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Checking references for intended status: Informational ---------------------------------------------------------------------------- No issues found here. Summary: 0 errors (**), 0 flaws (~~), 1 warning (==), 1 comment (--). Run idnits with the --verbose option for more detailed information about the items above. -------------------------------------------------------------------------------- 2 Quantum Internet Research Group W. Kozlowski 3 Internet-Draft S. Wehner 4 Intended status: Informational QuTech 5 Expires: May 6, 2020 R. Van Meter 6 Keio Univeristy 7 B. Rijsman 8 Individual 9 November 3, 2019 11 Architectural Principles for a Quantum Internet 12 draft-irtf-qirg-principles-02 14 Abstract 16 The vision of a quantum internet is to fundamentally enhance Internet 17 technology by enabling quantum communication between any two points 18 on Earth. To achieve this goal, a quantum network stack should be 19 built from the ground up as the physical nature of the communication 20 is fundamentally different. The first realisations of quantum 21 networks are imminent, but there is no practical proposal for how to 22 organise, utilise, and manage such networks. In this memo, we 23 attempt lay down the framework and introduce some basic architectural 24 principles for a quantum internet. This is intended for general 25 guidance and general interest, but also to provide a foundation for 26 discussion between physicists and network specialists. 28 Status of This Memo 30 This Internet-Draft is submitted in full conformance with the 31 provisions of BCP 78 and BCP 79. 33 Internet-Drafts are working documents of the Internet Engineering 34 Task Force (IETF). Note that other groups may also distribute 35 working documents as Internet-Drafts. The list of current Internet- 36 Drafts is at https://datatracker.ietf.org/drafts/current/. 38 Internet-Drafts are draft documents valid for a maximum of six months 39 and may be updated, replaced, or obsoleted by other documents at any 40 time. It is inappropriate to use Internet-Drafts as reference 41 material or to cite them other than as "work in progress." 43 This Internet-Draft will expire on May 6, 2020. 45 Copyright Notice 47 Copyright (c) 2019 IETF Trust and the persons identified as the 48 document authors. All rights reserved. 50 This document is subject to BCP 78 and the IETF Trust's Legal 51 Provisions Relating to IETF Documents 52 (https://trustee.ietf.org/license-info) in effect on the date of 53 publication of this document. Please review these documents 54 carefully, as they describe your rights and restrictions with respect 55 to this document. Code Components extracted from this document must 56 include Simplified BSD License text as described in Section 4.e of 57 the Trust Legal Provisions and are provided without warranty as 58 described in the Simplified BSD License. 60 Table of Contents 62 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 3 63 2. Model of communication . . . . . . . . . . . . . . . . . . . 3 64 2.1. Qubit . . . . . . . . . . . . . . . . . . . . . . . . . . 4 65 2.2. Multiple qubits . . . . . . . . . . . . . . . . . . . . . 4 66 3. Entanglement as the fundamental service . . . . . . . . . . . 6 67 4. Achieving quantum connectivity . . . . . . . . . . . . . . . 7 68 4.1. Challenges . . . . . . . . . . . . . . . . . . . . . . . 7 69 4.1.1. The measurement problem . . . . . . . . . . . . . . . 8 70 4.1.2. No-cloning theorem . . . . . . . . . . . . . . . . . 8 71 4.1.3. Fidelity . . . . . . . . . . . . . . . . . . . . . . 8 72 4.2. Bell pairs . . . . . . . . . . . . . . . . . . . . . . . 9 73 4.3. Teleportation . . . . . . . . . . . . . . . . . . . . . . 10 74 4.4. The life cycle of entanglement . . . . . . . . . . . . . 10 75 4.4.1. Link generation . . . . . . . . . . . . . . . . . . . 10 76 4.4.2. Entanglement swapping . . . . . . . . . . . . . . . . 11 77 4.4.3. Direct transmission vs. entanglement swapping . . . . 13 78 5. Architecture of a quantum internet . . . . . . . . . . . . . 13 79 5.1. New challenges . . . . . . . . . . . . . . . . . . . . . 13 80 5.2. Classical communication . . . . . . . . . . . . . . . . . 15 81 5.3. Abstract model of the network . . . . . . . . . . . . . . 15 82 5.3.1. Elements of a quantum network . . . . . . . . . . . . 15 83 5.3.2. Putting it all together . . . . . . . . . . . . . . . 16 84 5.4. Network boundaries . . . . . . . . . . . . . . . . . . . 17 85 5.4.1. Boundaries between different physical architectures . 17 86 5.4.2. Boundaries between different administrative regions . 18 87 5.5. Physical constraints . . . . . . . . . . . . . . . . . . 18 88 5.5.1. Memory lifetimes . . . . . . . . . . . . . . . . . . 18 89 5.5.2. Rates . . . . . . . . . . . . . . . . . . . . . . . . 18 90 5.5.3. Communication qubit . . . . . . . . . . . . . . . . . 19 91 5.5.4. Homogeneity . . . . . . . . . . . . . . . . . . . . . 19 92 5.6. Architectural principles . . . . . . . . . . . . . . . . 19 93 5.6.1. Goals of a quantum internet . . . . . . . . . . . . . 20 94 5.6.2. The principles of a quantum internet . . . . . . . . 22 95 6. Security Considerations . . . . . . . . . . . . . . . . . . . 24 96 7. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 25 97 8. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . 25 98 9. Informative References . . . . . . . . . . . . . . . . . . . 25 99 Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 26 101 1. Introduction 103 Quantum networks are distributed systems of quantum devices that 104 utilise fundamental quantum mechanical phenomena such as 105 superposition, entanglement, and quantum measurement to achieve 106 capabilities beyond what is possible with classical networks. 107 Depending on the stage of a quantum network [5] such devices may be 108 simple photonic devices capable of preparing and measuring only one 109 quantum bit (qubit) at a time, all the way to large-scale quantum 110 computers of the future. A quantum network is not meant to replace 111 classical networks, but rather form an overall hybrid classical 112 quantum network supporting new capabilities which are otherwise 113 impossible to realise. This new networking paradigm offers promise 114 for a range of new applications such as secure communications [1], 115 distributed quantum computation [2], or quantum sensor networks [3]. 116 The field of quantum communication has been a subject of active 117 research for many years and the most well-known application of 118 quantum communication, quantum key distribution (QKD) for secure 119 communications, has already been deployed at short (roughly 100km) 120 distances. 122 Fully quantum networks capable of transmitting and managing entangled 123 quantum states in order to send, receive, and manipulate distributed 124 quantum information are now imminent [4] [5]. Whilst a lot of effort 125 has gone into physically realising and connecting such devices, and 126 making improvements to their speed and error tolerance there are no 127 worked out proposals for how to run these networks. To draw an 128 analogy with a classical network, we are at a stage where we can 129 start to physically connect our devices and send data, but all 130 sending, receiving, buffer management, connection synchronisation, 131 and so on, must be managed by the application itself at what is even 132 lower than assembly level where no common interfaces yet exist. 133 Furthermore, whilst physical mechanisms for forwarding quantum states 134 exist, there are no robust protocols for managing such transmissions. 