![Page 1: Fundamental rate-loss tradeoff for optical quantum key](https://reader030.vdocuments.mx/reader030/viewer/2022012723/61b50baa9a96407cb73d7f8b/html5/thumbnails/1.jpg)
Masahiro Takeoka (NICT)
Saikat Guha (BBN)
Mark M. Wilde (LSU)
Fundamental rate-loss tradeoff for optical quantum key distribution
Quantum Krispy Kreme Seminar @LSU January 30, 2015
![Page 2: Fundamental rate-loss tradeoff for optical quantum key](https://reader030.vdocuments.mx/reader030/viewer/2022012723/61b50baa9a96407cb73d7f8b/html5/thumbnails/2.jpg)
‐Motivation
‐Main result
‐ Problem setting (generic QKD protocol) and proof outline
‐ Future outlook
Outline
![Page 3: Fundamental rate-loss tradeoff for optical quantum key](https://reader030.vdocuments.mx/reader030/viewer/2022012723/61b50baa9a96407cb73d7f8b/html5/thumbnails/3.jpg)
‐ Quantum Key Distribution can generate a shared key perfectly secret against any eavesdropper.
‐ Various (repeaterless) QKD protocols have been proposed so far.
‐ In all known protocols, the key rate decreases linearly with respect to the channel loss.
Motivation
![Page 4: Fundamental rate-loss tradeoff for optical quantum key](https://reader030.vdocuments.mx/reader030/viewer/2022012723/61b50baa9a96407cb73d7f8b/html5/thumbnails/4.jpg)
Various QKD protocols
Scarani et al., Rev. Mod. Phys. 81, 1301 (2009)
Channel loss [dB]
Secret key ra
te
![Page 5: Fundamental rate-loss tradeoff for optical quantum key](https://reader030.vdocuments.mx/reader030/viewer/2022012723/61b50baa9a96407cb73d7f8b/html5/thumbnails/5.jpg)
0 10 20 30 40 50 6010-6
10-4
0.01
1
Loss dB
Secr
etke
yrat
ebi
tspe
rmod
e
- Secret key generation rate for the single-photon effcient BB84Scarani et al., RMP 81, 1301 (2009)
Example1: Ideal single-photon BB84
- Ideal case
(efficient BB84 protocol)Lo et al., J. Crypt. 18, 133 (2006)
Key rate (per mode, pulse)
![Page 6: Fundamental rate-loss tradeoff for optical quantum key](https://reader030.vdocuments.mx/reader030/viewer/2022012723/61b50baa9a96407cb73d7f8b/html5/thumbnails/6.jpg)
0 10 20 30 40 50 6010-6
10-4
0.01
1
Loss dB
Secr
etke
yrat
ebi
tspe
rmod
e
Scarani et al., RMP 81, 1301 (2009)
Example2: CV-QKD (GG02)
- Ideal case
![Page 7: Fundamental rate-loss tradeoff for optical quantum key](https://reader030.vdocuments.mx/reader030/viewer/2022012723/61b50baa9a96407cb73d7f8b/html5/thumbnails/7.jpg)
0 10 20 30 40 50 6010-6
10-4
0.01
1
Loss dB
Secr
etke
yrat
ebi
tspe
rmod
e
Garcia-Patron et al., PRL 102, 210501 (2009)Pirandola et al., PRL 102, 050503 (2009)
Example3: Reverse Coherent Information
: von Neumann entropy of
Reverse coherent information
For a lossy channel
with (infinitely strong) two-mode squeezed vacuum
Garcia-Patron et al., PRL 102, 210501 (2009)
![Page 8: Fundamental rate-loss tradeoff for optical quantum key](https://reader030.vdocuments.mx/reader030/viewer/2022012723/61b50baa9a96407cb73d7f8b/html5/thumbnails/8.jpg)
Question
Is this a fundamental rate‐loss tradeoff in any optical QKD?
Are there yet‐to‐be‐discovered optical QKD protocols that could circumvent the linear rate‐loss tradeoff(without repeaters or trusted notes)?
![Page 9: Fundamental rate-loss tradeoff for optical quantum key](https://reader030.vdocuments.mx/reader030/viewer/2022012723/61b50baa9a96407cb73d7f8b/html5/thumbnails/9.jpg)
Our result
‐We show that this is the rate‐loss trade off is a fundamental limit.
