On the Stochastic Analysis of a Quantum Entanglement Distribution Switch

Abstract

We study a quantum entanglement distribution switch that serves k users in a star topology. We model variants of the system as continuous-time Markov chains (CTMCs) and obtain expressions for switch capacity, expected number of qubits stored in memory at the switch, and the quantum memory occupancy distribution. We obtain a number of analytic results for systems in which measurements are imperfect, the links are homogeneous or heterogeneous and for switches that have an infinite or finite number of quantum memories, or buffers. In addition, we model the effect of decoherence of quantum states and associated cutoff times on their storage using a simple model. From numerical observations, we discover that decoherence-associated cutoff times have little effect on capacity and expected number of stored qubits for homogeneous systems. For heterogeneous systems, especially those operating near the boundaries of their stability regions (i.e., systems that are nearly unstable), 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. The analytical results obtained in this work can serve as a useful guide toward the future design of quantum switches-e.g., by allowing the designer to determine how many quantum memories suffice for a given number of users-as well as provide valuable insight on the performance of these and similar devices.

Policies for elementary links in a quantum network

“…Such an extension would involve multiple cooperating agents, in contrast to the independent agents considered in this work, and can in principle be formulated for an arbitrary network topology. A simple, but relevant example of a network topology, which has also been considered recently, is the starshaped network used for the so-called "quantum entanglement switch" [80][81][82][83]. As we might expect, these extra elements of entanglement distillation and swapping will make analytic analysis (as done in this work) intractable.…”

“…One of the goals of this work is to explicitly formalize the approaches taken in the aforementioned works within the context of decision processes, because this allows us to systematically study different policies and calculate quantities that are relevant for quantum networks, such as entanglement distribution rates and fidelities of the quantum states of the links. This work is complementary to prior work that uses Markov chains to analyze waiting times and entanglement distribution rates for a chain of quantum repeaters [65,[130][131][132]; we also refer to the work on entanglement switches in [80][81][82][83], which use both discrete-time and continuous-time Markov chains. This work is also complementary to prior work that analyzes the quantum state in a quantum repeater chain with noisy quantum memories [133][134][135][136][137].…”

Analysis of a tripartite entanglement distribution switch

“…When n = 2, the switch uses Bell-state measurements (BSMs) and when n ≥ 3, it uses n-qubit Greenberger-Horne-Zeilinger (GHZ) basis measurements [29]. For additional background on quantum switches the interested reader is referred to [36, Section 2] and [37,Section II].…”

“…In this paper we consider the situation when three user's qubits are entangled (i.e., n = 3), links are homogeneous, and all buffers are infinite. The situation when n = 2 was studied in [37] under a variety of assumptions.…”

“…Exploiting this property, the state of the switch can be represented by a birth and death process thereby yielding the stationary distribution -and from it the main performance measures of interest -explicitly or in closed-form both when the switch has a finite or infinite memory [37]. When the memory is infinite it is shown in [37] that the system is stable when k > 2 and, among other results, that the system capacity -defined as the maximum achievable number of successful assemblies per time unit -is q k l=1 µ l /2 when the switch has an infinite memory. Results are also obtained in [37] when quantum state decoherence and an associated cut-off policy are added to the model, which amounts to assuming that stored entanglements have an exponential lifetime with a constant rate.…”