Quantum Communication Networks and Quantum repeaters...4 What is Quantum Communication? “Quantum...
Transcript of Quantum Communication Networks and Quantum repeaters...4 What is Quantum Communication? “Quantum...
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Quantum Communication Networksand Quantum repeaters
Winter School on Quantum Computing at Emory (WiSQCE)
Paul Hilaire, Edwin Barnes, Sophia Economou
January 9th 2020
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Privacy of long-distancecommunications usingquantum mechanics
Winter School on Quantum Computing at Emory (WiSQCE)
Paul Hilaire, Edwin Barnes, Sophia Economou
January 9th 2020
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Communications
B
0 𝑜𝑟 1
A
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What is Quantum Communication?
“Quantum communication is the art of transferring a quantum state from one place to another”
N. Gisin and R. Thew, Nat. Photon. (2007)
Bob
Ψ = 𝛼 0 + 𝛽|1⟩
Alice
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What is a Quantum Network?
A network which facilitates the transfer of quantum information.
Routing Q informationCommon language
Network of quantum networks:Quantum internet!
S. Wehner et al., Science (2018)
H. J. Kimble, Nature (2008)
Sharing the network
Complex topology
Different devices
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Why Quantum Communications?
Distributedquantum computing
Improved accuracySecure communications
Security QuantumComputing
Sensing
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Why Quantum Communications?
Security
QuantumComputing
SensingQKD
TelescopeobservationD. Gottesman et al., PRL (2012)
Stolen from a talk of R. Van Meter…
DistributedQuantum computing
R. Beals et al., Proc. R. Soc. A (2013)
Accurateclock synchro
P. Komar et al., Nat. Phys. (2014)
BlindQC
A. Broadbent et al., IEEE. (2009)
Bizantine agreementLeader election
M. Fitzi et al., Phys. Rev. Lett. (2001)
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Outline
Part 1: General tutorial
Part 2: Recent results (if time)
To show that you are able to read state-of-the-art QC papers now!
To learn more about quantum communications (QC)
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Outline
Quantum Network and distributed quantum computing
Quantum Key Distribution (QKD)
Long-distance Quantum Communications with Quantum Repeaters
Free-space long-distance Quantum Communications
Practical implementations
Part 1: General tutorial
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Outline
Quantum Network and distributed quantum computing
Quantum Key Distribution (QKD)
Long-distance Quantum Communications with Quantum Repeaters
Free-space long-distance Quantum Communications
Practical implementations
Part 1: General tutorial
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How to distribute a message?
Asymmetric encryption (RSA)
𝑛 = 𝑝 × 𝑞
DecryptEncrypt
𝑛 = 𝑝 × 𝑞
Easy
HardShor’s algorithm
Protected by computational hardness
P. W. Shor , SIAM review (1999)
One-way functionwith trapdoor
One-time padSymmetric key
Provably secure
Eve, the eavesdropper
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Symmetric key
“00101…”
How to distribute the key in a secure way?
“00101…”
𝑥 = 𝑚⊕ 𝑘 𝑚 = 𝑥 ⊕ 𝑘
…01100101011110…
⊕
𝑚…00110011110111……01010110101001… 𝑘
𝑥
0⊕ 0 = 00⊕ 1 = 11⊕ 0 = 11⊕ 1 = 0
𝑥 ⊕ 𝑘 = 𝑚⊕ 𝑘⊕ 𝑘
𝑥 ⊕ 𝑘 = 𝑚⊕ 0
𝑥 ⊕ 𝑘 = 𝑚
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Quantum key distribution
Provably secure symmetric key distribution, using the law quantum mechanics.
No cloning-theoremQuantum superposition Quantum measurement
𝜓 ⊗ |0⟩ → 𝜓 ⊗ |𝜓⟩|𝜓⟩ = 𝛼 0 + 𝛽|1⟩ 𝑃0(|𝜓⟩) = |𝛼|²
𝜓 → |0⟩ W.K. Wootters and W.H. Zurek. Nature (1982)
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Photonic qubit
• …
• Polarization: 𝑉 = |1⟩𝐻 = |0⟩
𝜔𝑏 = |1⟩• Energy: 𝜔𝑟 = |0⟩
𝑡1 = |1⟩𝑡0 = |0⟩
𝑡
• Time:
𝑡
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Quantum superposition and measurement
𝐴 =1
2( 𝐻 − |𝑉⟩)
𝐷 =1
2𝐻 + 𝑉 )
Polarizer
𝑉
𝜆/2𝑉
𝐻
PBS
𝑉
𝐻
Modify incident state:Ψ → |𝑉⟩
Measurement → |𝑉⟩
𝜆/2𝐴
𝐷
PBS
X basis
Z basis
One cannot make a measurement without perturbating the system.