136 2. Model of communication 138 In order to understand the framework for quantum networking a basic 139 understanding of quantum information is necessary. The following 140 sections aim to introduce the bare minimum necessary to understand 141 the principles of operation of a quantum network. This exposition 142 was written with a classical networking audience in mind. It is 143 assumed that the reader has never before been exposed to any quantum 144 physics. We refer to e.g. [10] for an in-depth introduction to 145 quantum information. 147 2.1. Qubit 149 The differences between quantum computation and classical computation 150 begin at the bit-level. A classical computer operates on the binary 151 alphabet { 0, 1 }. A quantum bit, a qubit, exists over the same 152 binary space, but unlike the classical bit, it can exist in a so- 153 called superposition of the two possibilities: 155 a |0> + b |1>, 157 where |X> denotes a quantum state, here the binary 0 and 1, and the 158 coefficients a and b are complex numbers called probability 159 amplitudes. Physically, such a state can be realised using a variety 160 of different technologies such as electron spin, photon polarisation, 161 atomic energy levels, and so on. 163 Upon measurement, the qubit loses its superposition and irreversibly 164 collapses into one of the two basis states, either |0> or |1>. Which 165 of the two states it ends up in is not deterministic, but it can be 166 determined from the readout of the measurement, a classical bit, 0 or 167 1 respectively. The probability of measuring the state in the |0> 168 state is |a|^2 and similarly the probability of measuring the state 169 in the |1> state is |b|^2, where |a|^2 + |b|^2 = 1. This randomness 170 is not due to our ignorance of the underlying mechanisms, but rather 171 it is a fundamental feature of a quantum mechanical system [6]. 173 The superposition property plays an important role in fundamental 174 gate operations on qubits. Since a qubit can exist in a 175 superposition of its basis states, the elementary quantum gates are 176 able to act on all states of the superposition at the same time. For 177 example, consider the NOT gate: 179 NOT (a |0> + b |1>) -> a |1> + b |0>. 181 2.2. Multiple qubits 183 When multiple qubits are combined in a single quantum state the space 184 of possible states grows exponentially and all these states can 185 coexist in a superposition. For example, the general form of a two- 186 qubit register is 188 a |00> + b |01> + c |10> + d |11> 189 where the coefficients have the same probability amplitude 190 interpretation as for the single qubit state. Each state represents 191 a possible outcome of a measurement of the two-qubit register. For 192 example, |01>, denotes a state in which the first qubit is in the 193 state |0> and the second is in the state |1>. 195 Performing single qubit gates affects the relevant qubit in each of 196 the superposition states. Similarly, two-qubit gates also act on all 197 the relevant superposition states, but their outcome is far more 198 interesting. 200 Consider a two-qubit register where the first qubit is in the 201 superposed state (|0> + |1>)/sqrt(2) and the other is in the 202 state |0>. This combined state can be written as: 204 (|0> + |1>)/sqrt(2) x |0> = (|00> + |10>)/sqrt(2), 206 where x denotes a tensor product (the mathematical mechanism for 207 combining quantum states together). Let us now consider the two- 208 qubit CNOT gate. The CNOT gate takes as input two qubits, a control 209 and target, and applies the NOT gate to the target if the control 210 qubit is set. The truth table looks like 212 +----+-----+ 213 | IN | OUT | 214 +----+-----+ 215 | 00 | 00 | 216 | 01 | 01 | 217 | 10 | 11 | 218 | 11 | 10 | 219 +----+-----+ 221 Now, consider performing a CNOT gate on the ensemble with the first 222 qubit being the control. We apply a two-qubit gate on all the 223 superposition states: 225 CNOT (|00> + |10>)/sqrt(2) -> (|00> + |11>)/sqrt(2). 227 What is so interesting about this two-qubit gate operation? The 228 final state is *entangled*. There is no possible way of representing 229 that quantum state as a product of two individual qubits, they are no 230 longer independent and their behaviour cannot be fully described 231 without accounting for the other qubit. The states of the two 232 individual qubits are now correlated beyond what is possible to 233 achieve classically. Neither qubit is in a definite |0> or |1> 234 state, but if we perform a measurement on either one, the outcome of 235 the partner qubit will *always* yield the exact same outcome. The 236 final state, whether it's |00> or |11>, is fundamentally random as 237 before, but the states of the two qubits following a measurement will 238 always be identical. 240 Once a measurement is performed, the two qubits are once again 241 independent. The final state is either |00> or |11> and both of 242 these states can be trivially decomposed into a product of two 243 individual qubits. The entanglement has been consumed and if the 244 same measurement is to be repeated, the entangled state must be 245 prepared again. 247 3. Entanglement as the fundamental service 249 Entanglement is the fundamental building block of quantum networks. 250 To see this, consider the state from the previous section: 252 (|00> + |11>)/sqrt(2). 254 Neither of the two qubits is in a definite |0> or |1> state and we 255 need to know the state of the entire register to be able to fully 256 describe the behaviour of the two qubits. 258 Entangled qubits have interesting non-local properties. Consider 259 sending one of the qubits to another device. This device could in 260 principle be anywhere: on the other side of the room, in a different 261 country, or even on a different planet. Provided negligible noise 262 has been introduced, the two qubits will forever remain in the 263 entangled state until a measurement is performed. The physical 264 distance does not matter at all for entanglement. 266 This lies at the heart of quantum networking, because it is possible 267 to leverage the non-classical correlations provided by entanglement 268 in order to design completely new types of application protocols that 269 are not possible to achieve with just classical communication. 270 Examples of such applications are quantum cryptography, blind quantum 271 computation, or distributed quantum computation. 273 Entanglement has two very special features from which one can derive 274 some intuition about the types of applications enabled by a quantum 275 network. 277 The first stems from the fact that entanglement enables stronger than 278 classical correlations, leading to opportunities for tasks that 279 require coordination. As a trivial example consider the problem of 280 consensus between two nodes who want to agree on the value of a 281 single bit. They can use the quantum network to prepare the state 282 (|00> + |11>)/sqrt(2) with each node holding one of the two qubits. 283 Once any of the two nodes performs a measurement the state of the two 284 qubits collapses to either |00> or |11> so whilst the outcome is 285 random and does not exist before measurement, the two nodes will 286 always measure the same value. We can also build the more general 287 multi-qubit state (|00...> + |11...>)/sqrt(2) and perform the same 288 algorithm between an arbitrary number of nodes. These stronger than 289 classical correlations generalise to more complicated measurement 290 schemes as well. 292 The second feature of entanglement is that it cannot be shared, in 293 the sense that if two qubits are maximally entangled with each other, 294 than it is physically impossible for any other system to have any 295 share of this entanglement. Hence, entanglement forms a sort of 296 private and inherently untappable connection between two nodes once 297 established. 