# On the Stochastic Analysis of a Quantum Entanglement Switch

## TL;DR

This paper analyzes a quantum entanglement switch using Markov chain models, comparing discrete and continuous approaches, and examines the impact of decoherence, buffer size, and heterogeneity on system performance.

## Contribution

It introduces continuous-time Markov chain models for quantum switches, providing analytical results and comparing them with discrete models to understand system behavior.

## Key findings

- CTMC approximates DTMC well for homogeneous links and infinite buffers.
- Decoherence has minimal impact on capacity in homogeneous systems.
- Buffer size increase from one to two qubits improves performance, with diminishing returns beyond that.

## Abstract

We study a quantum entanglement switch that serves $k$ users in a star topology. We model variants of the system using Markov chains and standard queueing theory and obtain expressions for switch capacity and the expected number of qubits stored in memory at the switch. While it is more accurate to use a discrete-time Markov chain (DTMC) to model such systems, we quickly encounter practical constraints of using this technique and switch to using continuous-time Markov chains (CTMCs). Using CTMCs allows us to obtain a number of analytic results for systems in which the links are homogeneous or heterogeneous and for switches that have infinite or finite buffer sizes. In addition, we can model the effects of decoherence of quantum states fairly easily using CTMCs. We also compare the results we obtain from the DTMC against the CTMC in the case of homogeneous links and infinite buffer, and learn that the CTMC is a reasonable approximation of the DTMC. From numerical observations, we discover that decoherence has little effect on capacity and expected number of stored qubits for homogeneous systems. For heterogeneous systems, especially those operating close to stability constraints, buffer size and decoherence can have significant effects on performance metrics. We also learn that in general, increasing the buffer size from one to two qubits per link is advantageous to most systems, while increasing the buffer size further yields diminishing returns.