Quantum Internet Research Group W. Kozlowski Internet-Draft S. Wehner Intended status: Informational QuTech Expires: September 10, 2020 R. Van Meter Keio University B. Rijsman Individual A. S. Cacciapuoti M. Caleffi University of Naples Federico II March 9, 2020 Architectural Principles for a Quantum Internet draft-irtf-qirg-principles-03 Abstract The vision of a quantum internet is to fundamentally enhance Internet technology by enabling quantum communication between any two points on Earth. To achieve this goal, a quantum network stack should be built from the ground up as the physical nature of the communication is fundamentally different. The first realisations of quantum networks are imminent, but there is no practical proposal for how to organise, utilise, and manage such networks. In this memo, we attempt lay down the framework and introduce some basic architectural principles for a quantum internet. This is intended for general guidance and general interest, but also to provide a foundation for discussion between physicists and network specialists. Status of This Memo This Internet-Draft is submitted in full conformance with the provisions of BCP 78 and BCP 79. Internet-Drafts are working documents of the Internet Engineering Task Force (IETF). Note that other groups may also distribute working documents as Internet-Drafts. The list of current Internet- Drafts is at https://datatracker.ietf.org/drafts/current/. Internet-Drafts are draft documents valid for a maximum of six months and may be updated, replaced, or obsoleted by other documents at any time. It is inappropriate to use Internet-Drafts as reference material or to cite them other than as "work in progress." This Internet-Draft will expire on September 10, 2020. Kozlowski, et al. Expires September 10, 2020 [Page 1]

Internet-Draft Principles for a Quantum Internet March 2020 Copyright Notice Copyright (c) 2020 IETF Trust and the persons identified as the document authors. All rights reserved. This document is subject to BCP 78 and the IETF Trust's Legal Provisions Relating to IETF Documents (https://trustee.ietf.org/license-info) in effect on the date of publication of this document. Please review these documents carefully, as they describe your rights and restrictions with respect to this document. Code Components extracted from this document must include Simplified BSD License text as described in Section 4.e of the Trust Legal Provisions and are provided without warranty as described in the Simplified BSD License. Table of Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 3 2. Quantum information . . . . . . . . . . . . . . . . . . . . . 4 2.1. Qubit . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2. Multiple qubits . . . . . . . . . . . . . . . . . . . . . 5 3. Entanglement as the fundamental resource . . . . . . . . . . 6 4. Achieving quantum connectivity . . . . . . . . . . . . . . . 8 4.1. Challenges . . . . . . . . . . . . . . . . . . . . . . . 8 4.1.1. The measurement problem . . . . . . . . . . . . . . . 8 4.1.2. No-cloning theorem . . . . . . . . . . . . . . . . . 8 4.1.3. Fidelity . . . . . . . . . . . . . . . . . . . . . . 9 4.2. Direct transmission . . . . . . . . . . . . . . . . . . . 9 4.3. Bell pairs . . . . . . . . . . . . . . . . . . . . . . . 10 4.4. Teleportation . . . . . . . . . . . . . . . . . . . . . . 10 4.5. The life cycle of entanglement . . . . . . . . . . . . . 11 4.5.1. Elementary link generation . . . . . . . . . . . . . 11 4.5.2. Entanglement swapping . . . . . . . . . . . . . . . . 12 4.5.3. Distillation . . . . . . . . . . . . . . . . . . . . 13 4.5.4. Delivery . . . . . . . . . . . . . . . . . . . . . . 14 5. Architecture of a quantum internet . . . . . . . . . . . . . 14 5.1. New challenges . . . . . . . . . . . . . . . . . . . . . 14 5.2. Classical communication . . . . . . . . . . . . . . . . . 15 5.3. Abstract model of the network . . . . . . . . . . . . . . 16 5.3.1. Elements of a quantum network . . . . . . . . . . . . 16 5.3.2. Putting it all together . . . . . . . . . . . . . . . 17 5.4. Network boundaries . . . . . . . . . . . . . . . . . . . 18 5.4.1. Boundaries between different physical architectures . 18 5.4.2. Boundaries between different administrative regions . 18 5.5. Physical constraints . . . . . . . . . . . . . . . . . . 19 5.5.1. Memory lifetimes . . . . . . . . . . . . . . . . . . 19 5.5.2. Rates . . . . . . . . . . . . . . . . . . . . . . . . 19 5.5.3. Communication qubits . . . . . . . . . . . . . . . . 20 Kozlowski, et al. Expires September 10, 2020 [Page 2]

Internet-Draft Principles for a Quantum Internet March 2020 5.5.4. Homogeneity . . . . . . . . . . . . . . . . . . . . . 20 6. Architectural principles . . . . . . . . . . . . . . . . . . 20 6.1. Goals of a quantum internet . . . . . . . . . . . . . . . 20 6.2. The principles of a quantum internet . . . . . . . . . . 23 7. Comparison with classical networks . . . . . . . . . . . . . 25 8. Security Considerations . . . . . . . . . . . . . . . . . . . 27 9. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 27 10. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . 27 11. Informative References . . . . . . . . . . . . . . . . . . . 27 Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 29 1. Introduction Quantum networks are distributed systems of quantum devices that utilise fundamental quantum mechanical phenomena such as superposition, entanglement, and quantum measurement to achieve capabilities beyond what is possible with classical networks. Depending on the stage of a quantum network [5] such devices may be simple photonic devices capable of preparing and measuring only one quantum bit (qubit) at a time, all the way to large-scale quantum computers of the future. A quantum network is not meant to replace classical networks, but rather form an overall hybrid classical quantum network supporting new capabilities which are otherwise impossible to realise. This new networking paradigm offers promise for a range of new applications such as secure communications [1], distributed quantum computation [2], or quantum sensor networks [3]. The field of quantum communication has been a subject of active research for many years and the most well-known application of quantum communication, quantum key distribution (QKD) for secure communications, has already been deployed at short (roughly 100km) distances. Fully quantum networks capable of transmitting and managing entangled quantum states in order to send, receive, and manipulate distributed quantum information are now imminent [4] [5]. Whilst a lot of effort has gone into physically realising and connecting such devices, and making improvements to their speed and error tolerance there are no worked out proposals for how to run these networks. To draw an analogy with a classical network, we are at a stage where we can start to physically connect our devices and send data, but all sending, receiving, buffer management, connection synchronisation, and so on, must be managed by the application itself at what is even lower than assembly level where no common interfaces yet exist. Furthermore, whilst physical mechanisms for transmitting quantum states exist, there are no robust protocols for managing such transmissions. Kozlowski, et al. Expires September 10, 2020 [Page 3]