299 It is impossible to entangle two qubits without ever having them 300 directly interact with each other (e.g. by performing a local two- 301 qubit gate, such as the CNOT). A local - or mediated - interaction 302 is necessary to create entanglement and thus such states cannot be 303 created between two quantum nodes that cannot transmit quantum states 304 to each other. Therefore, it is the transmission of qubits that 305 draws the line between a genuine quantum network and a collection of 306 quantum computers connected over a classical network. 308 A quantum network is defined as a collection of nodes that is able to 309 exchange qubits and distribute entangled states amongst themselves. 310 A quantum node that is able only to communicate classically with 311 another quantum node is not a member of a quantum network. 313 More complex services and applications can be built on top of 314 entangled states distributed by the network, see e.g. [5]> 316 4. Achieving quantum connectivity 318 This section explains the meaning of quantum connectivity and the 319 necessary physical processes at an abstract level. 321 4.1. Challenges 323 A quantum network cannot be built by simply extrapolating all the 324 classical models to their quantum analogues. One cannot just send 325 qubits like one can send bits over a wire. There are several 326 technological as well as fundamental challenges that make classical 327 approaches unsuitable in a quantum context. 329 4.1.1. The measurement problem 331 In classical computers and networks we can read out the bits stored 332 in memory at any time. This is helpful for a variety of purposes 333 such as copying, error detection and correction, and so on. This is 334 not possible with qubits. 336 A measurement of a qubit's state will destroy its superposition and 337 with it any entanglement it may have been part of. Once a qubit is 338 being processed, it cannot be read out until a suitable point in the 339 computation, determined by the protocol handling the qubit, has been 340 reached. Therefore, we cannot use the same methods known from 341 classical computing for the purposes of error detection and 342 correction. 344 4.1.2. No-cloning theorem 346 Since directly reading the state of a qubit is not possible, one 347 could ask the question if we can simply copy a qubit without looking 348 at it. Unfortunately, this is fundamentally not possible in quantum 349 mechanics. 351 The no-cloning theorem states that it is impossible to create an 352 identical copy of an arbitrary unknown quantum state. Therefore, it 353 is also impossible to use the same mechanisms that worked for 354 classical networks for signal amplification, retransmission, and so 355 on as they all rely on the ability to copy the underlying data. 356 Since any physical channel will always be lossy, connecting nodes 357 within a quantum network is a challenging endeavour and its 358 architecture must at its core address this very issue. 360 4.1.3. Fidelity 362 In general, it is expected that a classical packet arrives at its 363 destination without any errors introduced by hardware noise along the 364 way. This is verified at various levels through a variety of 365 checksums. Since we cannot read or copy a quantum state a similar 366 approach is out of question for quantum networks. 368 To describe the quality of a quantum state a physical quantity called 369 fidelity is used. Fidelity takes a value between 0 and 1 -- higher 370 is better, and less than 0.5 means the state is unusable. It 371 measures how close a quantum state is to the state we desire it to be 372 in. It expresses the probability that one state will pass a test to 373 identify as the other. Fidelity is an important property of a 374 quantum system that allows us to quantify how much a particular state 375 has been affected by noise from various sources (gate errors, channel 376 losses, environment noise). 378 Interestingly, quantum applications do not need perfect fidelity to 379 be able to execute -- as long as it is above some application- 380 specific threshold, they will simply operate at lower rates. 381 Therefore, rather than trying to ensure that we always deliver 382 perfect states (a technologically challenging task) applications will 383 specify a minimum threshold for the fidelity and the network will try 384 its best to deliver it. 386 4.2. Bell pairs 388 Conceptually, the most straightforward way to distribute an entangled 389 state is to simply transmit one of the qubits directly to the other 390 end across a series of nodes while performing sufficient forward 391 quantum error correction to bring losses down to an acceptable level. 392 Despite the no-cloning theorem and the inability to directly measure 393 a quantum state error-correcting mechanisms for quantum communication 394 exist [7]. However, quantum error correction makes very high demands 395 on both resources (physical qubits needed) and their initial 396 fidelity. Implementation is very challenging and quantum error 397 correction is not expected to be used until later generations of 398 quantum networks. 400 An alternative relies on the observation that we do not need to be 401 able to distribute any arbitrary entangled quantum state. We only 402 need to be able to distribute any one of what are known as the Bell 403 pair states. Bell pair states are the entangled two-qubit states: 405 |00> + |11>, 406 |00> - |11>, 407 |01> + |10>, 408 |01> - |10>, 410 where the constant 1/sqrt(2) normalisation factor has been ignored 411 for clarity. Any of the four Bell pair state above will do as it is 412 possible to transform any Bell pair into another Bell pair with local 413 operations performed on only one of the qubits. That is, either of 414 the nodes that hold the two qubits of the Bell pair can apply a 415 series of single qubit gates to just their qubit in order to 416 transform the ensemble between the different variants. 418 Distributing a Bell pair between two nodes is much easier than 419 transmitting an arbitrary quantum state over a network. Since the 420 state is known handling errors becomes easier and small-scale error- 421 correction (such as entanglement distillation discussed in a later 422 section) combined with reattempts becomes a valid strategy. 424 The reason for using Bell pairs specifically as opposed to any other 425 two-qubit state, is that they are the maximally entangled two-qubit 426 set of basis states. Maximal entanglement means that these states 427 have the strongest non-classical correlations of all possible two- 428 qubit states. Furthermore, since single-qubit local operations can 429 never increase entanglement, less entangled states would impose some 430 constraints on distributed quantum algorithms. This makes Bell pairs 431 particularly useful as a generic building block for distributed 432 quantum applications. 434 4.3. Teleportation 436 The observation that we only need to be able to distribute Bell pairs 437 relies on the fact that this enables the distribution of any other 438 arbitrary entangled state. This can be achieved via quantum state 439 teleportation. Quantum state teleportation consumes an unknown 440 quantum state that we want to transmit and recreates it at the 441 desired destination. This does not violate the no-cloning theorem as 442 the original state is destroyed in the process. 