node-to-node entanglement distribution can be implemented by an optical fiber network or via a wireless optical system (free-space channel  or a quantum-based satellite communication channel ). A physical link is characterized by a particular link loss . For a standard-quality optical fiber , the average link loss is , while the maximum of the tolerable link loss for an optical fiber system is [1, 6].
In an practical entangled quantum network, the level entangled connections refers to the case when the source and target quantum nodes are not directly connected by a physical link, but by an entangled connection that spans several quantum repeaters. An level entangled connection is formulated by several node-to-node interactions through the physical links in the physical layer.
3 Entanglement Access
3.1 Entanglement Fidelity Criterion
First, we characterize the entanglement fidelity criterion for a given node pair. Using the criterion, we then derive the probability of the existence of single-path and multipath sets with end-to-end connection-disjoint entangled paths [65, 66] between source and target nodes. The end-to-end connection-disjoint entangled paths share no any common entangled connection between a source node and a receiver node .
A given entangled connection is characterized by a particular fidelity
, whose quantity classifies the entangled connection, such that, where is a critical lower bound on the fidelity of entanglement.
Let refer to the entangled connection between a node pair , and let be the difference of the fidelity of entanglement in quantum nodes and , as
where is a maximal allowed fidelity distance, , and . Since the entangled connections are assumed to be time-varying in the network [65, 66], the probability that holds at a given time for an entangled connection is as
is the probability density function of entanglement fidelity distance.
3.1.1 Single-Path Entanglement Accessibility
Let refer to a single path between and , , where is a demand, and are the sender and destination nodes associated with the demand of user , is the number of users. The single entangled path setting means that entanglement can be distributed from to through only one given path in the network . Let it be assumed that consists of entangled connections; then the probability that a given single path exists between and with the fidelity criterion is expressed as
3.1.2 Multipath Entanglement Accessibility
Let , , refer to the th multipath between and , which means that entanglement can be distributed from to through a set of end-to-end connection-disjoint entangled paths as in the network . The probability  that and share a common entanglement with the fidelity criterion is as
where is a path index, and is the number of entangled connections associated with .
4 Integrated Criterion on Connection Probability and Fidelity
The integrated criterion extends the results of Section 3 to include the criterion on the probability of the existence of a given entangled connection between a node pair of a path. Using the integrated criterion on the connection probability and entanglement fidelity, we derive the probability of existence of single-path and multipath sets with end-to-end connection-disjoint entangled paths  between source and target nodes.
In our model, the fidelity of shared entanglement evolves in time for a given node pair . For each quantum node at a time , let be defined as
where is the probability of an -level entangled connection with a node determined in node at a time , while is the fidelity of entanglement determined in node at a time .
For a node pair , according to local quantities, the following distance can be defined:
where , and are the connection probability quantities determined in nodes and , and the fidelity distance is described by
where , and are the fidelity quantities determined in nodes and .
A distance of and for a node pair at a particular time is expressed via , as
Since the connection probability and the entanglement fidelity parameters evolve over time, after from an initial time , the quantity of a given node evolves as
where is expressed as
where is the probability density function of distance function , and and are expressed as the connection probability and entanglement fidelity evolution functions of node .
For a given node pair , the particular upper bound on the maximal allowable distance between and at time leads to a limit while can be referred to as entangled:
If exceeds , then the difference of the local entangled connection probabilities and entanglement fidelities are above a critical limit; therefore, the node pair is referred to as unentangled.
which leads to
A representation of and for a node pair is depicted in Fig. 1. The connection probability is assumed to be different in the nodes at a particular time.
4.1 Single-Path Entanglement Accessibility
After these derivations, the probability of a single path in function of the connection probability and entanglement fidelity in the nodes of the path (e.g., connection probability criterion and fidelity criterion for all entangled connections of the path) is as follows.