Internet-Draft Principles for a Quantum Internet March 2020 2. Quantum information In order to understand the framework for quantum networking a basic understanding of quantum information is necessary. The following sections aim to introduce the bare minimum necessary to understand the principles of operation of a quantum network. This exposition was written with a classical networking audience in mind. It is assumed that the reader has never before been exposed to any quantum physics. We refer to e.g. [10] for an in-depth introduction to quantum information. 2.1. Qubit The differences between quantum computation and classical computation begin at the bit-level. A classical computer operates on the binary alphabet { 0, 1 }. A quantum bit, a qubit, exists over the same binary space, but unlike the classical bit, it can exist in a so- called superposition of the two possibilities: a |0> + b |1>, where |X> denotes a quantum state, here the binary 0 and 1, and the coefficients a and b are complex numbers called probability amplitudes. Physically, such a state can be realised using a variety of different technologies such as electron spin, photon polarisation, atomic energy levels, and so on. Upon measurement, the qubit loses its superposition and irreversibly collapses into one of the two basis states, either |0> or |1>. Which of the two states it ends up in is not deterministic, but it can be determined from the readout of the measurement, a classical bit, 0 or 1 respectively. The probability of measuring the state in the |0> state is |a|^2 and similarly the probability of measuring the state in the |1> state is |b|^2, where |a|^2 + |b|^2 = 1. This randomness is not due to our ignorance of the underlying mechanisms, but rather it is a fundamental feature of a quantum mechanical system [6]. The superposition property plays an important role in fundamental gate operations on qubits. Since a qubit can exist in a superposition of its basis states, the elementary quantum gates are able to act on all states of the superposition at the same time. For example, consider the NOT gate: NOT (a |0> + b |1>) -> a |1> + b |0>. Kozlowski, et al. Expires September 10, 2020 [Page 4]

Internet-Draft Principles for a Quantum Internet March 2020 2.2. Multiple qubits When multiple qubits are combined in a single quantum state the space of possible states grows exponentially and all these states can coexist in a superposition. For example, the general form of a two- qubit register is a |00> + b |01> + c |10> + d |11> where the coefficients have the same probability amplitude interpretation as for the single qubit state. Each state represents a possible outcome of a measurement of the two-qubit register. For example, |01>, denotes a state in which the first qubit is in the state |0> and the second is in the state |1>. Performing single qubit gates affects the relevant qubit in each of the superposition states. Similarly, two-qubit gates also act on all the relevant superposition states, but their outcome is far more interesting. Consider a two-qubit register where the first qubit is in the superposed state (|0> + |1>)/sqrt(2) and the other is in the state |0>. This combined state can be written as: (|0> + |1>)/sqrt(2) x |0> = (|00> + |10>)/sqrt(2), where x denotes a tensor product (the mathematical mechanism for combining quantum states together). Let us now consider the two- qubit CNOT gate. The CNOT gate takes as input two qubits, a control and target, and applies the NOT gate to the target if the control qubit is set. The truth table looks like +----+-----+ | IN | OUT | +----+-----+ | 00 | 00 | | 01 | 01 | | 10 | 11 | | 11 | 10 | +----+-----+ Now, consider performing a CNOT gate on the ensemble with the first qubit being the control. We apply a two-qubit gate on all the superposition states: CNOT (|00> + |10>)/sqrt(2) -> (|00> + |11>)/sqrt(2). Kozlowski, et al. Expires September 10, 2020 [Page 5]

Internet-Draft Principles for a Quantum Internet March 2020 What is so interesting about this two-qubit gate operation? The final state is *entangled*. There is no possible way of representing that quantum state as a product of two individual qubits, they are no longer independent and their behaviour cannot be fully described without accounting for the other qubit. The states of the two individual qubits are now correlated beyond what is possible to achieve classically. Neither qubit is in a definite |0> or |1> state, but if we perform a measurement on either one, the outcome of the partner qubit will *always* yield the exact same outcome. The final state, whether it's |00> or |11>, is fundamentally random as before, but the states of the two qubits following a measurement will always be identical. Once a measurement is performed, the two qubits are once again independent. The final state is either |00> or |11> and both of these states can be trivially decomposed into a product of two individual qubits. The entanglement has been consumed and if the same measurement is to be repeated, the entangled state must be prepared again. 3. Entanglement as the fundamental resource Entanglement is the fundamental building block of quantum networks. To see this, consider the state from the previous section: (|00> + |11>)/sqrt(2). Neither of the two qubits is in a definite |0> or |1> state and we need to know the state of the entire register to be able to fully describe the behaviour of the two qubits. Entangled qubits have interesting non-local properties. Consider sending one of the qubits to another device. This device could in principle be anywhere: on the other side of the room, in a different country, or even on a different planet. Provided negligible noise has been introduced, the two qubits will forever remain in the entangled state until a measurement is performed. The physical distance does not matter at all for entanglement. This lies at the heart of quantum networking, because it is possible to leverage the non-classical correlations provided by entanglement in order to design completely new types of application protocols that are not possible to achieve with just classical communication. Examples of such applications are quantum cryptography, blind quantum computation, or distributed quantum computation. Kozlowski, et al. Expires September 10, 2020 [Page 6]

Internet-Draft Principles for a Quantum Internet March 2020 Entanglement has two very special features from which one can derive some intuition about the types of applications enabled by a quantum network. The first stems from the fact that entanglement enables stronger than classical correlations, leading to opportunities for tasks that require coordination. As a trivial example consider the problem of consensus between two nodes who want to agree on the value of a single bit. They can use the quantum network to prepare the state (|00> + |11>)/sqrt(2) with each node holding one of the two qubits. Once any of the two nodes performs a measurement the state of the two qubits collapses to either |00> or |11> so whilst the outcome is random and does not exist before measurement, the two nodes will always measure the same value. We can also build the more general multi-qubit state (|00...> + |11...>)/sqrt(2) and perform the same algorithm between an arbitrary number of nodes. These stronger than classical correlations generalise to more complicated measurement schemes as well. The second feature of entanglement is that it cannot be shared, in the sense that if two qubits are maximally entangled with each other, than it is physically impossible for any other system to have any share of this entanglement. Hence, entanglement forms a sort of private and inherently untappable connection between two nodes once established. Entanglement is created through local interactions between two qubits or as a product of the way the qubits were created (e.g. entangled photon pairs). To create a distributed entangled state one can then physically send one of the qubits to a remote node. It is also possible to directly entangle qubits that are physically separated, but this still requires local interactions between some other qubits that the separated qubits are initially entangled with. Therefore, it is the transmission of qubits that draws the line between a genuine quantum network and a collection of quantum computers connected over a classical network. A quantum network is defined as a collection of nodes that is able to exchange qubits and distribute entangled states amongst themselves. A quantum node that is able only to communicate classically with another quantum node is not a member of a quantum network. More complex services and applications can be built on top of entangled states distributed by the network, see e.g. [5]> Kozlowski, et al. Expires September 10, 2020 [Page 7]