444 To achieve this, an entangled pair needs to be distributed between 445 the source and destination before teleportation commences. The 446 source then entangles the transmission qubit with its end of the pair 447 and performs a read out on the two qubits (the sum of these 448 operations is called a Bell state measurement). This consumes the 449 Bell pair's entanglement turning the source and destination qubits 450 into independent states. The measurements yields two classical bits 451 which the source sends to the destination over a classical channel. 452 Based on the value of the received two classical bits, the 453 destination performs one of four possible corrections (called the 454 Pauli corrections) on its end of the pair which turns it into the 455 unknown quantum state that we wanted to transmit. 457 The unknown quantum state that was transmitted never entered the 458 network itself. Therefore, the network needs to only be able to 459 reliably produce Bell pairs between any two nodes in the network. 461 4.4. The life cycle of entanglement 463 Reducing the problem of quantum connectivity to one of generating a 464 Bell pair has facilitated the problem, but it has not solved it. In 465 this section we discuss, how these entangled pairs are generated in 466 the first place, and how its two qubits are delivered to the end- 467 points. 469 4.4.1. Link generation 471 [waiting for contrib] 473 4.4.2. Entanglement swapping 475 The problem with generating entangled pairs directly across a link is 476 that its efficiency decreases with its length. Beyond a few 10s of 477 kms the rate is effectively zero and due to the no-cloning theorem we 478 cannot simply amplify the signal. The solution is entanglement 479 swapping. 481 A Bell pair between any two nodes in the network can be constructed 482 by combining the pairs generated along each individual link on the 483 path between the two end-points. Each node along the path can 484 consume the two pairs on the two links that it is connected to in 485 order to produce a new entangled Pair between the two remote ends. 486 This process is known as entanglement swapping. Pictorially it can 487 be represented as follows: 489 +---------+ +---------+ +---------+ 490 | A | | B | | C | 491 | |------| |------| | 492 | X1~~~~~~~~~~X2 Y1~~~~~~~~~~Y2 | 493 +---------+ +---------+ +---------+ 495 where X1 and X2 are the qubits of the entangled pair X and Y1 and Y2 496 are the qubits of entangled pair Y. The entanglement is denoted with 497 ~~. In the diagram above nodes A and B share the pair X and nodes B 498 and C share the pair Y, but we want entanglement between A and C. 500 To achieve this goal we simply teleport the qubit X2 using the pair 501 Y. This requires node B to performs a Bell state measurement on the 502 qubits X2 and Y1 which result in the destruction of the entanglement 503 between Y1 and Y2. However, X2 is transmitted and recreated in Y2's 504 place carrying with it its entanglement with X1. The end-result is 505 shown below: 507 +---------+ +---------+ +---------+ 508 | A | | B | | C | 509 | |------| |------| | 510 | X1~~~~~~~~~~~~~~~~~~~~~~~~~~~X2 | 511 +---------+ +---------+ +---------+ 513 Depending on the needs of the network and/or application a final 514 Pauli correction at the recipient node may not be necessary since the 515 result of this operation is also a Bell pair. However, the two 516 classical bits that form the read out from the measurement at node B 517 must still be communicated, because they carry information about 518 which of the four Bell pairs was actually produced. If a correction 519 is not performed, the recipient must be informed which Bell pair was 520 received. 522 This process of teleporting Bell pairs using other entangled pairs is 523 called entanglement swapping. 525 4.4.2.1. Distillation 527 Neither the generation of Bell pairs nor the swapping operations are 528 noiseless operations. Therefore, with each link and each swap the 529 fidelity of the state degrades. However, it is possible to create 530 higher fidelity Bell pair states from two or more lower fidelity 531 pairs through a process called distillation or purification. 533 To purify a quantum state, a second (and sometimes third) quantum 534 state is used as a "test tool" to test a proposition about the first 535 state, e.g., "the parity of the first state is even." When the test 536 succeeds, confidence in the state is improved, and thus the fidelity 537 is improved. The test tool states are destroyed in the process, so 538 resource demands increase substantially when distillation is used. 539 When the test fails, the tested state must also be discarded. 540 Purification makes low demands on fidelity and resources, but 541 distributed protocols incur round-trip delays [11]. 543 4.4.2.2. Delivery 545 The bare minimum requirements of an application for every Bell pair 546 delivered to the two end-nodes are: 548 1. Information about which of the four Bell pairs was delivered. 549 The network may choose to not perform Pauli corrections at all 550 and simply notify the application of which state the delivered 551 pair is in or it may perform the Pauli corrections and always 552 deliver the same state. 554 2. An identifier that allows the applicatqion to unambiguously 555 determine which qubits at the two end-points belong to which 556 entangled pair. 558 3. An estimate of the fidelity of the delivered pair. This should 559 be above the minimum threshold determined by the application. 560 However, this will only be an estimate and not a guarantee. This 561 has security implications for applications which will be 562 discussed in the section on security. 564 There are several other features an application might want to be able 565 to request (e.g. multiple pairs delivered together close in time, but 566 doesn't matter when they are delivered), but they are beyond the 567 scope of this memo. 569 4.4.3. Direct transmission vs. entanglement swapping 571 Direct state transmission whilst simpler conceptually is much more 572 demanding to implement reliably in practice which means that any 573 near-term practical realisation is more likely to succeed if it is 574 based on the Bell pair and entanglement swapping architecture. All 575 near-term experimental implementations of quantum repeaters are based 576 on this approach. Therefore, this is the architecture that we will 577 focus on in the rest of this memo. 579 Nevertheless, the direct transmission proposal may be relevant in the 580 future as it has better fault-tolerance properties and much better 581 scaling with transmission distance. It might even be beneficial to 582 utilise a hybrid approach that combines the fault-tolerance of direct 583 transmission with the generic nature of Bell pairs which lends itself 584 to paralellisation and resource provisioning. That is, we still use 585 Bell pairs for transmission of user data, but direct transmission may 586 be used for some of hops for the purposes of Bell pair generation 587 rather than just relying solely on entanglement swapping. 589 5. Architecture of a quantum internet 591 It is evident from the previous sections that the fundamental service 592 provided by a quantum network significantly differs from that of a 593 classical network. Therefore, it is not surprising that the 594 architecture of a quantum internet will itself be very different from 595 that of the classical Internet. 597 5.1. New challenges 599 This subsection covers the major fundamental challenges building 600 quantum networks. Here, we only describe the fundamental 601 differences, technological limitations are described later. 