QIRG C. Wang
Internet-Draft A. Rahman
Intended status: Informational InterDigital Communications, LLC
Expires: July 31, 2020 January 28, 2020
Applications and Use Cases for the Quantum Internet
draft-wang-qirg-quantum-internet-use-cases-01
Abstract
The Quantum Internet has the potential to improve Internet protocol
and application functionality by incorporating quantum information
technology into the infrastructure of the overall Internet. In this
document, we provide an overview of some applications expected to be
used on the Quantum Internet, and then categorize them using the
standard telecommunications classification of control plane versus
data plane functionality. Other classification schemes are also
possible and discussed briefly. We then provide detailed use cases
for selected applications which can help steer the development of the
requisite Quantum Internet functionality.
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Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 2
2. Conventions used in this document . . . . . . . . . . . . . . 3
3. Terms and Acronyms List . . . . . . . . . . . . . . . . . . . 3
4. Quantum Internet Applications . . . . . . . . . . . . . . . . 4
4.1. Overview . . . . . . . . . . . . . . . . . . . . . . . . 4
4.2. Control vs Data Plane Classification . . . . . . . . . . 5
4.2.1. Control Plane Applications . . . . . . . . . . . . . 5
4.2.2. Data Plane Applications . . . . . . . . . . . . . . . 6
4.3. Other Possible Classifications . . . . . . . . . . . . . 6
5. Selected Quantum Internet Use Cases . . . . . . . . . . . . . 6
5.1. Secure Communication Setup . . . . . . . . . . . . . . . 6
5.2. Distributed Quantum Computing . . . . . . . . . . . . . . 9
5.3. Secure Quantum Computing with Privacy Preservation . . . 10
6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 12
7. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 13
8. Security Considerations . . . . . . . . . . . . . . . . . . . 13
9. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . 13
10. Informative References . . . . . . . . . . . . . . . . . . . 13
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 16
1. Introduction
The classical Internet has been constantly growing since it first
became commercially popular in the early 1990's. It essentially
consists of a large number of end-nodes (e.g., laptops, smart phones,
network servers) connected by routers. The end-nodes run
applications that provide some value added service for the end-users
such as processing and transmission of voice, video or data. The
physical connections between the various nodes in the Internet
include Digital Subscriber Lines (DSLs), fiber optics, etc. Bits are
transmitted across the classical Internet in packets.
Research and experimentation have picked up over the last few years
for developing a Quantum Internet [Wehner]. It is anticipated that
the Quantum Internet will provide intrinsic benefits such as better
end-user and network security. The Quantum Internet will have end-
nodes, which may be connected by quantum repeaters/routers. These
quantum end-nodes will also run value-added applications which will
be discussed later. The physical connections between the various
nodes in the Quantum Internet are expected to be primarily fiber
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optics and free-space optics. Unlike the classical Internet, qubits
(and not classical bits or packets) are expected to be transmitted
across the Quantum Internet due to the underlying physics
[I-D.irtf-qirg-principles].
The Quantum Internet is not anticipated to replace the classical
Internet. Instead the Quantum Internet will be integrated into the
classical Internet to form a new hybrid Internet. The process of
integrating the Quantum Internet with the classical Internet is
similar to, but with more profound implications, as the process of
introducing any new communication and networking paradigm into the
existing Internet.
2. Conventions used in this document
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
"SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this
document are to be interpreted as described in [RFC2119].
3. Terms and Acronyms List
This document assumes that the reader is familiar with the quantum
information technology related terms and concepts that are described
in [I-D.irtf-qirg-principles]. In addition, the following terms and
acronyms are defined here for clarity:
o Bit - Binary Digit (i.e., fundamental unit of information in a
classical computer).
o Classical Internet - The existing, deployed Internet (circa 2020)
where bits are transmitted in packets between nodes to convey
information. The classical Internet supports applications which
may be enhanced by the Quantum Internet. For example, the end-to-
end security of a classical Internet application may be improved
by secure communication setup using a quantum application.
o Hybrid Internet - The "new" or evolved Internet to be formed due
to a merger of the classical Internet and the Quantum Internet.
o NISQ - Noisy Intermediate-Scale Quantum
o Packet - Formatted unit of multiple related bits.
o Quantum End-node - An end-node hosts user applications and
interfaces with the rest of the Internet. Typically, an end-node
may serve in a client, server, or peer-to-peer role as part of the
application. If the end-node is part of the Quantum Internet it
must be able to generate/transmit and/or receive/process qubits.