‐We prove that the secret key agreement capacity (private capacity) of a lossy optical channel assisted by two‐way public classical communication is upper bounded by:
0 10 20 30 40 50 6010-6
10-4
0.01
1
Loss dB
Secr
etke
yrat
ebi
tspe
rmod
e
(Rev. coh. info. LB)
MT, S. Guha, M. M. Wilde, Nat. Commun. 5; 5235 (2014)
![Page 10: Fundamental rate-loss tradeoff for optical quantum key](https://reader030.vdocuments.mx/reader030/viewer/2022012723/61b50baa9a96407cb73d7f8b/html5/thumbnails/10.jpg)
‐ Generic point‐to‐point QKD protocol and its capacity (secret key agreement capacity assisted by two‐way public classical communication)
‐ Squashed entanglement of a quantum channel as an upper bound on the two‐way assisted SKA capacity
‐ Pure‐loss optical channel
‐ Loss and noise optical channel
‐ Summary
Rest of the talk
![Page 11: Fundamental rate-loss tradeoff for optical quantum key](https://reader030.vdocuments.mx/reader030/viewer/2022012723/61b50baa9a96407cb73d7f8b/html5/thumbnails/11.jpg)
‐ Generic point‐to‐point QKD protocol and its capacity (secret key agreement capacity assisted by two‐way public classical communication)
‐ Squashed entanglement of a quantum channel as an upper bound on the two‐way assisted SKA capacity
‐ Pure‐loss optical channel
‐ Loss and noise optical channel
‐ Summary
Rest of the talk
![Page 12: Fundamental rate-loss tradeoff for optical quantum key](https://reader030.vdocuments.mx/reader030/viewer/2022012723/61b50baa9a96407cb73d7f8b/html5/thumbnails/12.jpg)
General secret key agreement assisted by unlimited two-way public classical communication
secr
et k
eys secret keys
General secret key agreement assisted by unlimited two-way public classical communication
General point-to-point QKD
![Page 13: Fundamental rate-loss tradeoff for optical quantum key](https://reader030.vdocuments.mx/reader030/viewer/2022012723/61b50baa9a96407cb73d7f8b/html5/thumbnails/13.jpg)
Decoding
Alice Bob
Eve
General point-to-point QKD
![Page 14: Fundamental rate-loss tradeoff for optical quantum key](https://reader030.vdocuments.mx/reader030/viewer/2022012723/61b50baa9a96407cb73d7f8b/html5/thumbnails/14.jpg)
Decoding
Alice Bob
Eve
General point-to-point QKD
![Page 15: Fundamental rate-loss tradeoff for optical quantum key](https://reader030.vdocuments.mx/reader030/viewer/2022012723/61b50baa9a96407cb73d7f8b/html5/thumbnails/15.jpg)
Decoding
Alice Bob
two‐way classical communication
Eve
General point-to-point QKD
![Page 16: Fundamental rate-loss tradeoff for optical quantum key](https://reader030.vdocuments.mx/reader030/viewer/2022012723/61b50baa9a96407cb73d7f8b/html5/thumbnails/16.jpg)
Decoding
Alice Bob
Eve
General point-to-point QKD
![Page 17: Fundamental rate-loss tradeoff for optical quantum key](https://reader030.vdocuments.mx/reader030/viewer/2022012723/61b50baa9a96407cb73d7f8b/html5/thumbnails/17.jpg)
Decoding
Alice Bob
two‐way classical communication
Eve
General point-to-point QKD
![Page 18: Fundamental rate-loss tradeoff for optical quantum key](https://reader030.vdocuments.mx/reader030/viewer/2022012723/61b50baa9a96407cb73d7f8b/html5/thumbnails/18.jpg)
Decoding
Alice Bob
Eve
n‐quantum channels
General point-to-point QKD
![Page 19: Fundamental rate-loss tradeoff for optical quantum key](https://reader030.vdocuments.mx/reader030/viewer/2022012723/61b50baa9a96407cb73d7f8b/html5/thumbnails/19.jpg)
Decoding
Alice Bob
two‐way classical communication
Eve
n‐quantum channels
General point-to-point QKD
kse
cret
key
s ksecret keys
Secret key generation rate: R=k/n
![Page 20: Fundamental rate-loss tradeoff for optical quantum key](https://reader030.vdocuments.mx/reader030/viewer/2022012723/61b50baa9a96407cb73d7f8b/html5/thumbnails/20.jpg)
‐ Generic point‐to‐point QKD protocol and its capacity (secret key agreement capacity assisted by two‐way public classical communication)
‐ Squashed entanglement of a quantum channel as an upper bound on the two‐way assisted SKA capacity