603 1. There is no quantum equivalent of a payload carrying packet. 605 In most classical networks, including Ethernet, Internet Protocol 606 (IP), and Multi-Protocol Label Switching (MPLS) networks, user 607 data is grouped into packets. In addition to the user data each 608 packet also contains a series of headers which contain the 609 control information that lets routers and switches forward it 610 towards its destination. Packets are the fundamental unit in a 611 classical network. 613 In a quantum network the entangled pairs of qubits are the basic 614 unit of networking. These pairs are handled individually -- they 615 are not grouped into packets and they do not carry any headers. 616 Therefore, quantum networks will have to send all control 617 information via separate classical channels which the repeaters 618 will have to correlate with the qubits stored in their memory. 620 2. An entangled pair is only useful if the locations of both qubits 621 are known. 623 A classical network packet logically exists only at one location 624 at any point in time. If a packet is modified in some way, 625 headers or payload, this information does not need to be conveyed 626 to anybody else in the network. The packet can be simply 627 forwarded as before. 629 In contrast, entanglement is a phenomenon in which two or more 630 qubits exist in a physically distributed state. Operations on 631 one of the qubits change the mutual state of the pair. Since the 632 owner of a particular qubit cannot just read out its state, it 633 must coordinate all its actions with the owner of the pair's 634 other qubit. Therefore, the owner of any qubit that is part of 635 an entangled pair must know the location of its counterpart. 636 Location, in this context, need not be the explicit spatial 637 location. A relevant pair identifier, a means of communication 638 between the pair owners, and an association between the pair ID 639 and the individual qubits is sufficient. 641 3. Generating entanglement requires temporary state. 643 Packet forwarding in a classical network is largely a stateless 644 operation. When a packet is received, the router looks up its 645 forwarding table and sends the packet out of the appropriate 646 output. There is no need to keep any memory of the packet any 647 more. 649 A quantum repeater must be able to make decisions about qubits 650 that it receives and is holding in its memory. Since qubits do 651 not carry headers, the receipt of an entangled pair conveys no 652 control information based on which the repeater can make a 653 decision. The relevant control information will arrive 654 separately over a classical channel. This implies that a 655 repeater must store temporary state as the control information 656 and the qubit it pertains to will, in general, not arrive at the 657 same time. 659 4. Generating end-to-end entanglement is a parallelisable operation. 661 Classical packets carry user data from source destination by 662 performing a series of hops across the network. This process is 663 necessarily sequential -- it is impossible to forward a packet 664 ahead of time as the user data it carries cannot be known in 665 advance. A quantum network does not carry any user data. It is 666 only responsible for generating entangled pairs in any of the 667 generic Bell states. The process of creating an end-to-end Bell 668 pair is by its nature parallelisable -- all of the individual 669 link pairs can be generated independently of one another. 670 Furthermore, there is no ordering requirement on the entanglement 671 swapping operations either, they can happen in any order as long 672 as the network can keep track of which pairs were swapped so that 673 it can correctly identify the two ends of the final Bell pair. 674 This parallelism must be exploited to make the most efficient use 675 of the quantum network's resources. 677 5.2. Classical communication 679 In this memo we have already covered two different roles that 680 classical communication must perform: 682 o communicate classical bits of information as part of distributed 683 protocols such as entanglement swapping and teleportation, 685 o communicate control information within a network - this includes 686 both background protocols such as routing as well as signalling 687 protocols to set up end-to-end entanglement generation. 689 Classical communication is a crucial building block of any quantum 690 network. All nodes in a quantum network are assumed to have 691 classical connectivity with each other (within typical administrative 692 domain limts). Therefore, quantum routers will need to manage two 693 data planes in parallel, a classical one and a quantum one. 694 Additionally, it must be able to correlate information between them 695 so that the control information received on a classical channel can 696 be applied to the qubits managed by the quantum data plane. 698 5.3. Abstract model of the network 700 5.3.1. Elements of a quantum network 702 Collecting all the pieces described so far, a quantum network will 703 consist of the following elements: 705 o Quantum repeaters - A quantum repeater is a node in the network 706 that is capable of generating entangled pairs with its directly 707 connected neighbours and performing entanglement swap operations 708 on them. 710 o Quantum routers - A quantum router is a quantum repeater that is 711 connected to more than two quantum repeaters as neighbours. This 712 distinguishes it from quantum repeaters composed into a linear 713 chain to connect two quantum routers (since no-cloning prohibits 714 quantum signal amplification). 716 o End-nodes - End-nodes in a quantum network must be able to receive 717 and handle an entangled pair, but they do not need to be able to 718 perform an entanglement swap (and thus are not necessarily quantum 719 repeaters). End-nodes are also not required to have any quantum 720 memory as certain quantum applications can be realised by having 721 the end-node measure its qubit as soon as it is received. 723 o Non-quantum nodes - Not all nodes in a quantum network need to 724 have a quantum data plane. A non-quantum node is any device that 725 can handle classical network traffic. 727 o Quantum links - A quantum link is a link which can be used to 728 generate an entangled pair between two directly connected quantum 729 repeaters. It may include a dedicated classical channel that is 730 to be used solely for the purpose of coordinating the entanglement 731 generation on this quantum link. 733 o Classical links - A classical link is a link between any node in 734 the network that is capable of carrying classical network traffic. 736 5.3.2. Putting it all together 738 A two-hop path in a generic quantum network can be represented as: 740 | App |-------------------CC-------------------| App | 741 || || 742 ------ ------ ------ 743 | EN |----QC & CC----| QR |----QC & CC----| EN | 744 ------ ------ ------ 746 App - user-level application 747 QR - quantum repeater 748 EN - end-node 749 QC - quantum channel 750 CC - classical channel 752 An application running on two end-nodes attached to a network will at 753 some point need the network to generate entangled pairs for its use. 754 This will require negotiation between the end-nodes, because they 755 must both open a communication end-point (a quantum socket) which the 756 network can use to identify the two ends of the connection. The two 757 end-nodes use the classical connectivity available in the network to 758 achieve this goal. 760 When the network receives a request to generate end-to-end entangled 761 pairs it uses the classical communication channels to coordinate and 762 claim the resources necessary to fulfil this request. This may be 763 some combination of prior control information (e.g. routing tables) 764 and signalling protocols, but the details of how this is achieved are 765 an active research question and thus beyond the scope of this memo. 767 During or after the control information is distributed the network 768 performs the necessary quantum operations such as generating 769 entangled over individual links, performing entanglement swaps, and 770 further signalling to transmit the swap outcomes and other control 771 information. Since none of the entangled pairs carry any user data, 772 some of these operations can be performed before the request is 773 received in anticipation of the demand. 