On the Capacity Region of Bipartite and Tripartite Entanglement Switching

Multi-qubit entangled states are fundamental building blocks for quantum computation, sensing, and security. Consequently, there is a need for a quantum network that can generate such entanglement on demand between pairs and groups of users [9, 24, 25, 29, 30]. In this article, we study the performance of the simplest multi-user network, a star-topology quantum switch connecting k users, where each user is connected to the switch via a separate link. Bipartite, two-qubit maximally entangled states, i.e., Bell pairs (or EPR states) are generated at a constant rate across each link, with the qubits stored at local quantum memories at each end of the links. As these link-level entangled states start appearing, the switch uses two-qubit Bell-state measurements (BSM) between pairs of locally held qubits and three-qubit Greenberger-Horne-Zeilinger (GHZ) basis measurements between triples of locally held qubits to provide two-qubit and three-qubit entanglement to pairs and triples of users, respectively [23]. The capacity of such a switch to provide these two types of entanglement to the users depends on the switching mechanism, the number of quantum memories and their decoherence rates (assuming that quantum state decoherence imposes a qubit storage cut-off policy implemented at the switch), and the total number of links or users.

We define the bipartite and tripartite capacities of the switch as the highest possible rate at which the device can generate Bell pairs and 3-qubit GHZ states via entanglement swapping, respectively, under a specified entanglement switching policy. For a given number of end nodes and quantum memories, the set of all switching policies define the capacity region of the switch. In this article, we study the capacity region when the switch can store either \(B=1\) or \(B=2\) qubits for each link at any given time. The number of quantum memories available to a link is referred to as its buffer size. We consider a simple time division multiplexing (TDM) policy between the two types of entangled states, along with a class of randomized switching policies. In a TDM policy, the switch performs BSMs a fixed fraction of the time, and three-way GHZ basis measurements for the remaining portion of the time. When properly configured, the randomized switching policies provide higher capacities than TDM. However, the relative difference between the two sets of policies goes to zero as \(k \rightarrow \infty\). We also observe that increasing the number of memories from one to two increases capacity but that the increase diminishes as k increases. Since the locally stored qubits at each end of the link are subject to a noise process that reduces the entanglement between the two qubits, we also explore the effect that decoherence has on capacity. Specifically, we assume that the switch implements a cut-off policy for qubit storage to cope with finite coherence time of quantum memories. Throughout this manuscript, we often implicitly refer to storage cut-off times when referring to studying a system with quantum state decoherence. In the cases of \(B=1\) with and without decoherence, we have simple closed form expressions for capacity whereas for the case of \(B=2\), we provide a partial analysis but our results are mainly numerical.

The remainder of this article is organized as follows: in Section 2, we provide relevant background and related work. In Section 3, we formulate the problem and propose a method for solving it. In Section 4, we present the case where the system has a per-link buffer of size one, and provide analytical and numerical results. In Section 5, we present numerical results for the case where the system has a per-link buffer of size two and observe similar behavior to the buffer size one case. For the \(B=2\) scenario, we also provide a partial analysis. In Section 6, we introduce a simple technique for modeling quantum state decoherence and associated cut-off times on qubit storage, and use it to examine the effect of decoherence on the bipartite-tripartite capacity region for systems with per-link buffer sizes one and two. For the former, we also have analytical results. We make concluding remarks in Section 7.

Bell states are an integral part of a diverse set of distributed quantum applications, including Quantum Key Distribution (QKD) [2, 10], superdense coding [3], teleportation [1], and distributed quantum computation [15]. Similarly, GHZ states can be used to implement a variety of quantum protocols, such as cryptographic conferencing [12], quantum sensing [11], and multipartite generalization of superdense coding [14]. The advantage of these applications is that they offer functionality that cannot be achieved classically, e.g., information-theoretic security. However, quantum distributed tasks typically require reliable transport of quantum states; this can be a significant challenge due to the exponential rate-versus-distance decay [26]. Quantum repeaters positioned between communicating parties alleviate this issue [13]. In this work, we use the term “quantum switch” instead of “repeater” to indicate that the former is equipped with entanglement switching logic.