‐ Pure‐loss optical channel
‐ Loss and noise optical channel
‐ Summary
Rest of the talk
![Page 21: Fundamental rate-loss tradeoff for optical quantum key](https://reader030.vdocuments.mx/reader030/viewer/2022012723/61b50baa9a96407cb73d7f8b/html5/thumbnails/21.jpg)
Upper bound on the key rate: squashed entanglement of a quantum channel
: Squashed entanglement of a quantum channel
: Squashed entanglement (of a bipartite state )
Christandl, Winter, J. Math. Phys. 45, 829 (2004)
MT, Guha, Wilde, IEEE Trans. Info. Theory 60, 4987 (2014)
![Page 22: Fundamental rate-loss tradeoff for optical quantum key](https://reader030.vdocuments.mx/reader030/viewer/2022012723/61b50baa9a96407cb73d7f8b/html5/thumbnails/22.jpg)
Squashed entanglementSquashed entanglement: Esq(A;B)
conditional quantum mutual information
Christandl, Winter, J. Math. Phys. 45, 829 (2004)
A B
E
Channel S should be chosen to squash the quantum correlations between Alice and Bob (the squashing channel)E’
- Entanglement measure for a bipartite state (LOCC monotone, …)- Inspired by secrecy capacity upper bound in classical theory (intrinsic information)
![Page 23: Fundamental rate-loss tradeoff for optical quantum key](https://reader030.vdocuments.mx/reader030/viewer/2022012723/61b50baa9a96407cb73d7f8b/html5/thumbnails/23.jpg)
Squashed entanglement of a quantum channel
Definition:
where
AA’
B
![Page 24: Fundamental rate-loss tradeoff for optical quantum key](https://reader030.vdocuments.mx/reader030/viewer/2022012723/61b50baa9a96407cb73d7f8b/html5/thumbnails/24.jpg)
Main theorem
Proof outline
Esq(N) is an upper bound on the secret key generation rate R
1. Secret key distillation upper bound Christandl, et al., arXiv:quant-ph/0608119
2. New subadditivity inequality for the squashed entanglement
Theorem1
MT, Guha, Wilde, IEEE Trans. Info. Theory 60, 4987 (2014)
![Page 25: Fundamental rate-loss tradeoff for optical quantum key](https://reader030.vdocuments.mx/reader030/viewer/2022012723/61b50baa9a96407cb73d7f8b/html5/thumbnails/25.jpg)
Proof outline
1. Secret key distillation upper bound
Christandl, et al., arXiv:quant-ph/0608119
1. Monotonicity (does not increase under LOPC)
2. Continuity: if then
3. Normalization:
4. Subadditivity on tensor product states:
: private state Horodecki et al, PRL 94, 160502 (2005)
Theorem 3.7. Squashed entanglement is an upper bound on the distillable key rate from a tensor product state
The statement is proved by using the following four properties:
The similar technique is applicable to the channel scenario except 4.
![Page 26: Fundamental rate-loss tradeoff for optical quantum key](https://reader030.vdocuments.mx/reader030/viewer/2022012723/61b50baa9a96407cb73d7f8b/html5/thumbnails/26.jpg)
Proof outline
4. Subadditivity on tensor product states:
Product state
Could be entangledover n‐channel use!
?Can be replaced by
![Page 27: Fundamental rate-loss tradeoff for optical quantum key](https://reader030.vdocuments.mx/reader030/viewer/2022012723/61b50baa9a96407cb73d7f8b/html5/thumbnails/27.jpg)
AB1
B2
?
Proof outline
Subadditivity of Esq(N) ?
is true if one can show something like
AB1
AB2
![Page 28: Fundamental rate-loss tradeoff for optical quantum key](https://reader030.vdocuments.mx/reader030/viewer/2022012723/61b50baa9a96407cb73d7f8b/html5/thumbnails/28.jpg)
Monogamy of entanglement
Proof outline
Subadditivity of Esq ?is true if one can show something like
AB1
B2
AB1
AB2
Koashi and Winter, Phys. Rev. A 69, 022309 (2004)
![Page 29: Fundamental rate-loss tradeoff for optical quantum key](https://reader030.vdocuments.mx/reader030/viewer/2022012723/61b50baa9a96407cb73d7f8b/html5/thumbnails/29.jpg)
Proof outline
New subadditivity-like inequality
Proof consists of a chain of (in)equalities based on
-Duality of conditional entropy for
-Strong subadditivity for
is not possible.
However, we are able to show the following inequality:
For any five-party pure state
holds.