Internet-Draft Principles for a Quantum Internet March 2020 4. Achieving quantum connectivity This section explains the meaning of quantum connectivity and the necessary physical processes at an abstract level. 4.1. Challenges A quantum network cannot be built by simply extrapolating all the classical models to their quantum analogues. One cannot just send qubits like one can send bits over a wire. There are several technological as well as fundamental challenges that make classical approaches unsuitable in a quantum context. 4.1.1. The measurement problem In classical computers and networks we can read out the bits stored in memory at any time. This is helpful for a variety of purposes such as copying, error detection and correction, and so on. This is not possible with qubits. A measurement of a qubit's state will destroy its superposition and with it any entanglement it may have been part of. Once a qubit is being processed, it cannot be read out until a suitable point in the computation, determined by the protocol handling the qubit, has been reached. Therefore, we cannot use the same methods known from classical computing for the purposes of error detection and correction. 4.1.2. No-cloning theorem Since directly reading the state of a qubit is not possible, one could ask the question if we can simply copy a qubit without looking at it. Unfortunately, this is fundamentally not possible in quantum mechanics. The no-cloning theorem states that it is impossible to create an identical copy of an arbitrary unknown quantum state. Therefore, it is also impossible to use the same mechanisms that worked for classical networks for signal amplification, retransmission, and so on as they all rely on the ability to copy the underlying data. Since any physical channel will always be lossy, connecting nodes within a quantum network is a challenging endeavour and its architecture must at its core address this very issue. Kozlowski, et al. Expires September 10, 2020 [Page 8]

Internet-Draft Principles for a Quantum Internet March 2020 4.1.3. Fidelity In general, it is expected that a classical packet arrives at its destination without any errors introduced by hardware noise along the way. This is verified at various levels through a variety of checksums. Since we cannot read or copy a quantum state a similar approach is out of question for quantum networks. To describe the quality of a quantum state a physical quantity called fidelity is used. Fidelity takes a value between 0 and 1 -- higher is better, and less than 0.5 means the state is unusable. It measures how close a quantum state is to the state we desire it to be in. It expresses the probability that one state will pass a test to identify as the other. Fidelity is an important property of a quantum system that allows us to quantify how much a particular state has been affected by noise from various sources (gate errors, channel losses, environment noise). Interestingly, quantum applications do not need perfect fidelity to be able to execute -- as long as it is above some application- specific threshold, they will simply operate at lower rates. Therefore, rather than trying to ensure that we always deliver perfect states (a technologically challenging task) applications will specify a minimum threshold for the fidelity and the network will try its best to deliver it. 4.2. Direct transmission Conceptually, the most straightforward way to distribute an entangled state is to simply transmit one of the qubits directly to the other end across a series of nodes while performing sufficient forward quantum error correction to bring losses down to an acceptable level. Despite the no-cloning theorem and the inability to directly measure a quantum state, error-correcting mechanisms for quantum communication exist [7]. However, quantum error correction makes very high demands on both resources (physical qubits needed) and their initial fidelity. Implementation is very challenging and quantum error correction is not expected to be used until later generations of quantum networks. An alternative relies on the observation that we do not need to be able to distribute any arbitrary entangled quantum state. We only need to be able to distribute any one of what are known as the Bell pair states[12]. Kozlowski, et al. Expires September 10, 2020 [Page 9]

Internet-Draft Principles for a Quantum Internet March 2020 4.3. Bell pairs Bell pair states are the entangled two-qubit states: |00> + |11>, |00> - |11>, |01> + |10>, |01> - |10>, where the constant 1/sqrt(2) normalisation factor has been ignored for clarity. Any of the four Bell pair states above will do, as it is possible to transform any Bell pair into another Bell pair with local operations performed on only one of the qubits. That is, either of the nodes that hold the two qubits of the Bell pair can apply a series of single qubit gates to just their qubit in order to transform the ensemble between the different variants. Distributing a Bell pair between two nodes is much easier than transmitting an arbitrary quantum state over a network. Since the state is known handling errors becomes easier and small-scale error- correction (such as entanglement distillation discussed in a later section) combined with reattempts becomes a valid strategy. The reason for using Bell pairs specifically as opposed to any other two-qubit state, is that they are the maximally entangled two-qubit set of basis states. Maximal entanglement means that these states have the strongest non-classical correlations of all possible two- qubit states. Furthermore, since single-qubit local operations can never increase entanglement, less entangled states would impose some constraints on distributed quantum algorithms. This makes Bell pairs particularly useful as a generic building block for distributed quantum applications. 4.4. Teleportation The observation that we only need to be able to distribute Bell pairs relies on the fact that this enables the distribution of any other arbitrary entangled state. This can be achieved via quantum state teleportation. Quantum state teleportation consumes an unknown quantum state that we want to transmit and recreates it at the desired destination. This does not violate the no-cloning theorem as the original state is destroyed in the process. To achieve this, an entangled pair needs to be distributed between the source and destination before teleportation commences. The source then entangles the transmission qubit with its end of the pair and performs a read out on the two qubits (the sum of these operations is called a Bell state measurement). This consumes the Kozlowski, et al. Expires September 10, 2020 [Page 10]

Internet-Draft Principles for a Quantum Internet March 2020 Bell pair's entanglement turning the source and destination qubits into independent states. The measurements yields two classical bits which the source sends to the destination over a classical channel. Based on the value of the received two classical bits, the destination performs one of four possible corrections (called the Pauli corrections) on its end of the pair which turns it into the unknown quantum state that we wanted to transmit. The unknown quantum state that was transmitted was never fed into the network itself. Therefore, the network needs to only be able to reliably produce Bell pairs between any two nodes in the network. 4.5. The life cycle of entanglement Reducing the problem of quantum connectivity to one of generating a Bell pair has facilitated the problem, but it has not solved it. In this section we discuss, how these entangled pairs are generated in the first place, and how its two qubits are delivered to the end- points. 4.5.1. Elementary link generation In a quantum network, entanglement is always first generated locally (at a node or an auxiliary element) followed by a movement of one or both of the entangled qubits across the link through quantum channels. In this context, photons (particles of light) are the natural candidate for entanglement carriers, the so-called flying qubits. The rationale for this choice is related to the advantages provided by photons such as moderate interaction with the environment leading to moderate decoherence, convenient control with standard optical components, and high-speed low-loss transmissions. However, since photons cannot be stored, a transducer must transfer the flying qubit's state to a qubit suitable for information processing and/or storage (often referred to as a matter qubit). Since this process may fail, in order to generate and store entanglement efficiently, we must be able to distinguish successful attempts from failures. Entanglement generation schemes that are able to announce successful generation are called heralded entanglement generation schemes. There exist three basic schemes for heralded entanglement generation on a link through coordinated action of the two nodes at the two ends of the link [13]: o "At mid-point" scheme: the key idea is that an entangled pair source sends an entangled photon through a quantum channel to each of the nodes, where transducers are invoked to transfer the Kozlowski, et al. Expires September 10, 2020 [Page 11]