A mathematical model for a quantum switch was originally introduced in Reference [31]. There, the authors study a switch that serves only BSMs, but they account for quantum state decoherence, link heterogeneity, and arbitrary buffer sizes (including infinite). In Reference [22], the authors study a multipartite entanglement distribution switch that serves n-partite GHZ states to users, for \(n\ge 3\). In this work, links are assumed to be identical and the effects of state decoherence negligible. In References [22, 31], the authors model the switch as a continuous-time Markov chain (CTMC). In Reference [32], the authors use a discrete-time Markov chain (DTMC) instead, but they focus on the simplest and most idealized scenario of bipartite entanglement switching with an infinite number of quantum memories, unit quantum state fidelities, and identical links. In this work, the authors discover that the DTMC is a logically more accurate model of a quantum switch; at the same time, the authors illustrate the modeling and analytical challenges that arise with using a DTMC, rendering more complex problem formulations (e.g., those that include heterogeneous links, finite buffers, and storage cut-offs) intractable.

In contrast to this prior work, we no longer assume that the switch serves only one type of entangled state, i.e., we allow n to be either two or three, and our goal is to design and evaluate a suitable switching policy. Another difference is that the quantum switch is assumed to have an infinite number of quantum memories in Reference [22], while we consider finite buffer sizes that scale with the number of links. We opt for continuous-time Markov chains as the modeling technique, because of the findings of Reference [32]. It is worth noting that results of the CTMC analyses of References [22, 31] have been partially validated using NetSquid, a discrete-event simulation framework for quantum networks [8]. In particular, the authors of Reference [8] showed that CTMCs yield accurate results for the capacity of a quantum switch when there is no quantum state decoherence (storage cut-off times were not simulated).

Decoherence-associated quantum storage cut-off times [7, 17, 18, 19, 27, 28] play an important role when it comes to the fidelity (“quality”) of quantum states. The qubit storage cut-off time has varying definitions in the literature: while some define it as the quantum memory lifetime (or coherence time), others view it as a configurable parameter that specifies the amount of time a qubit is to be held in memory; in either case, when the storage cut-off time expires, the qubit is deemed unusable and is discarded. We model this (deterministic) qubit discarding procedure by a probabilistic one, by assuming that the qubit is discarded after an exponentially distributed amount of time with mean \(1/\alpha\). This approach was previously explored in Reference [33], where the authors argued that such an approximation is reasonable for realistic use cases (i.e., in scenarios where the rate at which qubits are discarded is at least an order of magnitude smaller than the entanglement generation rate). In this work, we do not study the effects of decoherence and of BSM and GHZ¹ measurements on the quality of the resulting quantum states and focus only on the effects of storage cut-offs on the capacity region of the switch.

We consider a switch that connects k users over k separate, identical links. Creation of end-to-end entanglement requires two steps. First, two-qubit Bell states are generated pairwise between a qubit stored locally at the switch and a qubit owned by a user. This can be accomplished using a number of available methods, see, e.g., Reference [21] and references therein for an overview, as well as Appendix A for additional detail. Once such link-level two-qubit entangled states have been created, the switch performs joint (entangling) measurements (over \(j \ge 2\) locally held qubits that are entangled with qubits held by j distinct users), which, if successful, produces a j-qubit maximally entangled state between the corresponding j users. Link-level entanglement generation, as well as entangling measurements, when realized with practical systems, are inherently probabilistic [13]. We assume that only two-user (two-qubit) and three-user (three-qubit) entangled states are created, i.e., BSMs and 3-qubit GHZ basis measurements are done at the switch.² For simplicity, we assume that these \(j = 2\) or 3 qubit measurements at the switch take negligible time and always succeed. The reasoning for the former assumption is that entanglement generation with remote nodes is likely to be more time consuming than local quantum gates and measurements performed at the central switch node—see, e.g., References [8, 9] for detailed descriptions of timings; see also Reference [5] for GHZ state creation latency data obtained from a simulation. The purpose of the latter assumption is to reduce clutter during the analysis; it may be relaxed if we allow BSMs and 3-qubit GHZ basis measurements to succeed with probabilities \(q_1\) and \(q_2\),³ as the only consequence would be that the bipartite and tripartite capacities \(C_2\) and \(C_3\) would be scaled by their respective factors.