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A quantum end-node, if it has quantum memory and quantum computing
capabilities, can be regarded as a quantum computer. A quantum
end-node must also be able to interface to the classical Internet
for control purposes and thus also be able to receive, process,
and transmit classical bits/packets.
o Quantum Internet - A new type of network enabled by quantum
information technology where qubits are transmitted between nodes
to convey information. (Note: qubits must be sent individually
and not in packets). The Quantum Internet will be merged into the
classical Internet to form a new hybrid Internet. The Quantum
Internet will use both quantum channels, and classical channels
provided by the classical Internet. The Quantum Internet may
either improve classical applications or may enable new quantum
applications.
o QC - Quantum Computer
o QKD - Quantum Key Distribution
o Qubit - Quantum Bit (i.e., fundamental unit of information in a
quantum computer). It is similar to a classic bit in that the
state of a qubit is either "0" or "1" after it is measured and is
denoted as its basis state |0> or |1>. However, the qubit is
different than a classic bit in that the qubit is in a linear
combination of both states before it is measured and termed to be
in superposition. A photon or an electron can be used to
represent a qubit.
4. Quantum Internet Applications
4.1. Overview
The Quantum Internet is expected to be extremely beneficial for a
subset of existing and new applications. We use "applications" in
the widest sense of the word and include functionality typically
contained in Layers 4 (Transport) to Layers 7 (Application) of the
Open System Interconnect (OSI) model.
The expected applications using Quantum Internet are still being
developed as we are in the formative stages of the Quantum Internet
[Castelvecchi] [Wehner]. However, an initial (and non-exhaustive)
list of the applications to be supported on the Quantum Internet can
be identified and classified using different schemes. We concentrate
on the telecom centric classification of control plane versus data
plane. We also briefly discuss other possible classification
schemes.
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4.2. Control vs Data Plane Classification
Traditionally, in the Internet most applications are classified as
either control plane functionality or data plane functionality.
Similarly, we classify Quantum Internet applications using the
paradigm of control plane applications versus data plane applications
where:
o Control Plane - Network functions and processes that operate on
(1) control bits/packets or qubits (e.g., to setup up end-user
encryption); or (2) management bits/packets or qubits (e.g., to
configure nodes).
o Data Plane - Network functions and processes that operate on end-
user application bits/packets or qubits (e.g., voice, video,
data). Sometimes also referred to as the user plane.
Some examples of classic Internet control plane applications are
Domain Name Server (DNS), Session Information Protocol (SIP), and
Internet Control Message Protocol (ICMP). Furthermore, examples of
classic Internet data plane applications are E-mail, web browsing,
and video streaming.
4.2.1. Control Plane Applications
Control Plane Applications using Quantum Internet:
1. Secure communication setup - Refers to secure cryptographic key
distribution between two or more end-nodes. The most well-known
method is referred to as QKD [Renner].
2. Fast Byzantine negotiation - Refers to a quantum network based
method for fast agreement in Byzantine negotiations [Fitzi].
This can be used for the popular financial blockchain feature as
well as other distributed computing features which use Byzantine
negotiations.
3. Network clock synchronization - Refers to a world wide set of
atomic clocks connected by the Quantum Internet to achieve an
ultra precise clock signal [Komar].
4. Position verification - Refers to a method for an end-node to
prove that it is at a particular location to, for example, access
a specific service [Unruh].
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4.2.2. Data Plane Applications
Data Plane Applications using Quantum Internet:
1. Distributed quantum computing - Refers to a collection of remote
small capacity quantum computers (i.e., each supporting a few
qubits) that are connected and working together in a coordinated
fashion so as to simulate a virtual large capacity quantum
computer [Wehner].
2. Secure quantum computing with privacy preservation - Refers to
private, or blind, quantum computation, which provides a way for
a client to delegate a computation task to one or more remote
quantum computers without disclosing the source data to be
computed over [Fitzsimons].