Internet-Draft Principles for a Quantum Internet March 2020 entanglement from the flying qubits to the matter qubits. In this scheme, the transducers know if the transfers succeeded and are able to herald successful entanglement generation via a message exchange over the classical channel. o "At source" scheme: the key idea is that one of the two nodes sends a flying qubit that is entangled with one of its matter qubits. A transducer at the other end of the link will transfer the entanglement from the flying qubit to one of its matter qubits. Also in this scheme, the transducer knows if its transfer succeeded and is able to herald successful entanglement generation with a classical message sent to the other node. o "At both end-points" scheme: the key idea is that both nodes send a flying qubit that is entangled with one of their matter qubits. A detector somewhere in between the nodes performs a joint measurement on the two qubits, which stochastically projects the remote matter qubits into an entangled quantum state. In this scheme, the detector knows if the entanglement succeeded and is able to herald successful entanglement generation by sending a message to each node over the classical channel. The "mid-point" scheme is more robust to photon loss, but in the other schemes the nodes retain greater control over the entangled pair generation. 4.5.2. Entanglement swapping The problem with generating entangled pairs directly across a link is that its efficiency decreases with its length. Beyond a few 10s of kms in optical fibre or 1000 kms in free space (via satellite) the rate is effectively zero and due to the no-cloning theorem we cannot simply amplify the signal. The solution is entanglement swapping. A Bell pair between any two nodes in the network can be constructed by combining the pairs generated along each individual link on the path between the two end-points. Each node along the path can consume the two pairs on the two links that it is connected to in order to produce a new entangled Pair between the two remote ends. This process is known as entanglement swapping. Pictorially it can be represented as follows: +---------+ +---------+ +---------+ | A | | B | | C | | |------| |------| | | X1~~~~~~~~~~X2 Y1~~~~~~~~~~Y2 | +---------+ +---------+ +---------+ Kozlowski, et al. Expires September 10, 2020 [Page 12]

Internet-Draft Principles for a Quantum Internet March 2020 where X1 and X2 are the qubits of the entangled pair X and Y1 and Y2 are the qubits of entangled pair Y. The entanglement is denoted with ~~. In the diagram above nodes A and B share the pair X and nodes B and C share the pair Y, but we want entanglement between A and C. To achieve this goal we simply teleport the qubit X2 using the pair Y. This requires node B to performs a Bell state measurement on the qubits X2 and Y1 which result in the destruction of the entanglement between Y1 and Y2. However, X2 is transmitted and recreated in Y2's place carrying with it its entanglement with X1. The end-result is shown below: +---------+ +---------+ +---------+ | A | | B | | C | | |------| |------| | | X1~~~~~~~~~~~~~~~~~~~~~~~~~~~X2 | +---------+ +---------+ +---------+ Depending on the needs of the network and/or application a final Pauli correction at the recipient node may not be necessary since the result of this operation is also a Bell pair. However, the two classical bits that form the read out from the measurement at node B must still be communicated, because they carry information about which of the four Bell pairs was actually produced. If a correction is not performed, the recipient must be informed which Bell pair was received. This process of teleporting Bell pairs using other entangled pairs is called entanglement swapping. Quantum nodes that create long- distance entangled pairs via entanglement swapping are called quantum repeaters in academic literature [12] and we will use the same terminology in this memo. 4.5.3. Distillation Neither the generation of Bell pairs nor the swapping operations are noiseless operations. Therefore, with each link and each swap the fidelity of the state degrades. However, it is possible to create higher fidelity Bell pair states from two or more lower fidelity pairs through a process called distillation (sometimes also referred to as purification). To distil a quantum state, a second (and sometimes third) quantum state is used as a "test tool" to test a proposition about the first state, e.g., "the parity of the first state is even." When the test succeeds, confidence in the state is improved, and thus the fidelity is improved. The test tool states are destroyed in the process, so resource demands increase substantially when distillation is used. Kozlowski, et al. Expires September 10, 2020 [Page 13]

Internet-Draft Principles for a Quantum Internet March 2020 When the test fails, the tested state must also be discarded. Distillation makes low demands on fidelity and resources, but distributed protocols incur round-trip delays [11]. 4.5.4. Delivery Eventually the Bell pairs must be delivered to an application (or higher layer protocol) at the two end-nodes. A detailed list of such requirements is beyond the scope of this memo. At minimum, the end- nodes require information to map a particular Bell pair to the qubit in their local memory that is part of this entangled pair. 5. Architecture of a quantum internet It is evident from the previous sections that the fundamental service provided by a quantum network significantly differs from that of a classical network. Therefore, it is not surprising that the architecture of a quantum internet will itself be very different from that of the classical Internet. 5.1. New challenges This subsection covers the major fundamental challenges building quantum networks. Here, we only describe the fundamental differences, technological limitations are described later. 1. Bell pairs are not equivalent to payload carrying packets. In most classical networks, including Ethernet, Internet Protocol (IP), and Multi-Protocol Label Switching (MPLS) networks, user data is grouped into packets. In addition to the user data each packet also contains a series of headers which contain the control information that lets routers and switches forward it towards its destination. Packets are the fundamental unit in a classical network. In a quantum network the entangled pairs of qubits are the basic unit of networking. These pairs are handled individually -- they are not grouped into packets and they do not carry any headers. Therefore, quantum networks will have to send all control information via separate classical channels which the repeaters will have to correlate with the qubits stored in their memory. 2. An entangled pair is only useful if the locations of both qubits are known. A classical network packet logically exists only at one location at any point in time. If a packet is modified in some way, Kozlowski, et al. Expires September 10, 2020 [Page 14]