Each link attempts two-qubit entanglement in each time slot of length \(\tau\) seconds, and with probability p, establishes one entangled pair successfully. For simplicity, we model the time to successfully create a link entanglement as an exponential random variable with mean \(1/\mu = \tau /p\). Because all links are identical, they all have the same parameters p and \(\mu\). Once a link-level entanglement is successfully generated, the qubits are stored in memories, one at the user, the second at the switch. We assume that each link can store \(B=1\) or \(B=2\) qubits. We also assume that qubits at the switch can decohere and be discarded, and model decoherence (or quantum storage cut-off) time as an exponential r.v. with mean \(1/\alpha\). Last, when a qubit is stored at the switch, with its entangled pair stored at a user, we refer to this as a stored link entanglement.

We assume all possible bipartite and tripartite user entanglement is of interest and consider two classes of probabilistic policies, one for \(B=1\) and the second for \(B=2\), that provide the flexibility to generate both types of entanglement with arbitrary rates. Policies in both classes incorporate the oldest link entanglement first(OLEF) rule whereby when a link entanglement is created it is always matched up with stored link entanglement when possible rather than be stored. This has the nice consequence, when coupled with the assumption that links are homogeneous but statistically independent, that the system can be modeled by a continuous time Markov chain where the state simply tracks the number of stored EPR pairs for two users. The next section describes the class of policies for \(B=1\) and Section 5 for the class of \(B=2\) policies.

In this section, we assume that each link can store one qubit in the buffer, so that the per-link buffer size \(B = 1\). We model this system using a CTMC, and by obtaining its stationary distribution, we are able to compute the capacity region of the switch. We discover that it is always possible to configure a randomized policy that outperforms TDM, although as the number of links grows, the advantage of using such a policy diminishes.

In a system with per-link buffer size two, there are three additional states, as shown in Figure 4. Recall that in Section 4, we analyzed the Markov chain of buffer size one systems to find policies that are optimal, in the sense that the corresponding \((C_3,C_2)\) point within the capacity region is farthest from the TDM line. The goal of this part of the study—i.e., for \(B=2\) systems—is to show the existence of better policies than TDM, rather than to find such an optimal policy, as the former allows for a significantly simpler analysis. Hence, the design in Figure 4 does not encapsulate all possible switching policies: for instance, there is no \(r_1\) parameter here. Our exhaustive policy search over the entire parameter space for systems with \(B=1\) in Section 4 revealed that \(r_1\) is best set to zero; while it is possible that a positive \(r_1\) may be beneficial to a \(B=2\) system, we omit it from the model under the intuitive reasoning that the switch should favor states that allow for (more demanding in terms of the number of necessary EPR pairs) 3-GHZ measurements. In addition, note that if the system is in state \((1,1)\) and another entanglement is generated on one of the links that already has a stored qubit, the system is not allowed to use two of the qubits for a BSM. The reasoning is that since \(B=2\), there is enough space to keep the new qubit. Similarly, when the system is in state \((2,1)\) a BSM is only allowed if (i) another entanglement is generated on one of the \(k-2\) links that does not have a stored qubit, or (ii) another entanglement is generated on the link that already has two qubits stored. In the latter scenario, not performing a BSM would cause a qubit to be discarded. While this design does not grant the switch access to the full range of policies, it does enable us to find a class of policies that are more efficient than TDM.

A note on the notation in this and the following sections: with some abuse of notation, we use the variables \(C_2\) and \(C_3\) to represent the bipartite and tripartite capacities of a switch with per-link buffer size two.

In this section, we present a simple way to augment the model from Section 4 to account for the decoherence of quantum states and associated cut-off time on quantum storage. For switches with \(B=1\), we present both analytic and numerical results. We also augment the model from Section 5 to incorporate decoherence and storage cut-offs, but for switches with \(B=2\), we present only numerical results. Our decoherence model is described in Section 3. For \(B=1\), the resulting CTMC is illustrated in Figure 7.

