1 Introduction
In the quantum Internet [2, 23], one of the most important tasks is to establish entanglement [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] between the legal parties [11, 12, 13, 14, 15] so as to allow quantum communication beyond the fundamental limits of pointtopoint connections [41, 42, 43]. For the problem of entanglement distribution in quantum repeater networks several methods [7, 10, 11, 12, 13, 14, 15], and physical approaches have been introduced [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 47, 48, 49]. The current results are mainly focusing on the physicallayer of the quantum transmission [5, 6, 7, 8, 9], implementations of entanglement swapping and purification, or on the optimization of quantum memories and quantum error correction in the repeater nodes [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40]. However, if the legal users of the quantum network are associated with different priority levels, or if a differentiation of entanglement availability between the users is a necessity in a multiuser quantum network, then an efficient and easily implementable entanglement availability service is essential.
In this work, we define the entanglement availability differentiation (EAD) service for the quantum Internet. We introduce differentiation methods, Protocols 1 and 2, within the EAD framework. In Protocol 1, the differentiation is made in the amount of entanglement associated with the legal users. The metric used for the quantization of entanglement is the relative entropy of entanglement function [44, 45, 46]. In Protocol 2, the differentiation is made in the amount of time that is required to establish a maximally entangled system between the legal parties.
The EAD framework contains a classical phase (Phase 1) for the distribution of timing information between the users of the quantum network. Phase 2 consists of all quantum transmission and unitary operations. In Phase 2, the entanglement establishment is also performed between the parties according to the selected differentiation method.
The entanglement distribution phase of EAD utilizes Hamiltonian dynamics, which allows very efficient practical implementation for both the entanglement establishment and the differentiation of entanglement availability. Using the Hamiltonian dynamics approach as a core protocol of Step 2 of the EAD framework, the entanglement differentiation method requires only unitary operations at the transmitter and requires no entanglement transmission. The application time of the unitaries can be selected as arbitrarily small in the transmitter to achieve an efficient practical realization. The proposed EAD framework is particularly convenient for experimental quantum networking scenarios, quantum communication networks, and future quantum internet.
The novel contributions of our manuscript are as follows:

We define the entanglement availability differentiation (EAD) service for the quantum Internet.

The entanglement availability differentiation is achieved via Hamiltonian dynamics between the users of the quantum network.

The EAD framework can differentiate in the amount of entanglement with respect to the relative entropy of entanglement associated to the legal users (Protocol 1), and also in the time domain with respect to the amount of time that is required to establish a maximally entangled system (Protocol 2) between the legal parties.