Internet-Draft Principles for a Quantum Internet March 2020 headers or payload, this information does not need to be conveyed to anybody else in the network. The packet can be simply forwarded as before. In contrast, entanglement is a phenomenon in which two or more qubits exist in a physically distributed state. Operations on one of the qubits change the mutual state of the pair. Since the owner of a particular qubit cannot just read out its state, it must coordinate all its actions with the owner of the pair's other qubit. Therefore, the owner of any qubit that is part of an entangled pair must know the location of its counterpart. Location, in this context, need not be the explicit spatial location. A relevant pair identifier, a means of communication between the pair owners, and an association between the pair ID and the individual qubits is sufficient. 3. Generating entanglement requires temporary state. Packet forwarding in a classical network is largely a stateless operation. When a packet is received, the router looks up its forwarding table and sends the packet out of the appropriate output. There is no need to keep any memory of the packet any more. A quantum node must be able to make decisions about qubits that it receives and is holding in its memory. Since qubits do not carry headers, the receipt of an entangled pair conveys no control information based on which the repeater can make a decision. The relevant control information will arrive separately over a classical channel. This implies that a repeater must store temporary state as the control information and the qubit it pertains to will, in general, not arrive at the same time. 5.2. Classical communication In this memo we have already covered two different roles that classical communication must perform: o communicate classical bits of information as part of distributed protocols such as entanglement swapping and teleportation, o communicate control information within a network - this includes both background protocols such as routing as well as signalling protocols to set up end-to-end entanglement generation. Classical communication is a crucial building block of any quantum network. All nodes in a quantum network are assumed to have Kozlowski, et al. Expires September 10, 2020 [Page 15]

Internet-Draft Principles for a Quantum Internet March 2020 classical connectivity with each other (within typical administrative domain limts). Therefore, quantum routers will need to manage two data planes in parallel, a classical one and a quantum one. Additionally, it must be able to correlate information between them so that the control information received on a classical channel can be applied to the qubits managed by the quantum data plane. 5.3. Abstract model of the network 5.3.1. Elements of a quantum network We have identified quantum repeaters as the core building block of a quantum network. However, a quantum repeater will have to do more than just entanglement swapping in a functional quantum network. Its key responsibilities will include: 1. Creating link-local entanglement between neighbouring nodes. 2. Extending entanglement from link-local pairs to long-range pairs through entanglement swapping. 3. Performing distillation to manage the fidelity of the produced pairs 4. Participate in the management of the network (routing etc.). Not all quantum repeaters in the network will be the same, here we break them down further: o Quantum routers (controllable quantum nodes) - A quantum router is a quantum repeater with a control plane that participates in the management of the network and will make decisions about which qubits to swap to generate the requested end-to-end pairs. o Automated quantum nodes - An automated quantum node is a data plane only quantum repeater that does not participate in network management. Since the no-cloning theorem precludes the use of amplification long-range links will be established by chaining multiple such automated nodes together. o End-nodes - End-nodes in a quantum network must be able to receive and handle an entangled pair, but they do not need to be able to perform an entanglement swap (and thus are not necessarily quantum repeaters). End-nodes are also not required to have any quantum memory as certain quantum applications can be realised by having the end-node measure its qubit as soon as it is received. Kozlowski, et al. Expires September 10, 2020 [Page 16]

Internet-Draft Principles for a Quantum Internet March 2020 o Non-quantum nodes - Not all nodes in a quantum network need to have a quantum data plane. A non-quantum node is any device that can handle classical network traffic. Additionally, we need to identify two kinds of links that will be used in a quantum network: o Quantum links - A quantum link is a link which can be used to generate an entangled pair between two directly connected quantum repeaters. It may include a dedicated classical channel that is to be used solely for the purpose of coordinating the entanglement generation on this quantum link. o Classical links - A classical link is a link between any node in the network that is capable of carrying classical network traffic. 5.3.2. Putting it all together A two-hop path in a generic quantum network can be represented as: | App |-------------------CC-------------------| App | || || ------ ------ ------ | EN |----QC & CC----| QR |----QC & CC----| EN | ------ ------ ------ App - user-level application QR - quantum repeater EN - end-node QC - quantum channel CC - classical channel An application running on two end-nodes attached to a network will at some point need the network to generate entangled pairs for its use. This will require negotiation between the end-nodes, because they must both open a communication end-point (a quantum socket) which the network can use to identify the two ends of the connection. The two end-nodes use the classical connectivity available in the network to achieve this goal. When the network receives a request to generate end-to-end entangled pairs it uses the classical communication channels to coordinate and claim the resources necessary to fulfil this request. This may be some combination of prior control information (e.g. routing tables) and signalling protocols, but the details of how this is achieved are an active research question and thus beyond the scope of this memo. Kozlowski, et al. Expires September 10, 2020 [Page 17]

Internet-Draft Principles for a Quantum Internet March 2020 During or after the control information is distributed the network performs the necessary quantum operations such as generating entangled over individual links, performing entanglement swaps, and further signalling to transmit the swap outcomes and other control information. Since none of the entangled pairs carry any user data, some of these operations can be performed before the request is received in anticipation of the demand. The entangled pair is delivered to the application once it is ready, together with the relevant pair identifier. However, being ready does not necessarily mean once all link pairs and entanglement swaps are complete as some applications can start executing on an incomplete pair. In this case the remaining entanglement swaps will propagate the actions across the network to the other end. 5.4. Network boundaries Just like classical network, there will various boundaries will exist in quantum networks. 5.4.1. Boundaries between different physical architectures There are many different physical architectures for implementing quantum repeater technology. The different technologies differ in how they store and manipulate qubits in memory and how they generate entanglement across a link with their neighbours. Different architectures come with different trade-offs and thus a functional network will likely consist of a mixture of different types of quantum repeaters. For example, architectures based on optical elements and atomic ensembles are very efficient at generating entanglement, but provide little control over the qubits once the pair is generated. On the other hand nitrogen-vacancy architectures offer a much greater degree of control over qubits, but have a harder time generating the entanglement across a link. It is an open research question where exactly the boundary will lie. It could be that a single quantum repeater node provides some backplane connection between the architectures, but it also could be that special quantum links delineate the boundary. 5.4.2. Boundaries between different administrative regions Just like in classical networks, multiple quantum networks will connect into a global quantum internet. This necessarily implies the existence of borders between different administrative regions. How Kozlowski, et al. Expires September 10, 2020 [Page 18]

