Entanglement Availability Differentiation Service for the Quantum Internet

A fundamental concept of the quantum Internet is quantum entanglement. In a quantum Internet scenario where the legal users of the network have different priority levels or where a differentiation of entanglement availability between the users is a necessity, an entanglement availability service is essential. Here we define the entanglement availability differentiation (EAD) service for the quantum Internet. In the proposed EAD framework, the differentiation is either made in the amount of entanglement with respect to the relative entropy of entanglement associated with the legal users, or in the time domain with respect to the amount of time that is required to establish a maximally entangled system between the legal parties. The framework provides an efficient and easily-implementable solution for the differentiation of entanglement availability in experimental quantum networking scenarios.


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 AB (1) is an even mixture of the Bell states which contains no entanglement. It is also the situation in Step 2 for the subsystem AB of ρ ABC (3), thus the relative entropy of entanglement for ρ AB is zero, E (A : B) = 0. The initial ρ AB in (1) and (3) with eigenvalues v + = 1 2 , v − = 0, u + = 1 2 , u − = 0. In Step 3, dynamics generated by local Hamiltonian H AC = σ x A σ x C with energy E AC will lead to entanglement oscillations in AB. Thus, if U AC is applied exactly only for a well determined time t, the local unitary will lead to maximally entangled AB with a unit probability.
As a result, for subsystem AB, the entanglement E (A : B) oscillates [10] with the application time t of the unitary. In particular, the entanglement oscillation in AB generated by the energy E AC (6) of the Hamiltonian H AC (5). This oscillation has a period time T π , which exactly equals to 4t, thus where t is determined by Alice and Bob. In other words, time t identifies π/4, where π is the oscillation period. Therefore, after Step 3, the density σ ABC of the final ABC state is as where |ϕ (t) ABC at t is evaluated as that can be rewritten as where the sign change on U AC (|φ + |− ) is due to the |− eigenstate on C. Thus, at t = π/4, i.e., up to the global phase both states are the same. Therefore the |ϕ (π/4) ABC system state of ABC at t = π/4 is yielded as while the density matrix σ ABC of the final ABC system in matrix form is as (A.9) As one can verify, the resulting AB state |ξ (π/4) AB at t = π/4 is pure and maximally entangled, yielding relative entropy of entanglement with unit probability. The σ AB density matrix of the final AB state is which in matrix form is as The negativity for the σ T B AB partial transpose of σ AB yields which also immediately proves that AB is maximally entangled. For a comparison, for the density matrix of initial AB, (1), is N ρ T B AB = 0. Note that subsystem C requires no further storage in a quantum memory, since the output density σ ABC can be rewritten as where I is the identity operator, therefore the protocol does not require long-lived quantum memories.

A.1.1 Classical Correlations
The classical correlation is transmitted subsystem B of (1) in Step 1 is as follows. Since ρ AB is a Bell-diagonal state [49] of two qubits A and B it can be written as where terms σ j refer to the Pauli operators, while |β ab is a Bell-state while λ ab are the eigenvalues as The I quantum mutual information of Bell diagonal state ρ AB quantifies the total correlations in the joint system ρ AB as where S (ρ) = −Tr (ρ log 2 ρ) is the von Neumann entropy of ρ, and S ( B| A) = S (ρ AB ) − S (ρ A ) is the conditional quantum entropy. The C (ρ AB ) classical correlation function measures the purely classical correlation in the joint state ρ AB . The amount of purely classical correlation C (ρ AB ) in ρ AB can be expressed as follows [49]: is the post-measurement state of ρ B , the probability of result k is while d is the dimension of system ρ A and the q k make up a normalized probability distribution, where c = 1.

EAD Entanglement Availability Differentiation
POVM Positive-Operator Valued Measure

A.3 Notations
The notations of the manuscript are summarized in Table A.1.