The framework provides an efficient and easilyimplementable solution for the differentiation of entanglement availability in experimental quantum networking scenarios.
2 System Model
The proposed EAD service allows differentiation in the amount of entanglement shared between the users or the amount of time required for the establishment of maximally entangled states between the users. The defined service requires no entanglement transmission to generate entanglement between the legal parties. The differentiation service consists of two phases: a classical transmission phase (Phase 1) to distribute side information for the entanglement differentiation and a quantum transmission phase (Phase 2), which covers the transmission of unentangled systems between the users and the application of local unitary operations to generate entanglement between the parties.
The proposed entanglement availability differentiation methods are detailed in Protocol 1 and Protocol 2. The protocols are based on a core protocol (Protocol 0) that utilizes Hamiltonian dynamics for entanglement distribution in quantum communication networks (see Section A.1). The aim of the proposed entanglement differentiation protocols (Protocol 1 and Protocol 2) is different from the aim of the core protocol, since Protocol 0 serves only the purpose of entanglement distribution, and allows no entanglement differentiation in a multiuser quantum network. Protocol 0 is used only in the quantum transmission phase and has no any relation with a classical communication phase.
2.1 Classical Transmission Phase
In the classical transmission phase (Phase 1), the timing information of the local Hamiltonian operators are distributed among the legal parties by an encoder unit. The content of the timing information depends on the type of entanglement differentiation method. The Hamiltonian operators will be applied in the quantum transmission phase (Phase 2) to generate entangled systems between the users. Since each types of entanglement differentiation requires the distribution of different timing information between the users, the distribution of classical timing information will be discussed in detail in Section 3.
2.2 Quantum Transmission Phase
The quantum transmission phase (Phase 2) utilizes a core protocol for the entanglement distribution protocol of the EAD framework. The core protocol requires no entanglement transmission for the entanglement generation, only the transmission of an unentangled quantum system (i.e., separable state [10, 11, 12, 13, 14, 15]) and the application of a unitary operation for a welldefined time in the transmit user. The core protocol of the quantum transmission phase for a userpair is summarized in Protocol 0. It assumes the use of redundant quantum parity code [7] for the encoding^{2}^{2}2Actual coding scheme can be different.. For a detailed description of Protocol 0, see Section A.1.
(1) 
(2) 
(3) 
(4) 
(5) 
(6) 
system with probability
as(7) 
2.3 Framework
In our multiuser framework, the quantum transmission phase is realized by the core protocol of Phase 2; however, time of the Hamiltonian operator is selected in a different way among the users, according to the selected type of differentiation. For an th user , the application time of the local unitary is referred to as . Without loss of generality, the th transmit user is referred to as , and the th receiver user is .
In the system model, the user pairs can use the same physical quantum link, therefore in the physical layer the users can communicate over the same quantum channel. On the other hand, in a logical layer representation of the protocols, the communication between the user pairs formulate logically independent channels.
3 Methods of Entanglement Availability Differentiation
The EAD service defines different types of differentiation. The differentiation can be achieved in the amount of entanglement in terms of the relative entropy of entanglement between the users (Protocol 1: differentiation in the amount of entanglement). In this method, all users have knowledge of a global oscillation period [10] of time for the application of their local unitaries, but the users will get different amounts of entanglement as a result.
The differentiation is also possible in the amount of time that is required to establish a maximally entangled system between the users (Protocol 2: differentiation in the time domain). In this method, all users get a maximally entangled system as a result; however, the time that is required for the entanglement establishment is variable for the users, and there is also no global oscillation period of time.
3.1 Differentiation in the Amount of Entanglement
The differentiation of the entanglement amount between the users allows us to weight the entanglement amount between the users in terms of relative entropy of entanglement. Using the timing information distributed in Phase 1 between the transmit users
(8) 
where
(9) 
for an th transmit user , the protocol generates an initial system , transmits separable B to receiver , and applies the local unitary on subsystem for time (using the core protocol of Phase 2). Depending on the selected , the resulting subsystem between users and contains the selected amount of entanglement,
(10) 
3.1.1 Relative Entropy of Entanglement
In the proposed service framework, the amount of entanglement is quantified by the relative entropy of entanglement function. By definition, the relative entropy of entanglement function of a joint state of subsystems and is defined by the quantum relative entropy function, without loss of generality as
(11) 
where is the set of separable states .
3.1.2 Differentiation Service
The Phases 1 and 2 of the method of entanglement amount differentiation (Protocol 1) are as included in Protocol 1.
Description
In the quantum transmission phase, the entanglement oscillation in is generated by the energy of the Hamiltonian [10]. This oscillation has a period of time , which exactly equals to ,
(12) 
where is determined by Alice and Bob. In other words, time identifies , where is the oscillation period. Therefore, in Protocol 1, the density of the final state is as
(13) 
where at time is evaluated as
(14) 
which at (see (8)) of user , for a given is evaluated as
(15) 
where the sign change on is due to the eigenstate on , and where
(16) 
Thus, up to the global phase, both states are the same.
Therefore, the system state of at is yielded as
(17) 
therefore, the resulting time state at and , is a nonmaximally entangled system
(18) 
with entanglement between user and as
(19) 
3.2 Differentiation in the Time Domain