4.3. Other Possible Classifications
Applications may also be classified by the industry sector that they
serve. For example, applications may be classified as quantum
computing, quantum metrology, quantum chemistry, quantum
cryptography, etc. This is a valid and useful classification scheme.
However, since the classic Internet community is used to the control
plane versus data plane paradigm we will primarily use that approach
in this document.
5. Selected Quantum Internet Use Cases
The Quantum Internet will support a variety of applications and
deployment configurations. This section details a few key use cases.
In system engineering, a use case is typically made up of a set of
possible sequences of interactions between nodes and users in a
particular environment and related to a particular goal. This will
be the definition that we use in this section.
5.1. Secure Communication Setup
In this scenario, two banks (i.e., Bank #1 and Bank #2) need to have
secure communications for transmitting important financial
transaction records (see Figure 1). For this purpose, they first
need to securely exchange a classic secret cryptographic key (i.e., a
sequence of classical bits), which is triggered by an end-user banker
at Bank #1. This results in a source quantum node A at Bank #1 to
securely send a classic secret key to a destination quantum node B at
Bank #2. This is referred to as a secure communication setup. Note
that the quantum node A and B could be either a bare-bone quantum
end-node or a full-fledged quantum computer.
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One requirement for this secure communication setup process is that
it should not be vulnerable to any classic or quantum computing
attack. This can be realized using QKD [ETSI-QKD]. QKD can securely
distribute a secret key between two quantum nodes, without physically
transmitting it through the network and thus achieving the required
security. QKD is the most mature feature of the quantum information
technology, and has been commercially deployed in small-scale and
short-distance deployments.
In general, QKD (e.g., [BB84]) without using entanglement works as
follows:
1. The source quantum node A (e.g. Alice) transforms the secret key
to qubits. Basically, for each classical bit in the secret key,
the source quantum node A randomly selects one quantum
computational basis and uses it to prepare/generate a qubit for
the classical bit.
2. The source quantum node A sends qubits to the destination quantum
node B (e.g. Bob) via quantum channel.
3. The destination quantum node receives qubits and measures them
based on its random quantum basis.
4. The destination quantum node sends the measurement results (i.e.,
classic bits) to the source quantum node via any public classic
channel.
5. Both the source node and the destination node inform each other's
random quantum basis.
6. Both nodes discard any measurement bit under different quantum
basis and store all remaining bits as the secret key.
It is worth noting that:
1. There are some entanglement-based QKD protocols such as
[Treiber], which work differently than above steps. The
entanglement-based schemes, where entangled states are prepared
externally to Alice and Bob, are not normally considered
"prepare-and-measure" as defined in [Wehner]; other entanglement-
based schemes, where entanglement is generated within Alice can
still be considered "prepare-and-measure"; send-and-return
schemes can still be "prepare-and-measure", if the information
content, from which keys will be derived, is prepared within
Alice before being sent to Bob for measurement.
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2. There are many enhanced QKD protocols based on [BB84]. For
example, a series of loopholes have been identified due to the
imperfections of measurement devices; there are several solutions
to take into account these attacks such as measurement-device-
independent QKD [ZhangPeiyu]. These enhanced QKD protocol can
work differently than the steps of BB84 protocol [BB84].
3. For large-scale QKD, QKD Networks (QKDN) are required, which can
be regarded as a subset of a Quantum Internet. A QKDN may
consist of a QKD application layer, a QKD network layer, and a
QKD link layer [QinHao]. One or multiple trusted QKD relays
[ZhangQiang] may exist between the source quantum node A and the
destination quantum node B, which are connected by a QKDN.
Alternatively, a QKDN may rely on entanglement distribution and
entanglement-based QKD protocols; as a result, quantum-repeaters/
routers instead of trusted QKD relays are needed for large-scale
QKD.
As a result, the Quantum Internet in Figure 1 contains quantum
channels. And in order to support secure communication setup
especially in large-scale deployment, it also requires entanglement
generation and entanglement distribution
[I-D.van-meter-qirg-quantum-connection-setup], quantum repeaters/
routers, and/or trusted QKD relays.