Internet-Draft Principles for a Quantum Internet March 2020 these boundaries will be handled is also an open question and thus beyond the scope of this memo. 5.5. Physical constraints The model above has effectively abstracted away the particulars of the hardware implementation. However, certain physical constraints need to be considered in order to build a practical network. Some of these are fundamental constraints and no matter how much the technology improves, they will always need to be addressed. Others are artefacts of the early stages of a new technology. Here, we consider a highly abstract scenario and refer to [5] for pointers to the physics literature. 5.5.1. Memory lifetimes In addition to discrete operations being imperfect, storing a qubit in memory is also highly non-trivial. The main difficulty in achieving persistent storage is that it is extremely challenging to isolate a quantum system from the environment. The environment introduces an uncontrollable source of noise into the system which affects the fidelity of the state. This process is known as decoherence. Eventually, the state has to be discarded once its fidelity degrades too much. The memory lifetime depends on the particular physical setup, but the highest achievable values currently are on the order of seconds. These values have increased tremendously over the lifetime of the different technologies and are bound to keep increasing. However, if quantum networks are to be realised in the near future, they need to be able to handle short memory lifetimes, for example by reducing latency on critical paths. 5.5.2. Rates Entanglement generation on a link between two connected nodes is not a very efficient process and it requires many attempts to succeed. A fast repetition rate for Bell pair generation is achievable, but only a small fraction will succeed. Currently, the highest achievable rates of success between nodes capable of storing the resulting qubits are of the order of 10 Hz. Combined with short memory lifetimes this leads to very tight timing windows to build up network-wide connectivity. Kozlowski, et al. Expires September 10, 2020 [Page 19]

Internet-Draft Principles for a Quantum Internet March 2020 5.5.3. Communication qubits Most physical architectures capable of storing qubits are only able to generate entanglement using only a subset of its available qubits called communication qubits. Once a Bell pair has been generated using a communication qubit, its state can be transferred into memory. This may impose additional limitations on the network. In particular if a given node has only one communication qubit it cannot simultaneously generate Bell Pairs over two links. It must generate entanglement over the links one at a time. 5.5.4. Homogeneity Currently all hardware implementations are homogeneous and they do not interface with each other. In general, it is very challenging to combine different quantum information processing technologies at present. Coupling different technologies with each other is of great interest as it may help overcome the weaknesses of the different implementations, but this may take a long time to be realised with high reliability and thus is not a near-term goal. 6. Architectural principles Given that the most practical way of realising quantum network connectivity is using Bell Pair and entanglement swapping repeater technology what sort of principles should guide us in assembling such networks such that they are functional, robust, efficient, and most importantly: they work. Furthermore, how do we design networks so that they work under the constraints imposed by the hardware available today, but do not impose unnecessary burden on future technology. As this is a completely new technology that is likely to see many iterations over its lifetime, this memo must not serve as a definitive set of rules, but merely as a general set of recommended guidelines for the first generations of quantum networks based on principles and observations made by the community. The benefit of having a community built document at this early stage is that expertise in both quantum information and network architecture is needed in order to successfully build a quantum internet. 6.1. Goals of a quantum internet When outlining any set of principles we must ask ourselves what goals do we want to achieve as inevitably trade-offs must be made. So what sort of goals should drive a quantum network architecture? The following list has been inspired by the history of the classical Internet, but it will inevitably evolve with time and the needs of Kozlowski, et al. Expires September 10, 2020 [Page 20]

Internet-Draft Principles for a Quantum Internet March 2020 its users. The goals are listed in order of priority which in itself may also evolve as the community learns more about the technology. 1. Support distributed quantum applications This goal seems trivially obvious, but makes a subtle, but important point which highlights a key difference between quantum and classical networks. Ultimately, quantum data transmission is not the goal of a quantum network - it is only one possible component of more advanced quantum application protocols. Whilst transmission certainly could be used as a building block for all quantum applications, it is certainly not the most basic one possible. For example, QKD, the most well known quantum application protocol only relies on the stronger than classical correlations and inherent secrecy of entangled Bell pairs and does not transmit arbitrary quantum states. The primary purpose of a quantum internet is to support distributed quantum application protocols and it is of utmost importance that they can run well and efficiently. Thus, it is important to develop performance metrics meaningful to application to drive the development of quantum network protocols. For example, the Bell pair generation rate is meaningless if one does not also consider their fidelity. It is generally, much easier to generate pairs of lower fidelity, but quantum applications may have to make multiple re-attempts or even abort if the fidelity is too low. A review of the requirements for different known quantum applications can be found in [5]. 2. Support tomorrow's distributed quantum applications Early-stage quantum networks will be very limited in terms of their capabilities and will only be able to run a limited set of applications. As quantum repeater technology becomes more advanced and acquire more sophisticated capabilities, new applications will become possible. The different stages of this development are described in [5]. Therefore, it is important that any proposed architecture for general purpose quantum repeater networks should not constrain their potential capabilities for the benefit of being able to run early-stage applications more efficiently. For example, it is already possible to run QKD efficiently on metropolitan scales and such networks are already commercially available. However, they are not based on quantum repeaters and thus will not be able to easily transition more sophisticated applications. Kozlowski, et al. Expires September 10, 2020 [Page 21]

Internet-Draft Principles for a Quantum Internet March 2020 3. Support hardware heterogeneity There are multiple proposals for realising practical quantum repeater hardware and they all have their advantages and disadvantages. Some may offer higher Bell pair generation rates on individual links at the cost of more difficult entanglement swap operations. Other platforms may be good all around, but are more difficult to build. Whilst conceptually they are all capable of the same tasks the optimal network configuration will likely leverage the advantages of multiple platforms to optimise the provided service. Therefore, it is an explicit goal to incorporate varied hardware support from the beginning. 4. Be flexible with regards to hardware capabilities and limitations This goal encompasses two important points. First, the architecture should be able to function under the physical constraints imposed by the current generation hardware. Second, it should not make it difficult to run the network over any hardware that may come along in the future. The physical capabilities of repeaters will improve and redeploying a technology is extremely challenging. 5. Ensure security at the network level Whilst the priority for the first quantum networks should be to simply work, we cannot forget that ultimately they have to also be secure. This is particularly important for quantum networks given that one of the key driving factors for the technology is the enhance security offered by quantum entanglement. This has implications for the physical realisations (do they satisfy the idealised theoretical models) and also the design of the control stack. It is actually difficult to guarantee security at the network level and even if the network did provide such guarantees, the application would still need to perform its own verification similarly to how one ensures end-to-end security in classical networks. It turns out that as long as the underlying implementation corresponds to (or sufficiently approximates) theoretical models of quantum cryptography, quantum cryptographic protocols do not need the network to provide any guarantees about the authenticity, confidentiality, or integrity of the transmitted qubits or the generated entanglement. Instead, applications, Kozlowski, et al. Expires September 10, 2020 [Page 22]