775 The entangled pair is delivered to the application once it is ready, 776 together with the relevant pair identifier. However, being ready 777 does not necessarily mean once all link pairs and entanglement swaps 778 are complete as some applications can start executing on an 779 incomplete pair. In this case the remaining entanglement swaps will 780 propagate the actions across the network to the other end. 782 5.4. Network boundaries 784 Just like classical network, there will various boundaries will exist 785 in quantum networks. 787 5.4.1. Boundaries between different physical architectures 789 There are many different physical architectures for implementing 790 quantum repeater technology. The different technologies differ in 791 how they store and manipulate qubits in memory and how they generate 792 entanglement across a link with their neighbours. Different 793 architectures come with different trade-offs and thus a functional 794 network will likely consist of a mixture of different types of 795 quantum repeaters. 797 For example, architectures based on optical elements and atomic 798 ensembles are very efficient at generating entanglement, but provide 799 little control over the qubits once the pair is generated. On the 800 other hand nitrogen-vacancy architectures offer a much greater degree 801 of control over qubits, but have a harder time generating the 802 entanglement across a link. 804 It is an open research question where exactly the boundary will lie. 805 It could be that a single quantum repeater node provides some 806 backplane connection between the architectures, but it also could be 807 that special quantum links delineate the boundary. 809 5.4.2. Boundaries between different administrative regions 811 Just like in classical networks, multiple quantum networks will 812 connect into a global quantum internet. This necessarily implies the 813 existence of borders between different administrative regions. How 814 these boundaries will be handled is also an open question and thus 815 beyond the scope of this memo. 817 5.5. Physical constraints 819 The model above has effectively abstracted away the particulars of 820 the hardware implementation. However, certain physical constraints 821 need to be considered in order to build a practical network. Some of 822 these are fundamental constraints and no matter how much the 823 technology improves, they will always need to be addressed. Others 824 are artefacts of the early stages of a new technology. We here 825 consider a highly abstract scenario and refer to [5] for pointers to 826 the physics literature. 828 5.5.1. Memory lifetimes 830 In addition to discrete operations being imperfect, storing a qubit 831 in memory is also highly non-trivial. The main difficulty in 832 achieving persistent storage is that it is extremely challenging to 833 isolate a quantum system from the environment. The environment 834 introduces an uncontrollable source of noise into the system which 835 affects the fidelity of the state. This process is known as 836 decoherence. Eventually, the state has to be discarded once its 837 fidelity degrades too much. 839 The memory lifetime depends on the particular physical setup, but the 840 highest achievable values currently are on the order of seconds. 841 These values have increased tremendously over the lifetime of the 842 different technologies and are bound to keep increasing. However, if 843 quantum networks are to be realised in the near future, they need to 844 be able to handle short memory lifetimes. An architecture that 845 handles short lifetimes may also be more cost-efficient in the 846 future. 848 5.5.2. Rates 850 Entanglement generation on a link between two connected nodes is not 851 a very efficient process and it requires many attempts to succeed. A 852 fast repetition rate for Bell Pair generation is achievable, but only 853 one in a few thousands will succeed. Currently, the highest 854 achievable rates of success between nodes capable of storing the 855 resulting qubits are of the order of 10 Hz. Combined with short 856 memory lifetimes this leads to very tight timing windows to build up 857 network-wide connectivity. Achievable rates are likely to increase 858 with time, but just like with quantum memories, it may be more cost- 859 efficient in the future to provide low-rate links in some parts of 860 the network. 862 5.5.3. Communication qubit 864 Most physical architectures capable of storing qubits are only able 865 to generate entanglement using only a subset of its available qubits 866 called communication qubits. Once a Bell Pair has been generated 867 using a communication qubit, its state can be transferred into 868 memory. This may impose additional limitations on the network. In 869 particular if a given node has only one communication qubit it cannot 870 simultaneously generate Bell Pairs over two links. It must generate 871 entanglement over the links one at a time. 873 5.5.4. Homogeneity 875 Currently all hardware implementations are homogeneous and they do 876 not interface with each other. In general, it is very challenging to 877 combine different quantum information processing technologies at 878 present. Coupling different technologies with each other is of great 879 interest as it may help overcome the weaknesses of the different 880 implementations, but this may take a long time to be realised with 881 high reliability and thus is not a near-term goal. 883 5.6. Architectural principles 885 Given that the most practical way of realising quantum network 886 connectivity is using Bell Pair and entanglement swapping repeater 887 technology what sort of principles should guide us in assembling such 888 networks such that they are functional, robust, efficient, and most 889 importantly: they work. Furthermore, how do we design networks so 890 that they work under the constraints imposed by the hardware 891 available today, but do not impose unnecessary burden on future 892 technology. Redeploying network technology is a non-trivial process. 894 As this is a completely new technology that is likely to see many 895 iterations over its lifetime, this memo must not serve as a 896 definitive set of rules, but merely as a general set of recommended 897 guidelines based on principles and observations made by the 898 community. The benefit of having a community built document at this 899 early stage is that expertise in both quantum information and network 900 architecture is needed in order to successfully build a quantum 901 internet. 903 5.6.1. Goals of a quantum internet 905 When outlining any set of principles we must ask ourselves what goals 906 do we want to achieve as inevitably trade-offs must be made. So what 907 sort of goals should drive a quantum network architecture? The 908 following list has been inspired by the history of the classical 909 Internet, but it will inevitably evolve with time and the needs of 910 its users. The goals are listed in order of priority which in itself 911 may also evolve as the community learns more about the technology. 