In the time domain differentiation service, a transmit user generates the initial system , transmits separable to receiver , and applies the local unitary on subsystem for time (using the core protocol of Phase 2). Using the oscillation period distributed in Phase 1, the resulting subsystem after total time
(20) 
between users and , is a maximally entangled system, , for all .
3.2.1 Differentiation Service
The Phases 1 and 2 of the time domain differentiation method (Protocol 2) are as included in Protocol 2.
(21) 
Description
Let us focus on a particular of users and . The same results apply for all users of the network.
After the steps of Protocol 2, the density of the final state is as
(22) 
where at is evaluated as
(23) 
where the sign change on is due to the eigenstate on .
Thus, at , the system state is
(24) 
where
(25) 
Thus, up to the global phase both states are the same yielding relative entropy of entanglement between users and as
(26) 
with unit probability.
3.3 Comparative Analysis
4 Conclusions
Entanglement differentiation is an important problem in quantum networks where the legal users have different priorities or where differentiation is a necessity for an arbitrary reason. In this work, we defined the EAD service for the availability of entanglement in quantum Internet. In EAD, the differentiation is either made in the amount of entanglement associated with a legal user or in the amount of time that is required to establish a maximally entangled system. The EAD method requires a classical phase for the distribution of timing information between the users. The entanglement establishment is based on Hamiltonian dynamics, which allows the efficient implementation of the entanglement differentiation methods via local unitary operations. The method requires no entanglement transmission between the parties, and the application time of the unitaries can be selected as arbitrarily small via the determination of the oscillation periods to achieve an efficient practical realization. The EAD method is particularly convenient for practical quantum networking scenarios, quantum communication networks, and future quantum Internet.
Acknowledgements
L.GY. would like to thank Tomasz Paterek for useful discussions. This work was partially supported by the National Research Development and Innovation Office of Hungary (Project No. 20171.2.1NKP201700001), by the Hungarian Scientific Research Fund  OTKA K112125 and in part by the BME Artificial Intelligence FIKP grant of EMMI (BME FIKPMI/SC).
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Appendix A Appendix
a.1 Steps of the Core Protocol
The detailed discussion of the Core Protocol (Protocol 0) is as follows.
In Step 1, the input system (1) is an even mixture of the Bell states which contains no entanglement. It is also the situation in Step 2 for the subsystem of (3), thus the relative entropy of entanglement for is zero, . The initial in (1) and (3), is the unentangled, Belldiagonal state
(A.1) 
with eigenvalues
.In Step 3, dynamics generated by local Hamiltonian with energy will lead to entanglement oscillations in . Thus, if is applied exactly only for a well determined time , the local unitary will lead to maximally entangled with a unit probability.
As a result, for subsystem , the entanglement oscillates [10] with the application time of the unitary. In particular, the entanglement oscillation in generated by the energy (6) of the Hamiltonian (5). This oscillation has a period time , which exactly equals to , thus
(A.2) 
where is determined by Alice and Bob. In other words, time identifies , where is the oscillation period.
Therefore, after Step 3, the density of the final state is as
(A.3) 
where at is evaluated as
(A.4) 
that can be rewritten as
(A.5) 
where the sign change on is due to the eigenstate on .
Thus, at ,
(A.6) 
where
(A.7) 
i.e., up to the global phase both states are the same.
Therefore the system state of at is yielded as
(A.8) 
while the density matrix of the final system in matrix form is as
(A.9) 
As one can verify, the resulting state at is pure and maximally entangled,
(A.10) 
yielding relative entropy of entanglement
(A.11) 
with unit probability.
The density matrix of the final state is
(A.12) 
which in matrix form is as
(A.13) 
The negativity for the partial transpose of yields
(A.14) 
which also immediately proves that is maximally entangled. For a comparison, for the density matrix of initial , (1), is .
Note that subsystem requires no further storage in a quantum memory, since the output density can be rewritten as
(A.15) 
where is the identity operator, therefore the protocol does not require longlived quantum memories.
a.1.1 Classical Correlations
The classical correlation is transmitted subsystem of (1) in Step 1 is as follows. Since is a Belldiagonal state [49] of two qubits and it can be written as
(A.16) 
where terms refer to the Pauli operators, while is a Bellstate
(A.17) 
while are the eigenvalues as
(A.18) 
The quantum mutual information of Bell diagonal state quantifies the total correlations in the joint system as
(A.19) 
where is the von Neumann entropy of , and is the conditional quantum entropy.
The classical correlation function measures the purely classical correlation in the joint state . The amount of purely classical correlation in can be expressed as follows [49]:
(A.20) 
where
(A.21) 
is the postmeasurement state of , the probability of result is
(A.22) 
while is the dimension of system and the
make up a normalized probability distribution,
are rankone POVM (positiveoperator valued measure) elements of the POVM measurement operator [49], while is the binary entropy function, and(A.23) 
For the transmission of the subsystem is expressed as given by (1), thus the classical correlation during the transmission is
(A.24) 
where .
a.2 Abbreviations
 EAD

Entanglement Availability Differentiation
 POVM

PositiveOperator Valued Measure
a.3 Notations
The notations of the manuscript are summarized in Table A.1.
Notation  Description 
Initial system.  
Final system.  
,  Initial subsystems. 
Subsystem , , encoded via an redundant quantum parity code as
, where . 

Period time selected by Alice and Bob.  
Intermediate quantum repeaters between Alice and Bob.  
Pauli X matrix.  
Hamiltonian, .  
Energy of Hamiltonian .  
Unitary, applied by Alice on subsystem for a time , , where is a Hamiltonian, is the Pauli X matrix.  
Application time of unitary , determined by Alice and Bob.  
Identity operator.  
Reduced Planck constant.  
Relative entropy of entanglement.  
Oscillation period, , where is the period.  
Output subsystem at time ,
, where , are maximally entangled states. 

Partial transpose of output AB subsystem .  
Negativity for the partial transpose of . 
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