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Conjugate coding
Encode:
Basis PolarizationBit value
0
1 |𝐴⟩
|𝐻⟩
|𝑉⟩
|𝐷⟩
𝑿
𝒁
Choose:
0
1If you don’t know the basis,
You cannot recover fully the bit by measuring it
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No-Cloning Theorem
An unknown arbitrary quantum state cannot be copied.
|𝜙⟩
|𝜙⟩
|0⟩
|𝜙⟩𝑈
|Ψ⟩
|Ψ⟩
|0⟩
|Ψ⟩𝑈
𝑈( 𝜙 ⊗ |0⟩) = 𝜙 ⊗ |𝜙⟩𝑈( Ψ ⊗ |0⟩) = Ψ ⊗ |Ψ⟩
𝑈†𝑈 = 1
Impossible
Proof: Calculate A = ⟨Ψ| ⊗ 0 𝑈†𝑈 𝜙 ⊗ |0⟩
𝑈†𝑈 = 1 ⇒ 𝐴 = (⟨Ψ| ⊗ 0 )(|𝜙⟩ ⊗ |0⟩ )⇒ 𝐴 = ⟨Ψ|𝜙⟩
Ψ|𝜙 = ⟨Ψ 𝜙 2 Only true if Ψ = |𝜙⟩ or if Ψ|𝜙 = 0
W.K. Wootters and W.H. Zurek. Nature (1982)
𝑈( 𝜙 ⊗ |0⟩) = 𝜙 ⊗ |𝜙⟩⇒ 𝐴 = (⟨Ψ| ⊗ ⟨Ψ|)(|𝜙⟩ ⊗ |𝜙⟩) = ⟨Ψ 𝜙 2
Exercise?
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QKD with single photons
CH. Bennet and G. Brassard, Proc. IEEE (1984)
Prepare: Measure:
or XZ
BasisPolarization Value
1
0
1
0
|𝐴⟩
|𝐻⟩
|𝑉⟩
|𝐷⟩
𝑋
𝑍
𝑍
𝑋 00110101011011000
XZXZXZZXXZXXZZXXZ
00110101011010100
ZZXZXXZXZXXXZXZXX
Key: 0110011010
ZZXZXXZXZXXXZXZXX
00110101011010100 00110101011011000
XZXZXZZXXZXXZZXXZ
Key: 0110011010
…XXZ……XZZ…
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What about the eavesdropper?
CH. Bennet and G. Brassard, Proc. IEEE (1984)
Key: 0111010011
Measure: Measure 𝑍Prepare |𝐻⟩
𝑋
𝑍
0 𝑜𝑟 1 0 𝑜𝑟 1
0 0
0
Key: 0110011010
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What about errors?
Key: 0111010011
Measure 𝑍Prepare |𝐻⟩
Key: 0110011010
𝑍, 1𝑍, 0 𝑍, 0 or
Errors or Eavesdropper?
𝜖 ≤ 0.110 Unconditional security
𝜖 ≤ 0.146 Secure against individual attacks
ϵ Qubit Error Rate1 − ϵ
P.W. Shor et al., Phys. Rev. Lett. (2000)
N. Gisin et al., Rev. Mod. Phys (2002)
Classical error correction + privacy amplificationCH. Bennett et al., J. Cryptology. (1992)
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Very diverse field
AK. Eckert, Phys. Rev. Lett. (1991)
2-state
6-state
…
With decoy-state
“Quantum hacking”:Trojan horse attacks / Real life imperfection
Internet Interfacing
With attenuated laser
Original proposal BB84 (4-state with single photons)CH. Bennet and G. Brassard, Proc. IEEE (1984)
L. Lydersen et al., Nat. Photon. (2010)
HK. Lo et al., Phys. Rev. Lett. (2005)
Commercially available
With entangled Bell pairs (E91)
Real world QKD
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Challenge: Photon losses
Photon loss 𝜂𝑡 𝐿 = 𝑒−
𝐿𝐿𝑎𝑡𝑡
𝐿𝑎𝑡𝑡 ≈ 20𝑘𝑚
How to extend QKD to long distances?