913 1. Support distributed quantum applications 915 The primary purpose of a quantum internet is to run distributed 916 quantum protocols and it is of utmost importance that they can 917 run well and efficiently. Therefore, the needs of quantum 918 applications should always be considered first. The requirements 919 for different applications can be found in [5]. 921 If a network is able to distribute entanglement it is officially 922 quantum. However, if it is unable to distribute these states 923 with a sufficiently high fidelity at a reasonable rate for a 924 majority of potential applications it is not practical. 926 2. Support tomorrow's distributed quantum applications 928 There are many applications already proposed to run over a 929 quantum internet. However, more algorithms will be invented as 930 the community grows as well as the robustness and the reliability 931 of the technology. Any proposed architecture should not 932 constrain the capabilities of the network for short-term benefit. 934 3. Hardware heterogeneity 936 There are multiple proposals for realising practical quantum 937 repeaters and they all have their advantages and disadvantages. 938 It is also very likely that the most optimal technologies in the 939 future will be hybrid combinations of the many different 940 solutions currently under development. It should be an explicit 941 goal of the architecture to allow for a large variety of hardware 942 implementations. 944 4. Be flexible with regards to hardware capabilities and limitations 946 This goal encompasses two important points. First, the 947 architecture should be able to function under the physical 948 constraints imposed by the current generation hardware. Second, 949 it should not make it difficult to run the network over any 950 hardware that may come along in the future. The physical 951 capabilities of repeaters will improve and redeploying a 952 technology is extremely challenging. 954 5. Security 956 Whilst the priority for the first quantum networks should be to 957 simply work, we cannot forget that ultimately they have to also 958 be secure. This has implications for the physical realisations 959 (do they satisfy the idealised theoretical models) and also the 960 design of the control stack. 962 It is actually difficult to guarantee security at the network 963 level and even if the network did provide such guarantees, the 964 application would still need to perform its own verification 965 similarly to how one ensures end-to-end security in classical 966 networks. 968 It turns out that as long as the underlying implementation 969 corresponds to (or sufficiently approximates) theoretical models 970 of quantum cryptography, quantum cryptographic protocols do not 971 need the network to provide any guarantees about the 972 authenticity, confidentiality, or integrity of the transmitted 973 qubits or the generated entanglement. Instead, applications such 974 as QKD establish such guarantees using the classical network in 975 conjunction with he quantum one. This is much easier than 976 demanding that the network deliver secure entanglement, which 977 indeed is not needed for quantum applications. 979 Nevertheless, control protocols themselves should be security 980 aware in order to protect the operation of the network itself and 981 limit disruption. 983 6. Availability and resilience 985 A practical and usable network is able to continue to operate 986 despite losses and failures, and will be robust to malicious 987 actors trying to disable connectivity. These may be simply 988 considered different aspects of security, but it is worthwhile to 989 address them explicitly at the architectural level already. 991 7. Easy to manage and monitor 993 Quantum networks rely on complex physical phenomena and require 994 hardware that is challenging to build. Furthermore, the quantum 995 resources will at first be very scarce and potentially very 996 expensive. This entails a need for a robust management solution. 997 It is important that a good management solution needs to come 998 with adequate monitoring capabilities. 1000 Good management solutions may also be key to optimising the 1001 networks which in turn may be crucial in making them economically 1002 feasible. Unlike user data that is transmitted over classical 1003 networks, quantum networks only need to generate generic Bell 1004 Pairs. This leaves a lot of room for pre-allocating resources in 1005 an efficient manner. 1007 5.6.2. The principles of a quantum internet 1009 The principles support the goals, but are not goals themselves. The 1010 goals define what we want to build and the principles provide a 1011 guideline in how we might achieve this. The goals will also be the 1012 foundation for defining any metric of success for a network 1013 architecture, whereas the principles in themselves do not distinguish 1014 between success and failure. For more information about design 1015 considerations for quantum networks see [8] [9] . 1017 1. Bell Pairs are the fundamental building block 1019 The key service that a quantum network provides is the 1020 distribution of entanglement between the nodes in a network. 1021 This point additionally specifies that the entanglement is 1022 primarily distributed in the form of the entangled Bell Pair 1023 states which should be used as a building block in providing 1024 other services, including more complex entangled states. 1026 2. Fidelity is part of the service 1028 In addition to being able to deliver Bell Pairs to the 1029 communication end-points, the Bell Pairs must be of sufficient 1030 fidelity. Unlike in classical networks where errors should 1031 essentially be eliminated for most application protocols, many 1032 quantum applications only need imperfect entanglement to 1033 function. However, different applications will have different 1034 requirements for what fidelity they can work with. It is the 1035 network's responsibility to balance the resource usage with 1036 respect to the application's requirements. It may be that it is 1037 cheaper for the network to provide lower fidelity pairs that are 1038 just above the threshold required by the application than it is 1039 to guarantee high fidelity pairs to all applications regardless 1040 of their requirements. 1042 3. Bell Pairs are indistinguishable 1044 Any two Bell Pairs between the same two nodes are 1045 indistinguishable for the purposes of an application provided 1046 they both satisfy its required fidelity threshold. This point is 1047 crucial in enabling the reuse of resources of a network and for 1048 the purposes of provisioning resources to meet application 1049 demand. However, the qubits that make up the pair themselves are 1050 not indistinguishable and the two nodes operating on a pair must 1051 coordinate to make sure they are operating on qubits that belong 1052 to the same Bell Pair. 1054 4. Time as an expensive resource 1056 With the current technology, time is the most expensive resource. 1057 It is not the only resource that is in short supply (memory, and 1058 communication qubits are as well), but ultimately it is the 1059 lifetime of quantum memories that imposes the most difficult 1060 conditions for operating an extended network of quantum nodes. 