The analysis of this model is almost identical to that of the \(B=1\) system without decoherence. As with the latter, the capacity region is bounded above by two lines:To avoid ambiguity, let \(C_2^{\prime }\) and \(C_3^{\prime }\) denote the bipartite and tripartite capacities of a system with decoherence. As with the previous model, \(C_2^{\prime }\) is maximized at \(r_1=1\), \(r_2=r_3=0\); \(C_3^{\prime }\) is maximized at \(r_1=r_3=0\), \(r_2=1\), and the point farthest from TDM is obtained by setting \(r_1=0\), \(r_2=r_3=1\). The first bounding line passes through the points \((0,C_2^{\prime }(1,0,0))\) and \((C_3^{\prime }(0,1,1),C_2^{\prime }(0,1,1))\); and the second line passes through \((C_3^{\prime }(0,1,1),C_2^{\prime }(0,1,1))\) and \((C_3^{\prime }(0,1,0),0)\). Moreover, all points on the bounding lines are achievable, indicating that the bound is tight.

The capacities are given byNote that the denominator is quadratic in \(\alpha\). This causes both \(C_2^{\prime }\) and \(C_3^{\prime }\) to tend to zero as \(\alpha \rightarrow \infty\).

Figure 8 shows a comparison of the capacity regions for systems with \(B=1\), for three and ten links and different decoherence rates. For all cases, \(\mu\) is set to one: for qualitative results, we only need to concern ourselves with the value of \(\alpha\) relative to \(\mu\). In real scenarios, we expect \(\alpha\) to be at least one order of magnitude less than \(\mu\). From numerical results, we observe that the effect of decoherence on the capacity region is not significant, especially as the number of links grows. Analysis supports this observation, since we can show that

Figure 9 shows a comparison for systems with \(B=2\) and varying number of links and decoherence rates. Results are consistent with that of the case \(B=1\): the effects of decoherence on capacity are less apparent for larger k values.

In this work, we explored a set of policies for a quantum switch that can store up to two qubits per link and whose objective is to perform bipartite and tripartite joint measurements to distribute two and three qubit entanglement to pairs and triples of users. We presented analytical results for the case where the per-link buffer has size one. We presented a class of policies that achieve a larger capacity region than time-division multiplexing, but found that as the number of links grows, the advantage of using such policies diminishes. We also compared the capacity regions for systems with different per-link buffer sizes and observed that systems with fewer links benefit more from the extra storage space than systems with a larger number of links. Finally, we modeled decoherence and associated storage cut-off times for both types of systems and presented analytical results for the case with per-link buffer size one. Observations and analysis showed that as the number of links increases, the effects of decoherence and storage cut-offs become less apparent on systems.

We have stated earlier that successful entanglement generation at the elementary link level⁴—for our switch architecture, this refers to entanglement between the switch and an end node—is an inherently probabilistic event, and that each attempt succeeds with a probability p. In this Appendix, we discuss this process in more detail.

A major challenge for distributed quantum systems is the difficulty of safely transmitting quantum states across large distances. For optical fiber, the channel transmissivity is given by \(\eta =e^{-\gamma L}\), where L is the link length and \(\gamma\) the fiber attenuation coefficient—i.e., fiber losses scale exponentially with distance. A link’s entanglement generation success probability p is proportional to its transmissivity \(\eta\), and further has a dependence on the parameters of the entanglement generation scheme—some of these may be tunable, while others are inherent properties of the hardware used. An example of the former is the bright-state population of the single-photon entanglement generation protocol [6], and examples of the latter are photon emission probability and detector efficiency.

The precise method of entanglement generation at the elementary link level depends on the hardware available, as well as the manner in which qubits are encoded into photonic states. We describe here two types of link-level architectures:

The “identical links” assumption of our quantum switch network, along with an implicit assumption that all end nodes have identical hardware, ensure that p is the same for all links. We have also assumed that the switch has enough communication qubits to support simultaneous communication with all end nodes, and that storage qubits have sufficiently long coherence times to store entangled states until they can be consumed in entangling measurements.