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Outline
Quantum Network and distributed quantum computing
Quantum Key Distribution (QKD)
Long-distance Quantum Communications with Quantum Repeaters
Free-space long-distance Quantum Communications
Practical implementations
Part 1: General tutorial
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Free-space communications
𝐿𝑎𝑡𝑡 ≈ 20𝑘𝑚
Problem: Eye of sight
R. Ursin et al., Nat. Phys. (2007)
“Quantum Satellite”
J. Liao et al., Phys. Rev. Lett. (2018)
J. Yin et al., Science (2017)
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Quantum Satellite: Trusted repeater
J. Liao et al., Phys. Rev. Lett. (2018)
Trusting the repeater Untrusted repeater?
⊕=
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Entanglement
1
2𝐻𝑎𝐻𝑏 + 𝑉𝑎𝑉𝑏 =
1
2𝐷𝑎𝐷𝑏 + 𝐴𝑎𝐴𝑏
AK. Eckert, Phys. Rev. Lett. (1991)
Work for QKD
Measure:
or XZMeasure:
or XZ
Alice and Bob can verify that the source is sending the desired state
by measuring Bell inequalities
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Untrusted “quantum satellite”!J. Yin et al., Science (2017)
Untrusted repeater = … + Entanglement
Trusted repeater = Quantum superposition + Measurement
Quantum Satellite: Untrusted repeater
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Problems of satellite communications
Keep fibers?
Expensive
Weather dependent
Intermittent
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Outline
Quantum Network and distributed quantum computing
Quantum Key Distribution (QKD)
Long-distance Quantum Communications with Quantum Repeaters
Free-space long-distance Quantum Communications
Practical implementations
Part 1: General tutorial
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How to overcome fiber losses?
In classical communications
Repeater
No-cloning theorem
Quantum amplifier?
+ AmplifierDivide the distance
Quantum teleportation!
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Quantum teleportation
What it is.
Faster than light?
What it is NOT.
Bob
|Ψ′⟩
Alice
|Ψ⟩
Faster than light transfer of objects
Transfer of quantum informationWithout transfer of matter.
|Ψ⟩|Ψ⟩
|Ψ⟩
Classical information
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Quantum teleportation
ba
|Ψ′⟩
Ψ± =1
200 ± 11
Bell state measurement(BSM)
1
20𝑎0𝑏 + 1𝑎1𝑏
Φ± =1
201 ± 10
|Ψ⟩|Ψ⟩
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Quantum teleportation (math)
Total state
|𝜙⟩ = 𝛼 01 + 𝛽|11⟩ ⊗ 0203 + 1213 )State to transfer𝜓 = 𝛼 01 + 𝛽|11⟩
1
32 3
Ψ+ = 0203 + |1213⟩Shared entangled state
Ψ12± = 0102 ± 1112
Φ12± = 0112 ± 1102
Bell state measurement
Φ12− → 𝜓3 = −𝛽 03 + 𝛼 13 = 𝑍𝑋|𝜓⟩
Measurement outcome:
|Ψ12− ⟩ → 𝜓3 = 𝛼 03 − 𝛽 13 = 𝑍|𝜓⟩
|Ψ12+ ⟩ → 𝜓3 = 𝛼 03 + 𝛽 13 = |𝜓⟩
Φ12+ → 𝜓3 = 𝛽 03 + 𝛼 13 = 𝑋|𝜓⟩
Classical information
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Quantum teleportation (math)
𝑑𝑒𝑓 𝑡𝑒𝑙𝑒𝑝𝑜𝑟𝑡𝑎𝑡𝑖𝑜𝑛 𝑞 :𝑞𝑎 , 𝑞𝑏 = 𝑏𝑒𝑙𝑙()𝑏1, 𝑏2 = 𝑎𝑙𝑖𝑐𝑒 𝑞𝑎 , 𝑞 = 𝑏𝑒𝑙𝑙_𝑚𝑒𝑎𝑠(𝑞𝑎 , 𝑞)𝑟𝑒𝑡𝑢𝑟𝑛 𝑏𝑜𝑏(𝑞𝑏 , 𝑏1, 𝑏2)
𝑞𝑏
𝑞
𝑞𝑎
Bob
Alice H
𝑎𝑙𝑖𝑐𝑒 𝑞𝑎, 𝑞
X Z
𝑏𝑜𝑏(𝑞𝑏,𝑏1, 𝑏2)
H
𝑏𝑒𝑙𝑙()
𝑏2
𝑏1|Ψ⟩
|Ψ⟩
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Entanglement Swapping
2 43
1
BSM
Teleporting half of an entangled state?