1061 Current hardware has low rates of Bell Pair generation, short 1062 memory lifetimes, and access to a limited number of communication 1063 qubits. All these factors combined mean that even a short 1064 waiting queue at some node could be enough for the Bell Pairs to 1065 decohere. 1067 However, time is only expensive once quantum operations are 1068 underway. If no quantum operations are currently being processed 1069 then the network can use this time to prepare and provision 1070 resources. 1072 As hardware improves, the need for carefully timing quantum 1073 operations may become smaller. It is currently unknown what the 1074 cost of these improvements will be, but it is conceivable that 1075 there is value in having relatively cheap and undemanding links 1076 connected at the edges of a network which will have very short 1077 memory lifetimes and low rates of Bell Pair generation. 1079 5. Limit classical communication 1081 This point offers a practical guideline to the issue of timing. 1082 A bottleneck in many quantum networked algorithms is the 1083 classical communication needed between quantum operations to 1084 synchronise state. Ideally, classical control mechanisms that 1085 require increased memory lifetimes should be avoided. 1087 For example, some quantum protocols may need to perform a 1088 correction for the random outcome of a quantum measurement. For 1089 this, they will block the state from further operations until a 1090 classical message is received with the information necessary to 1091 perform the correction. The time during which the quantum state 1092 is blocked is effectively wasted. It reduces the time available 1093 for subsequent operations possibly rendering the state useless 1094 for an application. 1096 Trade-offs that allow a protocol to limit the number of blocking 1097 classical communication rounds once quantum operations have 1098 commenced will in general be worth considering. 1100 6. Parallelise quantum operations 1102 A further point to address the issue of timing constraints in the 1103 network. The Bell Pairs on the individual links need not be 1104 generated one after another along the path between the 1105 communication end-points. The order does not matter at all. 1106 Furthermore, the order of the swap operations is flexible as long 1107 as they don't reduce the fidelity too much. Parallelising these 1108 operations is key to optimising quantum protocols. 1110 7. Avoid time-based coordination when possible 1112 A solution to timing constraints is to synchronise clocks and 1113 agree on the timing of events. However, such solutions have 1114 several downsides. Whilst network clock synchronisation may be 1115 accurate enough for certain purposes it introduces an additional 1116 element of complexity, especially when multiple nodes in 1117 different networks must be synchronised. Furthermore, clock 1118 synchronisation will never be perfect and it is conceivable that 1119 hardware capabilities advance so much that time-based mechanisms 1120 under-utilise resources in the more efficient parts of the 1121 network. 1123 Nevertheless, it may not be possible to avoid clocks, but such 1124 solutions should be adequately justified. 1126 8. Pre-allocate resources 1128 Regardless of what application is running over the network it 1129 will have the same needs as any other application: a number of 1130 Bell Pairs of sufficient fidelity. Whilst the fidelity is a 1131 variable number, the indistinguishability of Bell Pairs means 1132 that there is lots of flexibility in how a network may provision 1133 resources to meet demand. The additional timing constraints mean 1134 that pre-allocation of resources will be central to a usable 1135 quantum network. 1137 6. Security Considerations 1139 Even though no user data enters a quantum network security is listed 1140 as an explicit goal for the architecture and this issue is addressed 1141 in the section on goals. Even though user data doesn't enter the 1142 network, it is still possible to attack the control protocols and 1143 violate the authenticity, confidentiality, and integrity of 1144 communication. However, as this is an informational memo it does not 1145 propose any concrete mechanisms to achieve these goals. 1147 In summary: 1149 As long as the underlying implementation corresponds to (or 1150 sufficiently approximates) theoretical models of quantum 1151 cryptography, quantum cryptographic protocols do not need the network 1152 to provide any guarantees about the authenticity, confidentiality, or 1153 integrity of the transmitted qubits or the generated entanglement. 1154 Instead, applications such as QKD establish such guarantees using the 1155 classical network in conjunction with he quantum one. This is much 1156 easier than demanding that the network deliver secure entanglement. 1158 7. IANA Considerations 1160 This memo includes no request to IANA. 1162 8. Acknowledgements 1164 The authors of this memo acknowledge funding received from the EU 1165 Flagship on Quantum Technologies through Quantum Internet Alliance 1166 project. 1168 The authors would further like to acknowledge Carlo Delle Donne, 1169 Matthew Skrzypczyk, and Axel Dahlberg for useful discussions on this 1170 topic prior to the submission of this memo. 1172 9. Informative References 1174 [1] Bennett, C. and G. Brassard, "Quantum cryptography: Public 1175 key distribution and coin tossing", Theoretical Computer 1176 Science 560, 7-11, 2014, 1177 . 1179 [2] Crepeau, C., Gottesman, D., and A. Smith, "Secure multi- 1180 party quantum computation. Proceedings of Symposium on 1181 Theory of Computing", Proceedings of Symposium on Theory 1182 of Computing , 2002, 1183 . 1185 [3] Giovanetti, V., Lloyd, S., and L. Maccone, "Quantum- 1186 enhanced measurements: beating the standard quantum 1187 limit", Science 306(5700), 1330-1336, 2004, 1188 . 1190 [4] Castelvecchi, D., "The Quantum Internet has arrived (and 1191 it hasn't)", Nature 554, 289-292, 2018, 1192 . 1194 [5] Wehner, S., Elkouss, D., and R. Hanson, "Quantum internet: 1195 A vision for the road ahead", Science 362, 6412, 2018, 1196 . 1199 [6] Aspect, A., Grangier, P., and G. Roger, "Experimental 1200 Tests of Realistic Local Theories via Bell's Theorem", 1201 Phys. Rev. Lett. 47 (7): 460-463, 1981, 1202 . 1205 [7] Muralidharan, S., Kim, J., Lutkenhaus, N., Lukin, M., and 1206 L. Jiang, "Ultrafast and Fault-Tolerant Quantum 1207 Communication across Long Distances", Phys. Rev. Lett. 112 1208 (25-27), 250501, 2014, . 1210 [8] Meter, R. and J. Touch, "Designing quantum repeater 1211 networks", IEEE Communications Magazine 51, 64-71, 2013, 1212 . 1214 [9] Dahlberg, A., Skrzypczyk, M., Coopmans, T., Wubben, L., 1215 Rozpedek, F., Pompili, M., Stolk, A., Pawelczak, P., 1216 Knegjens, R., de Oliveira Filho, J., Hanson, R., and S. 1217 Wehner, "A Link Layer Protocol for Quantum Networks", 1218 arXiv 1903.09778, 2019, 1219 . 1221 [10] Nielsen, M. and I. Chuang, "Quantum Computation and 1222 Quantum Information", Cambridge University Press , 2011. 1224 [11] Bennett, C., DiVincenzo, D., Smolin, J., and W. Wootters, 1225 "Mixed State Entanglement and Quantum Error Correction", 1226 Phys. Rev. A Vol. 54, Iss. 5, 1996, 1227 . 1229 Authors' Addresses 1230 Wojciech Kozlowski 1231 QuTech 1232 Building 22 1233 Lorentzweg 1 1234 Delft 2628 CJ 1235 Netherlands 1237 Email: w.kozlowski@tudelft.nl 1239 Stephanie Wehner 1240 QuTech 1241 Building 22 1242 Lorentzweg 1 1243 Delft 2628 CJ 1244 Netherlands 1246 Email: S.D.C.Wehner@tudelft.nl 1248 Rodney Van Meter 1249 Keio Univeristy 1250 5322 Endo 1251 Fujisawa, Kanagawa 252-0882 1252 Japan 1254 Email: rdv@sfc.wide.ad.jp 1256 Bruno Rijsman 1257 Individual 1259 Email: brunorijsman@gmail.com