Internet-Draft Principles for a Quantum Internet March 2020 such as QKD, establish such guarantees using the classical network in conjunction with he quantum one. This is much easier than demanding that the network deliver secure entanglement in the first place. Nevertheless, control protocols themselves should be security aware in order to protect the operation of the network itself and limit disruption. This will primarily involve securing the classical control and management traffic by means of authentication and possibly encryption. 6. Make them easy to manage and monitor The fundamental unit of quantum information, the qubit, cannot be actively monitored as any readout irreversibly destroys its contents. Furthermore, given one end of an entangled pair, it is impossible to tell where the other qubit is without any additional information. Therefore, monitoring quantum networks will be more challenging and more important if any meaningful network management solution is to be developed. 7. Ensure availability and resilience A practical and usable network is able to continue to operate despite losses and failures, and will be robust to malicious actors trying to disable connectivity. These may be simply considered different aspects of security, but it is worthwhile to address them explicitly at the architectural level already. The coexistence of two separate channels, a quantum and a classical one, will likely prove to be challenging. 6.2. The principles of a quantum internet The principles support the goals, but are not goals themselves. The goals define what we want to build and the principles provide a guideline in how we might achieve this. The goals will also be the foundation for defining any metric of success for a network architecture, whereas the principles in themselves do not distinguish between success and failure. For more information about design considerations for quantum networks see [8] [9] . 1. Bell Pairs are the fundamental building block The key service that a quantum network provides is the distribution of entanglement between the nodes in a network. This point additionally specifies that the entanglement is primarily distributed in the form of the entangled Bell Pair Kozlowski, et al. Expires September 10, 2020 [Page 23]

Internet-Draft Principles for a Quantum Internet March 2020 states which should be used as a building block in providing other services, including more complex entangled states. 2. Bell Pairs are indistinguishable Any two Bell Pairs between the same two nodes are indistinguishable for the purposes of an application provided they both satisfy its required fidelity threshold. This point is crucial in enabling the reuse of resources of a network and for the purposes of provisioning resources to meet application demand. However, the qubits that make up the pair themselves are not indistinguishable and the two nodes operating on a pair must coordinate to make sure they are operating on qubits that belong to the same Bell Pair. 3. Fidelity is part of the service In addition to being able to deliver Bell Pairs to the communication end-points, the Bell Pairs must be of sufficient fidelity. Unlike in classical networks where errors should essentially be eliminated for most application protocols, many quantum applications only need imperfect entanglement to function. However, different applications will have different requirements for what fidelity they can work with. It is the network's responsibility to balance the resource usage with respect to the application's requirements. It may be that it is cheaper for the network to provide lower fidelity pairs that are just above the threshold required by the application than it is to guarantee high fidelity pairs to all applications regardless of their requirements. 4. Time is part of the service With the current technology, time is the most expensive resource. It is not the only resource that is in short supply (memory, and communication qubits are as well), but ultimately it is the lifetime of quantum memories that imposes the most difficult conditions for operating an extended network of quantum nodes. Current hardware has low rates of Bell Pair generation, short memory lifetimes, and access to a limited number of communication qubits. All these factors combined mean that even a short waiting queue at some node could be enough for the Bell Pairs to decohere. Kozlowski, et al. Expires September 10, 2020 [Page 24]

Internet-Draft Principles for a Quantum Internet March 2020 7. Comparison with classical networks Creating end-to-end Bell pairs between remote end-points is a stateful distributed task that requires a lot of a-priori coordination. Therefore, a connection-oriented approach seems the most natural for quantum networks. In this section, we discuss a plausible quantum network architecture inspired by MPLS. This is not an architecture proposal, but a thought experiment to give the reader an idea of what components are necessary for a functional quantum network. We use classical MPLS as a basis as it is well known and understood in the networking community. In connection-oriented quantum networks, when two quantum application end-points wish to start creating end-to-end Bell pairs, they must first create a quantum virtual circuit (QVC). As an analogy, in MPLS networks end-points must establish a label switched path (LSP) before exchanging traffic. Connection-oriented quantum networks may also support virtual circuits with multiple end-points for creating multipartite entanglement. As an analogy, MPLS networks have the concept of multi-point LSPs for multicast. When a quantum application creates a quantum virtual circuit, it can indicate quality of service (QoS) parameters such as the required capacity in end-to-end Bell pairs per second (BPPS) and the required fidelity of the Bell pairs. As an analogy, in MPLS networks applications specify the required bandwidth in bits per second (BPS) and other constraints when they create a new LSP. Quantum networks need a routing function to compute the optimal path (i.e. the best sequence of routers and links) for each new quantum virtual circuit. The routing function may be centralized or distributed. In the latter case, the quantum network needs a distributed routing protocol. As an analogy, classical networks use routing protocols such as open shortest path first (OSPF) and intermediate-system to intermediate system (ISIS). Given the very scarce availability of resources in early quantum networks, a traffic engineering function is likely to be beneficial. Without traffic engineering, quantum virtual circuits always use the shortest path. In this case, the quantum network cannot guarantee that each quantum end-point will get its Bell pairs at the required rate or fidelity. This is analogous to "best effort" service in classical networks. With traffic engineering, quantum virtual circuits choose a path that is guaranteed to have the requested resources (e.g. bandwidth in BPPS) available, taking into account the capacity of the routers and links and taking into account the resources already consumed by other Kozlowski, et al. Expires September 10, 2020 [Page 25]

Internet-Draft Principles for a Quantum Internet March 2020 virtual circuits. As an analogy, both OSPF and ISIS have traffic engineering (TE) extensions to keep track of used and available resources, and can use constrained shortest path first (CSPF) to take resource availability and other constraints into account when computing the optimal path. The use of traffic engineering implies the use of call admission control (CAC): the network denies any virtual circuits for which it cannot guarantee the requested quality of service a-priori. Or alternatively, the network pre-empts lower priority circuits to make room for the new one. Quantum networks need a signaling function: once the path for a quantum virtual circuit has been computed, signaling is used to install the "forwarding rules" into the data plane of each quantum router on the path. The signaling may be distributed, analogous to the resource reservation protocol (RSVP) in MPLS. Or the signaling may be centralized, similar to OpenFlow. Quantum networks need an abstraction of the hardware for specifying the forwarding rules. This allows us to de-couple the control plane (routing and signaling) from the data plane (actual creation of Bell pairs). The forwarding rules are specified using abstract building blocks such as "creating local Bell pairs", "swapping Bell pairs", "distillation of Bell pairs". As an analogy, classical networks use abstractions that as based on match conditions (e.g. looking up header fields in tables) and actions (e.g. modifying fields or forwarding a packet to a specific interface). The data-plane abstractions in quantum networks will be very different from those in classical networks due to the fundamental differences in technology and the stateful nature of quantum networks. In fact, choosing the right abstractions will be one of the biggest challenges when designing interoperable quantum network protocols. In quantum networks, control plane traffic (routing and signaling messages) is exchanged over a classical channel, whereas data plane traffic (the actual Bell pair qubits) is exchanged over a separate quantum channel. This is in contrast to most classical networks, where control plane traffic and data plane traffic share the same channel and where a single packet contains both user fields and header fields. There is, however, a classical analogy to the way quantum networks work. Generalized MPLS (GMPLS) networks use separate channels for control plane traffic and data plane traffic. Furthermore, GMPLS networks support data planes where there is no such thing as data plane headers (e.g. DWDM or TDM networks). Kozlowski, et al. Expires September 10, 2020 [Page 26]