Ψ12+ ⊗ |Ψ34
+ ⟩BSM (2,3)
Ψ14± 𝑜𝑟|Φ14
± ⟩
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Dividing the distance
BSM
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𝑑𝑒𝑓 𝑒𝑛𝑡𝑎𝑛𝑔𝑙𝑒𝑚𝑒𝑛𝑡_𝑠𝑤𝑎𝑝𝑝𝑖𝑛𝑔 𝑞 :𝑞𝑎 , 𝑞𝑐1 = 𝑏𝑒𝑙𝑙()𝑞𝑏 , 𝑞𝑐2 = 𝑏𝑒𝑙𝑙()𝑏1, 𝑏2 = 𝑐ℎ𝑎𝑟𝑙𝑖𝑒 𝑞𝑐1, 𝑞𝑐2 = 𝑏𝑒𝑙𝑙_𝑚𝑒𝑎𝑠(𝑞𝑐1, 𝑞𝑐2)𝑟𝑒𝑡𝑢𝑟𝑛 𝑏𝑜𝑏(𝑞𝑏 , 𝑏1, 𝑏2)
Quantum teleportation (math)
𝑞𝑏
𝑞𝑐2
𝑞𝑐1
Bob
CharlieH
X Z
H
𝑏2
𝑏1
𝑞𝑎Alice
H
1
2|0𝑎0𝑏⟩ + |1𝑎1𝑏⟩
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Quantum relay
L
BSMBSM
BSM with linear optics:
Success probability of 1
2.
𝐿0 =𝐿
𝑁𝑄𝑅
𝜂𝑡𝐿
2𝑁𝑄𝑅
2𝑁𝑄𝑅
= 𝑒−
𝐿2𝑁𝑄𝑅𝐿𝑎𝑡𝑡
2𝑁𝑄𝑅
= 𝜂𝑡(𝐿)
BC. Jacobs et al., Phys. Rev. A (2002)
Ingredient missing…→ a memory!
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Distant spin-spin heralded entanglement
Spin-photon entanglement
↑ 0𝑝ℎ + ↓ |1𝑝ℎ⟩
“Failure”
Transfer measurement outcome
Classical signaling
Try swap entanglement
“Success”
SuccessFail
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Quantum Repeater
Idea:• Include memories• Fail? → Try again! Bahskar et al., Arxiv (2019)
Briegel et al., Phys. Rev. Lett. (1998)
N. Sangouard et al., Rev. Mod. Phys.(2011)
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Coherence time
How long can you store quantum information?
Quantum information is fragile
𝜓(𝑡) = 𝛼 0 + 𝛽𝑒𝑖𝛿𝜃(𝑡)|1⟩
Environment
𝜓(𝑡) = 𝛼 0 + 𝛽|1⟩
Induce errors
Store info for limited time 𝑇2
Error correctionL. Jiang et al., Phys. Rev. A (2009)
Quantum memory multiplexingCollins et al., Phys Rev Lett. (2007)
Entanglement purification
JW. Pan et al., Nature (2001)
W. Dür et al., Phys. Rev. A (1999)
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Plenty of proposals
Munro et al., Nat. Phys. (2012)
Hosegawa et al., Nat. Comm. (2019)
F. Ewert et al., Phys. Rev. Lett. (2016)
K. Azuma et al., Nat. Comm. (2015)
Not using quantum memories(Error correction instead)
Childress et al., Phys. Rev. Lett. (2006)
Collins et al., Phys Rev Lett. (2007)
S. Muralidharan et al., Phys. Rev. Lett. (2014)
Using quantum memories
Rozpcedek et al., Phys. Rev. A (2019)
S. Muralidharan et al., Sci. Rep. (2016)
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Outline
Quantum Network and distributed quantum computing
Quantum Key Distribution (QKD)
Long-distance Quantum Communications with Quantum Repeaters
Free-space long-distance Quantum Communications
Practical implementations
Part 1: General tutorial
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Secure computation
𝑓(|𝑥⟩) = ? ? ?
Personal data
Algorithm Solution |𝑦⟩
Quantum algorithm
IBM Quantum computer
Problem: No Quantum computational power
Alice:• Alice don’t want anybody to learn about
what she is doing. (𝑓, 𝑥 , 𝑦 )• Alice want to check that she’s receiving
the right solution.