+---------------+
| End User |
|(e.g., Banking |
| Application) |
+---------------+
^
| User Interface
| (e.g., GUI)
V
+-----------------+ /--------\ +-----------------+
| |--->( Quantum )--->| |
| Source | ( Internet ) | Destination |
| Quantum | \--------/ | Quantum |
| Node A | | Node B |
| (e.g., Bank #1) | /--------\ | (e.g., Bank #2) |
| | ( Classical) | |
| |<-->( Internet )<-->| |
+-----------------+ \--------/ +-----------------+
Figure 1: Secure Communication Setup
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5.2. Distributed Quantum Computing
In this scenario, Noisy Intermediate-Scale Quantum (NISQ) computers
distributed in different locations are available for sharing.
According to the definition in [Preskill], a NISQ computer can only
realize a small number of qubits and has limited quantum error
correction. In order to gain higher computation power before fully-
fledged quantum computers become available, NISQ computers can be
connected via classic and quantum channels. This scenario is
referred to as distributed quantum computing [Caleffi]
[Cacciapuoti01] [Cacciapuoti02].
As an example, scientists can leverage these connected NISQ computer
to solve highly complex scientific computation problems such as
analysis of chemical interactions for medical drug development (see
Figure 2). In this case, qubits will be transmitted among connected
quantum computers via quantum channels, while classic control
messages will be transmitted among them via classic channels for
coordination and control purpose . Qubits from one NISQ computer to
another NISQ computer are very sensitive and cannot be lost. For
this purpose, quantum teleportation can be leveraged to teleport
sensitive data qubits from one quantum computer A to another quantum
computer B.
Specifically, the following steps happen between A and B:
1. The quantum computer A locally generates some sensitive data
qubits to be teleported to the quantum computer B.
2. A shared entanglement is established between the quantum computer
A and the quantum computer B (i.e., there are two entangled
qubits: |q1> at A and |q2> at B).
3. Then, the quantum computer A performs a Bell measurement of the
entangled qubit |q1> and the sensitive data qubit.
4. The result from this Bell measurement will be encoded in two
classic bits, which will be physically transmitted via a classic
channel to the quantum computer B.
5. Based on the received two classic bits, the quantum computer B
modifies the state of the entangled qubit |q2> in the way to
generate a new qubit identical to the sensitive data qubit at the
quantum computer A.
In Figure 2, the Quantum Internet contains quantum channels and
quantum repeaters/routers [I-D.irtf-qirg-principles]. This use case
needs to support entanglement generation in order to enable quantum
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teleportation, entanglement distribution or quantum connection setup
[I-D.van-meter-qirg-quantum-connection-setup] in order to support
long-distance quantum teleportation.
+-----------------+
| End-User |
|(e.g., Scientist)|
+-----------------+
^
|User Interface (e.g. GUI)
|
+------------------+-------------------+
| |
| |
V V
+----------------+ /--------\ +----------------+
| |--->( Quantum )--->| |
| | ( Internet ) | |
| Quantum | \--------/ | Quantum |
| Computer A | | Computer B |
| (e.g., Site #1)| /--------\ | (e.g., Site #2)|
| | ( Classical) | |
| |<-->( Internet )<-->| |
+----------------+ \--------/ +----------------+
Figure 2: Distributed Quantum Computing
5.3. Secure Quantum Computing with Privacy Preservation
Secure computation with privacy preservation refers to the scenario:
1. A client node with source data delegates the computation of the
source data to a remote computation node.
2. Furthermore, the client node does not want to disclose any source
data to the remote computation node and thus preserve the source
data privacy.
3. Note that there is no assumption or guarantee that the remote
computation node is a trusted entity from the source data privacy
perspective.
As an example illustrated in Figure 3, the client node could be a
virtual voice-controlled home assistant device like Amazon's Alexa
product. The remote computation node could be a quantum computer in
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the cloud. A resident as an end-user uses voice to control the home
device. The home device captures voice-based commands from the end-
user. Then, the home device interfaces to a home quantum terminal
node (e.g., a home gateway), which interacts with the remote
computation node to perform computation over the captured voice-based
commands. The home quantum terminal could be either a bare-bone
quantum end-node or a full-fledged quantum computer.