Internet-Draft Principles for a Quantum Internet March 2020 8. Security Considerations Even though no user data enters a quantum network security is listed as an explicit goal for the architecture and this issue is addressed in the section on goals. Even though user data doesn't enter the network, it is still possible to attack the control protocols and violate the authenticity, confidentiality, and integrity of communication. However, as this is an informational memo it does not propose any concrete mechanisms to achieve these goals. In summary: As long as the underlying implementation corresponds to (or sufficiently approximates) theoretical models of quantum cryptography, quantum cryptographic protocols do not need the network to provide any guarantees about the authenticity, confidentiality, or integrity of the transmitted qubits or the generated entanglement. Instead, applications such as QKD establish such guarantees using the classical network in conjunction with he quantum one. This is much easier than demanding that the network deliver secure entanglement. 9. IANA Considerations This memo includes no request to IANA. 10. Acknowledgements The authors of this memo acknowledge funding received from the EU Flagship on Quantum Technologies through Quantum Internet Alliance project. The authors would further like to acknowledge Carlo Delle Donne, Matthew Skrzypczyk, and Axel Dahlberg for useful discussions on this topic prior to the submission of this memo. 11. Informative References [1] Bennett, C. and G. Brassard, "Quantum cryptography: Public key distribution and coin tossing", Theoretical Computer Science 560, 7-11, 2014, <http://www.sciepub.com/reference/53249>. [2] Crepeau, C., Gottesman, D., and A. Smith, "Secure multi- party quantum computation. Proceedings of Symposium on Theory of Computing", Proceedings of Symposium on Theory of Computing , 2002, <https://arxiv.org/abs/quant-ph/0206138>. Kozlowski, et al. Expires September 10, 2020 [Page 27]

Internet-Draft Principles for a Quantum Internet March 2020 [3] Giovanetti, V., Lloyd, S., and L. Maccone, "Quantum- enhanced measurements: beating the standard quantum limit", Science 306(5700), 1330-1336, 2004, <https://arxiv.org/abs/quant-ph/0412078>. [4] Castelvecchi, D., "The Quantum Internet has arrived (and it hasn't)", Nature 554, 289-292, 2018, <https://www.nature.com/articles/d41586-018-01835-3>. [5] Wehner, S., Elkouss, D., and R. Hanson, "Quantum internet: A vision for the road ahead", Science 362, 6412, 2018, <http://science.sciencemag.org/content/362/6412/ eaam9288.full>. [6] Aspect, A., Grangier, P., and G. Roger, "Experimental Tests of Realistic Local Theories via Bell's Theorem", Phys. Rev. Lett. 47 (7): 460-463, 1981, <https://journals.aps.org/prl/abstract/10.1103/ PhysRevLett.47.460>. [7] Muralidharan, S., Kim, J., Lutkenhaus, N., Lukin, M., and L. Jiang, "Ultrafast and Fault-Tolerant Quantum Communication across Long Distances", Phys. Rev. Lett. 112 (25-27), 250501, 2014, <https://arxiv.org/abs/1310.5291>. [8] Van Meter, R. and J. Touch, "Designing quantum repeater networks", IEEE Communications Magazine 51, 64-71, 2013, <https://ieeexplore.ieee.org/document/6576340>. [9] Dahlberg, A., Skrzypczyk, M., Coopmans, T., Wubben, L., Rozpedek, F., Pompili, M., Stolk, A., Pawelczak, P., Knegjens, R., de Oliveira Filho, J., Hanson, R., and S. Wehner, "A Link Layer Protocol for Quantum Networks", arXiv 1903.09778, 2019, <https://arxiv.org/abs/1903.09778>. [10] Nielsen, M. and I. Chuang, "Quantum Computation and Quantum Information", Cambridge University Press , 2011. [11] Bennett, C., DiVincenzo, D., Smolin, J., and W. Wootters, "Mixed State Entanglement and Quantum Error Correction", Phys. Rev. A Vol. 54, Iss. 5, 1996, <https://arxiv.org/abs/quant-ph/9604024>. [12] Briegel, H., Dur, W., Cirac, J., and P. Zoller, "Quantum Repeaters: The Role of Imperfect Local Operations in Quantum Communication", Phys. Rev. Lett. Vol. 81, Num. 26, 1998, <https://arxiv.org/abs/quant-ph/9803056>. Kozlowski, et al. Expires September 10, 2020 [Page 28]

Internet-Draft Principles for a Quantum Internet March 2020 [13] Cacciapuoti, A., Caleffi, M., Van Meter, R., and L. Hanzo, "When Entanglement meets Classical Communications: Quantum Teleportation for the Quantum Internet", , 2019, <https://arxiv.org/abs/1907.06197>. Authors' Addresses Wojciech Kozlowski QuTech Building 22 Lorentzweg 1 Delft 2628 CJ Netherlands Email: w.kozlowski@tudelft.nl Stephanie Wehner QuTech Building 22 Lorentzweg 1 Delft 2628 CJ Netherlands Email: s.d.c.wehner@tudelft.nl Rodney Van Meter Keio University 5322 Endo Fujisawa, Kanagawa 252-0882 Japan Email: rdv@sfc.wide.ad.jp Bruno Rijsman Individual Email: brunorijsman@gmail.com Kozlowski, et al. Expires September 10, 2020 [Page 29]

Internet-Draft Principles for a Quantum Internet March 2020 Angela Sara Cacciapuoti University of Naples Federico II Department of Electrical Engineering and Information Technologies Claudio 21 Naples 80125 Italy Email: angelasara.cacciapuoti@unina.it Marcello Caleffi University of Naples Federico II Department of Electrical Engineering and Information Technologies Claudio 21 Naples 80125 Italy Email: marcello.caleffi@unina.it Kozlowski, et al. Expires September 10, 2020 [Page 30]