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Towards quantum internet: Today
Quantum channel
Classical channel
𝑓, |𝑥⟩
QKD
|𝑦⟩𝑓 |𝑥⟩|𝑦⟩𝑓 |𝑥⟩Knows
𝑦 = 𝑓(|𝑥⟩)
𝑦
Can you do better?(IBM still knows everything!)
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Quantum channel
Towards quantum internet: Future
Classical channel
𝑓
|𝑥⟩
|𝑦⟩𝑓 |𝑥⟩Knows
𝑦 = 𝑓(|𝑥⟩)𝑦
Q Teleportation
IBM does not know your personal information but knows 𝑓.
Can we do better?
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Universal Blind Quantum Computation
Classical channel
𝑓, |𝑥⟩
|𝑦⟩𝑓 |𝑥⟩Knows
𝑦 = 𝑓(|𝑥⟩)𝑦
Q TeleportationQuantum channel
JF. Fitzsimons, npj QI (2017)
A. Broadbent et al., IEEE. (2009)
IBM knows nothing!Alice can check if IBM is cooperating
(|𝑦⟩ is the right answer)
Alice don’t need to trust IBM to work with it!
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Building blocks of a quantum internet
User with different capacities
UQRTQR
Quantum Repeaters
Fibers
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Challenges
Routing quantum information
Find common language
Sharing the network
Complex topology
Different devices/technology
AsynchronousResource-efficientInhomogeneity: different distances and resourcesTake into account errors
R. Van Meter Quantum networking (2014)
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Outline
Quantum Network and distributed quantum computing
Quantum Key Distribution (QKD)
Long-distance Quantum Communications with Quantum Repeaters
Free-space long-distance Quantum Communications
Practical implementations
Part 1: General tutorial
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QKD and trusted repeater network
M. Lucamarini, Optics Express (2013)
Detector Telecom fiber
Attenuatedlaser
Qubit encoding
Qubit decoding
Columbus (OH)
Soccer World CupSouth Africa 2010
Decoy
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Source of single photons
Multiplexing
Quantum emitters
Heralded single photon source
Spontaneous ParametricDownConversion (SPDC) sources
F. Kaneda and PG. Kwiat, Science. Advances (2019)
ME. Reimer and C. Cher, Nat. Photonics (2019)
Quantum dot
Conduction band
Valence band
N. Somaschi et al., Nat. Photonics (2016)
Low single-photon probability
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Source of entangled photons
Quantum emitters
Quantum dot
Conduction band
Valence band
1
2|𝐻𝑠𝑉𝑖⟩ + |𝑉𝑠𝐻𝑖⟩
Low probability source of entangled photons
Hui Wang et al., Phys. Rev. Lett. (2019)
A. Beveratos et al., Eur. Phys. J. D (2014)
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“Natural” quantum memories
Trapped ions
Requirements:• Entanglement with photons• Long-lived
C. Monroe et al., Phys. Rev. A (2014)
Long-lived memories (>1min)
Experimentally difficult to manipulate…
Atomic ensembles
N. Sangouard et al., Rev. Mod. Phys. (2011)
Not “real” qubitsStrong light-matter interactions Light-matter interactions enhanced by cavities
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Solid-state quantum memories
Short-lived memories…
Charged Quantum dotsDefects in diamonds
Distant spin-spin entanglement
Long-lived quantum memory (≈ 1𝑠)
B. Hensen et al., Nature (2015)
Spin-photon entanglement
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Photon conversion
Most quantum emitters emits optical single photons
(visible or NIR)
Low-loss fiber= telecom wavelength (1.5µm)
A. Dréau et al., Phys. Rev. App. (2018)
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Conclusion Part 1
Blind quantum computing:• Collaboration without trust• Entanglement
Quantum network
Quantum Key Distribution• First application of QIT• Quantum superposition + measurement• Small trusted network
Applications on privacy (but not only)
Can be a trusted oruntrusted network
Long-distance quantum communications
Via Quantum repeater(with entanglement swapping)
Via Satellite (free space)
Require multiple building blocksThat communicate together
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Questions?
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Outline
Part 1: General tutorial
Part 2: Recent results (if time)
To show that you are able to read state-of-the-art QC papers now!
