Dynamics of entangled networks of the quantum Internet

Entangled quantum networks are a fundamental of any global-scale quantum Internet. Here, a mathematical model is developed to quantify the dynamics of entangled network structures and entanglement flow in the quantum Internet. The analytical solutions of the model determine the equilibrium states of the entangled quantum networks and characterize the stability, fluctuation attributes, and dynamics of entanglement flow in entangled network structures. We demonstrate the results of the model through various entangled structures and quantify the dynamics.


A.3.1 Seamless Optimal Entanglement Flow at Fluctuating Entangled Connections
Lemma A.1 (Seamless optimal entanglement flow in the quantum Internet at fluctuating entangled connections). For the total Q paths of N, the F N entanglement flow is seamless optimal, F N = F * N , if f (B F (S * (N))) ≥ 1 and ϕ (E s ) ≤ ϕ * (E s ) for s = 1, . . . , ∑ Q j=1 S P j .
Proof. Recalling the condition of B F (F N ) ≥ B F (S * (N)), f (B F (S * (N))) ≥ 1 for f (B F (S * (N))) straightforwardly follows. Then, from the proof of Theorem 3, the L * (N) Laplacian of the entangled quantum network N is always symmetrizable for any seamless optimal entanglement flow F * N , which determines ϕ * (E s ) for s = 1, . . . , ∑ Q j=1 S P j . The proof is concluded here. The notations of the fluctuation dynamics analysis of the entangled structure are summarized in Fig. A. , for which ϕ * (E s ) = 0, s = 1, . . . , S * P j , where S * P j is an connection set of a j-th path P j .
As ∆ (F N ) = 0, the stable equilibrium states of the entangled structure are symmetrical for all f (B F (F N )), thus the stable equilibrium states for f (B F (F N )) > 1 are overlapped in the right figure. (b)-(d) As ∆ (F N ) increases, the entangled structure has one stable equilibrium state only for low values of f (B F (F N )). For a higher values of f (B F (F N )) the two stable equilibrium states are asymmetrical, thus the stable equilibrium states in the right figure are distinguished into red and blue dots for a particular f (B F (F N )). (e) As ∆ (F N ) → 1, f (B F (F N )) ≤ 3, the entangled structure has one stable state, while if f (B F (F N )) > 3, then the entangled network has two asymmetrical stable equilibrium states.
The proof is concluded here.

A.4 Notations
The notations of the manuscript are summarized in Table A.1. A A source user (quantum node) in the quantum network.

B
A destination user (quantum node).

R i
An i-th quantum repeater, i = 1, . . . , |V |, where |V | is the total number of quantum nodes of N.
R k A next neighbor of R k (towards destination).
l Level of entanglement.
L l (x, y) An l-level entangled connection between quantum nodes x and y, simplified as E (x, y).
d (x, y) L l Hop-distance at an L l -level entangled connection between quantum nodes x and y, d (x, y) L l = 2 l−1 .

O C
An oscillator with frequency f C , f C = 1 t C , serves as a reference clock. E (x, y) An l-level entangled connection between quantum nodes x and y. F N Entanglement flow in N.

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P j A j-th entangled path of F N , j = 1, . . . , Q, where Q is the total number of paths in N.
F P j (R i ) Average entanglement fidelity outputted via quantum repeater R i in the F N entanglement flow of N.
F P (R i ) Average entanglement fidelity outputted via quantum repeater R i in the F N entanglement flow of N.
The average fidelity of entanglement flow F N .
V P j Number of quantum nodes of a given entangled path P j .
S P j Number of entangled connections nodes of a given entangled path P j .
Average entanglement rate of P j of F N at a particular entanglement fidelity F.
Average entanglement throughput of an s-th entangled connection E s for a particular entanglement fidelity F, s = 1, . . . , S P j .

B (F N )
Average entanglement throughput of F N for the total Q paths of N, for a particular entanglement fidelity F.  φ (F (F N )) Normalized value of F (F N ).

H (∆ (F N ))
Hamiltonian of the average noise ∆ (F N ) of F N .
σ i A state of quantum node.
where the sign (x) function returns the sign of x (sign (0) is considered as negative.
Average number of entangled connections between the nodes.
E (F N ) Energy of the system S (N).
State of the entangled structure N at a particular t, t = 1, . . . , T .
Probability of transitionξ (F N ) →ξ (F N ) at a given S (N (t)).

S (N (t))
Statistical physics model of an entangled quantum network N (t).
ω A normalization term.

Ω (F N )
A coefficient of the stability analysis of the entangled structure N.
ζ (F N ) A coefficient of the stability analysis of the entangled structure N.
A coefficient of the stability analysis of the entangled structure N.
A coefficient of the stability analysis of the entangled structure N.
) stability function of the entangled quantum network N.
Derivative the Ψ F N (φ (F (F N ))) stability function of the entangled quantum network N.
) stability function of the entangled quantum network N.
An averaged fidelity for a given S N , at a particular ∆ (F N ).

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δ (R i ) State of an i-th quantum repeater R i . B F (P R i ) Entanglement rate consumption of entanglement purification P R i (sum of incoming and outcoming entanglement rates in R i associated with P R i ).

S P R i
Set of entangled connections of R i associated with entanglement purification P R i .

S P R i
Cardinality of set S P R i .
ω A ratio of the target average fidelity F * (F N ) of the entanglement flow F N of N and F * P (R i ).
Capability of a quantum repeater R i to improve the F (F N ) average fidelity of F N to a target F * (F N ) via an entanglement purification P R i .

C (N)
Capability of the entangled network N to improve the F (F N ) average fidelity of F N to a target F * (F N ) via P R i in the |V | nodes, i = 1, . . . , |V |, of N.
Total entanglement rate consumption of entanglement purification P N in N.
B F (P R i ) Average entanglement rate consumption at P N for a given node.

P R i
Entanglement purification in a local R i with an increased target entanglement rate consumption B * F P R i . B F (P N ) Total entanglement rate consumption of entanglement purification P N .

B F P R i
Average entanglement rate consumption of P N . λ An eigenvalue of L * (N).