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Stochastic Analysis of Entanglement-assisted Quantum Communication Channels

Karim S. Elsayed, Olga Izyumtseva, Wasiur R. KhudaBukhsh, Amr Rizk

TL;DR

A Functional Law of Large Numbers (FLLN) and a Functional Central Limit Theorem (FCLT) for the standard queue averaging the dynamics of the fast service queue are proved.

Abstract

In this paper, we present a queueing model for quantum communication networks, a rapidly growing field of research inspired by its technological promise and recent experimental successes. The model consists of a primary queue and a service queue where Bell pairs are formed and stored. The Bell pairs are by nature extremely short-lived rendering the service queue (the quantum queue) much faster than the primary queue. We study the asymptotic behaviour of this multi-scale queueing system utilizing the theory of stochastic averaging principle. We prove a Functional Law of Large Numbers (FLLN) and a Functional Central Limit Theorem (FCLT) for the standard queue averaging the dynamics of the fast service queue. Our proofs are probablistic and rely on the stochastic analysis of Stochastic Differential Equations (SDEs) driven by Poisson Random Measures.

Stochastic Analysis of Entanglement-assisted Quantum Communication Channels

TL;DR

A Functional Law of Large Numbers (FLLN) and a Functional Central Limit Theorem (FCLT) for the standard queue averaging the dynamics of the fast service queue are proved.

Abstract

In this paper, we present a queueing model for quantum communication networks, a rapidly growing field of research inspired by its technological promise and recent experimental successes. The model consists of a primary queue and a service queue where Bell pairs are formed and stored. The Bell pairs are by nature extremely short-lived rendering the service queue (the quantum queue) much faster than the primary queue. We study the asymptotic behaviour of this multi-scale queueing system utilizing the theory of stochastic averaging principle. We prove a Functional Law of Large Numbers (FLLN) and a Functional Central Limit Theorem (FCLT) for the standard queue averaging the dynamics of the fast service queue. Our proofs are probablistic and rely on the stochastic analysis of Stochastic Differential Equations (SDEs) driven by Poisson Random Measures.

Paper Structure

This paper contains 17 sections, 7 theorems, 130 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Mathematical contributions
  3. Notational conventions
  4. Stochastic Model
  5. FLLN
  6. Relative compactness
  7. Identification of the limit
  8. FCLT
  9. Convergence of the martingale term $\tilde{M}^{(n)}$
  10. Convergence of $\frac{1}{\sqrt{n}}(F(Y^{(n)}(t))-F(Y^{(n)}(0)))$
  11. Convergence of the term $\frac{1}{\sqrt{n}}\int^t_0\delta^{(n)}_F(Y^{(n)}(s))\mathrm{d} s$
  12. Tightness of the sequence $W_n$
  13. Identification of the limit
  14. Discussion
  15. Additional mathematical details
  16. ...and 2 more sections

Key Result

Lemma 1

Assume $Y^{(n)}_B(0) = 0$, and set $\rho \coloneqq \lambda/\mu$. Then, for $\beta > 0$, and $k \ge 1$, we have the following estimate

Figures (2)

  • Figure 1: Queueing system: messages are sent through the quantum channel with an accelerated service propensity function $r_4^{(n)}(x)$ only when there exist entanglements in Queue B, otherwise messages are served with $r_3^{(n)}(x)$ through the classical communication channel. Queue B gets depleted at rate $r_5^{(n)}(x)$ due to the quality decay of entanglements with time. Here, $x=(x_1,x_2)$ is the vector of queue lengths of the messages and entanglement queues and the functions $r_{1}^{(n)}(x), r_{2}^{(n)}(x), \ldots, r_{5}^{(n)}(x)$ are the propensity/intensity functions defined in \ref{['eq:propensity_1']} and \ref{['eq:propensity_2']} in \ref{['sec:ctmc_model']}.
  • Figure 2: (Left) Accuracy of the FLLN approximation for the scaled queue lengths $Y^{(n)}_A$ by the deterministic function $y_A$ solving the ODE in \ref{['eq:limit_ODE']} in \ref{['thm:relative_compactness']}. The stochastic simulations are performed using the standard Doob--Gillespie's algorithm Anderson:2011:CTMWilkinson2018SMS. A total of $100$ trajectories are shown in this figure. (Right) Steady state values of the scaled queue length at queue $A$ as a function of the arrival rate $\lambda_B$ at the queue $B$. The queue A dependent propensity functions are: (a) $r_1=\lambda_A, r_3(y_1)=\mu_A y_1, r_4(y_1)=M \mu_A y_1$ (b) $r_1=\lambda_A/(y_1+1), r_3(y_1)=\mu_A y_1, r_4(y_1)=M \mu_A y_1$. The parameter values used: $n=10^5, y_A(0)=0, \lambda_A = 5.0, \lambda_B=3.0, \mu_A=2.0, \mu_B=2.0$, and the speedup factor $M=2.0$.

Theorems & Definitions (17)

  • Lemma 1
  • proof : Proof of \ref{['lem:MGF_hitting_times']}
  • Remark 1
  • Theorem 1: FLLN
  • proof : Proof of \ref{['thm:relative_compactness']}
  • Remark 2
  • Lemma 2
  • proof : Proof of \ref{['lem:Poisson_eqn']}
  • Theorem 2: FCLT
  • proof : Proof of \ref{['thm:clt']}
  • ...and 7 more