On the Stochastic Analysis of a Quantum Entanglement Switch

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. Index Terms-entanglement, quantum switch, Markov chain. arXiv:1903.04420v2 [quant-ph] 18 Jul 2019 1 With a linear optical circuit, four unentangled ancilla single photons and photon number resolving detectors, with all the devices being lossless, q = 25/32 = 0.78 can be achieved for BSMs [16]. With other technologies q close to 1 can be achieved [17].

Entanglement formation in continuous-variable random quantum networks

“…Universal scaling laws and dynamical models of entanglement formation has been established, based on nonlinear surface growth models 12,13 . Recently, experimental probing 21,47,48 of scrambling is also made possible; from the quantum network perspective, protocol design [49][50][51][52] for entanglement establishment has also been a recent focus.…”

Designing a quantum network protocol

“…We propose that the signalling protocol is best suited to the task of schedule management. It is an open question how best to perform scheduling at a quantum node [89,90]. In early-stage network this synchronisation will have to be very precise and may need to allocate dedicated time bins to each circuit.…”

On the Analysis of a Multipartite Entanglement Distribution Switch

“…A set of randomized switching policies are explored, together comprising an achievable capacity region. In [49], we use Markov chains to analyze an entanglement switch that serves only bipartite end-to-end entangled states to users by performing Bell-state measurements. In addition to swapping failures, the model also accounts for nite bu er sizes, potentially non-identical links (implying di erent entanglement generation rates for di erent links), and incorporates the e ects of state decoherence.…”

“…e proof of Proposition 6.1 (see Appendix C) can be considerably shortened if instead of working with the Lyapunov functions min{|x| 2 ,T 2 } and min{|x| (2) ,T }, one selects the functions |x| 2 and |x| (2) , respectively. However, in addition to the stability assumption, this approach requires to assume that E[|Q| 2 ] < ∞ so that Lemma 4.2 can be invoked; see [49] for a proof. We conjecture that E[|Q| 2 ] < ∞ if the system is stable but have not been able to prove it.…”