To learn more about quantum communications (QC)
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Outline
Part 2: Recent results (if time)
“Experimental demonstration of memory-enhanced quantum communication”, M. Bhaskar et al., ArXiv (2019)
“Satellite-based entanglement distribution over 1200 kilometers”, J. Yin et al., Science (2017)
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Presentation Bhaskar et al. (2019)
Impressive results!Combination of QKD and quantum repeaters.Illustrate the variety of quantum protocol schemes
Why this paper?
Prepare:|±⟩ or |𝑖±⟩
Measure
QuantumMemory
Prepare:|±⟩ or |𝑖±⟩
Features:• Only one quantum memory • Cannot be cascaded (no entanglement swapping)• Works only for QKD• The memory sends receives the photons• Time-bin photon encoding / attenuated laser
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Protocol
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Quantum memory node
Quantum memory node:• SiV center 𝑆 = 1/2• Embedded in a cavity• With a measurement device
1 = |𝑙𝑎𝑡𝑒⟩
0 = |𝑒𝑎𝑟𝑙𝑦⟩
Photonic qubit:Time-bin encoding
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Cavity-mediated spin-photon interaction
Eigenstates: polaritons (𝜔𝑐 = 𝜔𝑠)𝐸± = 𝜔 ± 𝑔
| ↑⟩| ↓⟩
𝜎+
Cavity-QED (with a two-level system):𝐻 = 𝜔𝑐𝑎†𝑎 + 𝜔𝑠𝜎
+𝜎− + 𝑔(𝜎+𝑎 + 𝜎−𝑎†)
𝜅
E
R
𝐸− 𝐸+
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Spin-photon entanglement
1
2𝑒𝑎𝑟𝑙𝑦 + 𝑒𝑖𝜙1 𝑙𝑎𝑡𝑒
1
2↑ + ↓
1
2↓ +𝑚1𝑚2𝑒
𝑖𝜙2+𝜙1 ↑
1
2↓ 𝑒𝑎𝑟𝑙𝑦 + 𝑒𝑖𝜙1 ↑ 𝑙𝑎𝑡𝑒
1
2↓ + 𝑚1𝑒
𝑖𝜙1 ↑
After the second photon
Spin measurement in X basis:𝑚3
If 𝜙1 + 𝜙2 ∈ 0, 𝜋 :1
2↑ ± ↓
→ Asynchronous BSM(𝑚1𝑚2𝑚3 = ±1)
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Performances
Overall quantum bit error:𝑄𝐵𝐸𝑅 = 0.097 ± 0.006
Cavity-QED:𝐶 = 105 ± 11
Fidelity:𝐹 = 0.944 ± 0.008
≤ 0.110 Unconditional security
≤ 0.146 Secure against individual attacks
70𝑑𝐵 → 350 𝑘𝑚 of telecom fiber
Entanglement efficiency:𝜂 = 0.423 ± 0.004
4x improvement
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Outline
Part 2: Recent results (if time)
“Experimental demonstration of memory-enhanced quantum communication”, M. Bhaskar et al., ArXiv (2019)
“Satellite-based entanglement distribution over 1200 kilometers”, J. Yin et al., Science (2017)
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Presentation J. Yin et al. (2017)
Record of length for provably secure communicationsIllustration of the E91 protocol.Illustration of technological challenge
Why this paper?
Entangled photons
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Producing entangled photons from space
1
2|𝐻𝑠𝑉𝑖⟩ + |𝑉𝑠𝐻𝑖⟩
Laser pump
1
2|𝐻𝑠𝑉𝑖⟩ + |𝑉𝑠𝐻𝑖⟩
Source
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Send entangled photon from space
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Requirement of a tracking system
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ATP tracking system
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ATP tracking system
Beam divergence 1.2mrad
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Photon detection
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Verifying the satellite is not cheating
Set detectionbasis
Look at the Bell inequalityviolation
𝑆 = |𝐸(𝜙 1 , 𝜙 2 ) – 𝐸(𝜙 1 , 𝜙 2 ′) + 𝐸(𝜙 1 ′, 𝜙 2 ) + 𝐸(𝜙 1 ′, 𝜙 2 ′)| ≤ 2
𝑆 = 2.37 ± 0.09
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Conclusion
Distribution of entanglement over 1200 km12-18 orders of magnitude improvement compared to direct fibered transmission
Rate > 1𝐻𝑧
Only working 275s per day
Fidelity 86.9 ± 8.5%
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Thank you!