In this particular case, there is no privacy concern since the source
data (i.e., captured voice-based commands) will not be sent to the
remote computation node which could be compromised. Protocols
[Fitzsimons] for delegated quantum computing or blind quantum
computation can be leveraged to realize secure delegated computation
and guarantee privacy preservation simultaneously. Using delegated
quantum computing protocols, the client node does not need send the
source data but qubits with some measurement instructions to the
remote computation node (e.g., a quantum computer).
After receiving qubits and measurement instructions, the remote
computation node performs the following actions:
1. It first performs certain quantum operations on received qubits
and measure them according to received measurement instructions
to generate computation results (in classic bits).
2. Then it sends the computation results back to the client node via
classic channel.
3. In this process, the source data is not disclosed to the remote
computation node and the privacy is preserved.
In Figure 3, the Quantum Internet contains quantum channels and
quantum repeaters/routers for long-distance qubits transmission
[I-D.irtf-qirg-principles].
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+----------------+
| End-User |
|(e.g., Resident)|
+----------------+
^
| User Interface
| (e.g., voice commands)
V
+----------------+
| Home Device |
+----------------+
^
| Classic
| Channel
V
+----------------+ /--------\ +----------------+
| |--->( Quantum )--->| |
| Quantum | ( Internet ) | Remote |
| Terminal | \--------/ | Computation |
| Node | | Node |
| (e.g., Home | /--------\ | (e.g., QC |
| Gateway) | ( Classical) | in Cloud) |
| |<-->( Internet )<-->| |
+----------------+ \--------/ +----------------+
Figure 3: Secure Computation with Privacy Preservation
6. Conclusion
This document provides an overview of some expected applications for
the Quantum Internet and details selected use cases. The
applications are classified as either control plane or data plane
functionality as typical for Internet applications. One key take
away is that a variety of control plane applications will run on the
Quantum Internet. In contrast, the data plane applications running
on the Quantum Internet will be focused on supporting different forms
of remote quantum computing. This set of applications may, of
course, naturally expand over time as the Quantum Internet matures.
This document can also serve as an introductory text to persons
interested in learning about the practical uses of the Quantum
Internet. Finally, it is hoped that this document will help guide
further research and development of the specific Quantum Internet
functionality required to implement the applications and uses cases
described herein.
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7. IANA Considerations
This document requests no IANA actions.
8. Security Considerations
This document does not define an architecture nor a specific protocol
for the Quantum Internet. It focuses on detailing use cases and
describing typical Quantum Internet applications. However, some
useful observations can be made regarding security as follows.
It has been clearly identified that once large-scale quantum
computing becomes reality it will be able to theoretically break many
of the public-key (i.e., asymmetric) cryptosystems currently in use
because of the exponential increase of computing power with quantum
computing. This would negatively affect many of the security
mechanisms currently in use on the classic Internet. This has given
strong impetus for starting development of new cryptographic systems
that are secure against quantum computing attacks [NISTIR8240].
Paradoxically, development of a Quantum Internet will also mitigate
the threats posed by quantum computing attacks against public-key
cryptosystems. Specifically, the secure communication setup feature
of the Quantum Internet as described in Section 5.1 will be strongly
resistant to both classical and quantum computing attacks.
Finally, Section 5.3 provides a method to perform remote quantum
computing while preserving the privacy of the source data.
9. Acknowledgments
The authors want to thank Xavier de Foy, Wojciech Kozlowski, Ruidong
Li, and Gelard Patrick for their very useful reviews and comments to
the document.
10. Informative References
[BB84] Bennett, C. and G. Brassard, "Quantum Cryptography: Public
Key Distribution and Coin Tossing", 1984,
<http://researcher.watson.ibm.com/researcher/files/us-
bennetc/BB84highest.pdf>.
[Cacciapuoti01]
Cacciapuoti, A., "Quantum Internet: Networking Challenges
in Distributed Quantum Computing", IEEE Network, (Early
Access), 2019,
<https://ieeexplore.ieee.org/document/8910635>.
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[Cacciapuoti02]
Cacciapuoti, A., "When Entanglement meets Classical
Communications: Quantum Teleportation for the Quantum
Internet", 2019, <https://arxiv.org/abs/1907.06197>.