4.2 Multipath Entanglement Accessibility
For the multipath scenario, (7) can be written via (16) as follows. For a given set of end-to-end connection-disjoint entangled paths expressed as between and , the probability that and share a common entanglement with a connection probability criterion and fidelity criterion at time is expressed as
5 Control of Entanglement Access
The entanglement access control algorithm establishes a number of connection-disjoint entangled paths between a source node and a target node. Changing the number of connection-disjoint entangled paths allows us to modify both the probability of entanglement between the source and target nodes and the fidelity of available entanglement in the end nodes.
For the algorithm, a cost function  of a given entangled connection is defined as
Let be the actual quantum repeater network with quantum nodes. A given -level entangled connection between a node pair is expressed as .
Let and be the source and target quantum nodes of a demand associated with user . Using (20) and a given entangled path with a set of quantum repeaters , , and a set entangled connections, as
the cost of path is defined as
The entanglement access control algorithm outputs a set of , which contains the connection-disjoint entangled paths between and .
In function of , priority classes can be defined for the users of the quantum Internet. A high priority user gets a high value of , while lower priority users get lower values of . The actual value of for a particular user class can be determined in function of the current network conditions.
The steps are given in Algorithm 1.
A brief description of the entanglement access control algorithm is as follows.
Step 1 sets for a user by the priority class of the user. The determines the available value(s) of for .
In Step 2, entanglement is established between the source node of the given demand of the user and the neighboring quantum repeaters. The relevant metrics quantities are also calculated in this step.
Steps 4-5 deal with the intermediate quantum repeater nodes associated with the given demand. These steps also ensure that entanglement is distributed through the cheapest path from a source node towards , via a given intermediate repeater node . It is ensured in our model that if the intermediate repeater node also shares entanglement with a quantum repeater , then node will not establish entanglement with , since can be reached via , which is on the cheapest path .
Step 6 outputs the set of end-to-end connection-disjoint entangled paths between and and the path costs for all paths.
Step 7 determines the probability for user via (19), and updates the actual value of if needed.
Finally, Step 8 extends the steps for all users.
In Fig. 2 a multipath entanglement accessibility is depicted in a quantum Internet setting with heterogeneous entanglement levels. The network situation depicts connection-disjoint entangled paths that share no common entangled connection between a source node and a receiver node . The entangled paths are characterized by the derived formulas.
5.2 Computational Complexity
For a given quantum network with quantum nodes, the computational complexity of the entanglement access control algorithm for a given demand is at most , since the problem is analogous to the establishment of a path by message broadcasting .
5.3 Numerical Evidence
We provide a numerical evidence on the distribution of the and path probabilities.
Let us set for the lower bound on the fidelity of entanglement between all node pairs and , , and . Then, the maximal allowed fidelity distance is set as .
Then, let us assume that a single path between and consist of entangled connections with different entanglement levels between the nodes of the path . For the multipath scenario, each entangled path consist of entangled connections with different entanglement levels between the nodes of each entangled path of .
For simplicity let us assume that the number of entangled connections is set as for all entangled paths, and the distribution of the probabilities for the entangled connections of a single path at is as depicted in Fig. 3(a). The distribution of the probabilities for the entangled connections of a multipath with at is distributed as depicted in Fig. 3(b). The resulting and probabilities are depicted in Fig. 3(c).
The numerical analysis revealed that for a single entangled path at the particular connection-level values of the path (given in Fig. 3(a)). The multipath setting at connection-level values of Fig. 3(b), at doubles the success probability of the single path setting with , while at , the resulting probability is .
In this work, we defined a method to achieve entanglement access control in the quantum Internet. The algorithm utilizes different paths between the source and target nodes in function of a particular path cost function. The path cost function uses the local entanglement fidelities of the nodes and the probability of the existence of the entangled connections. Increasing the number of available paths leads to a multipath setting, which allows the parties to establish high fidelity entanglement with reliable entangled connections between the end nodes. The proposed scheme has moderate complexity, and it is particularly convenient for the entangled quantum network structure of the quantum Internet.
This work was partially supported by the National Research Development and Innovation Office of Hungary (Project No. 2017-1.2.1-NKP-2017-00001), by the Hungarian Scientific Research Fund - OTKA K-112125 and in part by the BME Artificial Intelligence FIKP grant of EMMI (BME FIKP-MI/SC).
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