[Caleffi] Caleffi, M., "Quantum internet: From Communication to
Distributed Computing!", NANOCOM, ACM, 2018,
<https://arxiv.org/abs/1907.06197>.
[Castelvecchi]
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>.
[ETSI-QKD]
ETSI GR QKD 003 V2.1.1, "Quantum Key Distribution (QKD):
Components and Internal Interfaces", 2018,
<https://www.etsi.org/deliver/etsi_gr/
QKD/001_099/003/02.01.01_60/gr_QKD003v020101p.pdf>.
[Fitzi] Fitzi, M. and et. al., "A Quantum Solution to the
Byzantine Agreement Problem", 2001,
<https://arxiv.org/pdf/quant-ph/0107127.pdf>.
[Fitzsimons]
Fitzsimons, J., "Private Quantum Computation: An
Introduction to Blind Quantum Computing and Related
Protocols", 2017,
<https://www.nature.com/articles/s41534-017-0025-3.pdf>.
[I-D.dahlberg-ll-quantum]
Dahlberg, A., Skrzypczyk, M., and S. Wehner, "The Link
Layer service in a Quantum Internet", draft-dahlberg-ll-
quantum-03 (work in progress), October 2019.
[I-D.irtf-qirg-principles]
Kozlowski, W., Wehner, S., Meter, R., and B. Rijsman,
"Architectural Principles for a Quantum Internet", draft-
irtf-qirg-principles-02 (work in progress), November 2019.
[I-D.van-meter-qirg-quantum-connection-setup]
Meter, R. and T. Matsuo, "Connection Setup in a Quantum
Network", draft-van-meter-qirg-quantum-connection-setup-01
(work in progress), September 2019.
[Komar] Komar, P. and et. al., "A Quantum Network of Clocks",
2013, <https://arxiv.org/pdf/1310.6045.pdf>.
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[NISTIR8240]
Alagic, G. and et. al., "Status Report on the First Round
of the NIST Post-Quantum Cryptography Standardization
Process", NISTIR 8240, 2019,
<https://nvlpubs.nist.gov/nistpubs/ir/2019/
NIST.IR.8240.pdf>.
[Preskill]
Preskill, J., "Quantum Computing in the NISQ Era and
Beyond", 2018, <https://arxiv.org/pdf/1801.00862>.
[QinHao] Qin, H., "Towards Large-Scale Quantum Key Distribution
Network and Its Applications", 2019,
<https://www.itu.int/en/ITU-T/Workshops-and-
Seminars/2019060507/Documents/Hao_Qin_Presentation.pdf>.
[Renner] Renner, R., "Security of Quantum Key Distribution", 2006,
<https://arxiv.org/pdf/quant-ph/0512258.pdf>.
[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", BCP 14, RFC 2119,
DOI 10.17487/RFC2119, March 1997,
<https://www.rfc-editor.org/info/rfc2119>.
[Treiber] Treiber, A. and et. al., "A Fully Automated Entanglement-
based Quantum Cyptography System for Telecom Fiber
Networks", New Journal of Physics, 11, 045013, 2009,
<https://doi.org/10.1364/OE.26.024260>.
[Unruh] Unruh, D., "Quantum Position Verification in the Random
Oracle Model", 2014, <https://link.springer.com/content/
pdf/10.1007/978-3-662-44381-1_1.pdf>.
[Wehner] Wehner, S., Elkouss, D., and R. Hanson, "Quantum internet:
A vision for the road ahead", Science 362, 2018,
<http://science.sciencemag.org/content/362/6412/
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[ZhangPeiyu]
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Wang & Rahman Expires July 31, 2020 [Page 15]
Internet-Draft Quantum Internet Use Cases January 2020
Authors' Addresses
Chonggang Wang
InterDigital Communications, LLC
1001 E Hector St
Conshohocken 19428
USA
Email: Chonggang.Wang@InterDigital.com
Akbar Rahman
InterDigital Communications, LLC
1000 Sherbrooke Street West
Montreal H3A 3G4
Canada
Email: rahmansakbar@yahoo.com
Wang & Rahman Expires July 31, 2020 [Page 16]