Lemma
MT, Guha, Wilde, IEEE Trans. Info. Theory 60, 4987 (2014)
![Page 30: Fundamental rate-loss tradeoff for optical quantum key](https://reader030.vdocuments.mx/reader030/viewer/2022012723/61b50baa9a96407cb73d7f8b/html5/thumbnails/30.jpg)
implies
Proof outline
Additivity of
![Page 31: Fundamental rate-loss tradeoff for optical quantum key](https://reader030.vdocuments.mx/reader030/viewer/2022012723/61b50baa9a96407cb73d7f8b/html5/thumbnails/31.jpg)
Proof outline
1. Monotonicity (does not increase under LOPC)
2. Continuity: if then
3. Normalization:
4. Additivity:
: private state
For the details of the proof, see
From the following four conditions:
One can show
MT, Guha, Wilde, IEEE Trans. Info. Theory 60, 4987 (2014)
![Page 32: Fundamental rate-loss tradeoff for optical quantum key](https://reader030.vdocuments.mx/reader030/viewer/2022012723/61b50baa9a96407cb73d7f8b/html5/thumbnails/32.jpg)
‐ Generic point‐to‐point QKD protocol and its capacity (secret key agreement capacity assisted by two‐way public classical communication)
‐ Squashed entanglement of a quantum channel as an upper bound on the two‐way assisted SKA capacity
‐ Pure‐loss optical channel
‐ Loss and noise optical channel
‐ Summary
Rest of the talk
![Page 33: Fundamental rate-loss tradeoff for optical quantum key](https://reader030.vdocuments.mx/reader030/viewer/2022012723/61b50baa9a96407cb73d7f8b/html5/thumbnails/33.jpg)
Lossy bosonic channel
Need to find a good squashing channel (in a heuristic way...)
Pure‐loss squashing channel
Alice Bob
Eve
Lossy bosonic channel
minimized at
maximized with
(Two‐mode squeezed vacuum)
![Page 34: Fundamental rate-loss tradeoff for optical quantum key](https://reader030.vdocuments.mx/reader030/viewer/2022012723/61b50baa9a96407cb73d7f8b/html5/thumbnails/34.jpg)
Lossy bosonic channel
Ns: a mean input power (average photon number of one share of the TMSV)
![Page 35: Fundamental rate-loss tradeoff for optical quantum key](https://reader030.vdocuments.mx/reader030/viewer/2022012723/61b50baa9a96407cb73d7f8b/html5/thumbnails/35.jpg)
‐ Generic point‐to‐point QKD protocol and its capacity (secret key agreement capacity assisted by two‐way public classical communication)
‐ Squashed entanglement of a quantum channel as an upper bound on the two‐way assisted SKA capacity
‐ Pure‐loss optical channel
‐ Loss and noise optical channel
‐ Summary
Rest of the talk
![Page 36: Fundamental rate-loss tradeoff for optical quantum key](https://reader030.vdocuments.mx/reader030/viewer/2022012723/61b50baa9a96407cb73d7f8b/html5/thumbnails/36.jpg)
Loss and thermal noise channel
Alice Bob
Channel model
Decomposition of the phase-insensitive Gaussian channel
Caruso, Giovannetti, Holevo, NJP 8, 310 (2006)Garcia-Patron et al., PRL 108, 110505 (2012)
Alice BobT
Pure‐loss ch
Amplifier ch
G
![Page 37: Fundamental rate-loss tradeoff for optical quantum key](https://reader030.vdocuments.mx/reader030/viewer/2022012723/61b50baa9a96407cb73d7f8b/html5/thumbnails/37.jpg)
Loss and thermal noise channel
Data processing inequality for quantum conditional mutual information
Out[17]=
0 5 10 15 20
0.050.10
0.501.00
5.0010.00
Loss dB
Secr
etke
yra
te
Esq UB NB=0
Esq UB NB=0.3
Esq UB NB=0.5
RCI LB NB=0RCI LB NB=0.3RCI LB NB=0.5
![Page 38: Fundamental rate-loss tradeoff for optical quantum key](https://reader030.vdocuments.mx/reader030/viewer/2022012723/61b50baa9a96407cb73d7f8b/html5/thumbnails/38.jpg)
Summary
‐ The secret key rate of any repeaterless QKD protocols in a lossy optical channel is upper bounded by
‐ The bound is based on the squashed entanglement of a quantum channel, which is a general upper bound on the two‐way classically assisted secret key agreement capacity.
‐ Open problems‐ True two‐way assisted capacity?‐ Tight bound for noisy channel?‐ Finite block code analysis
(weak converse)
(needs strong converse or second order analysis)
MT, Guha, Wilde, Nat. Commun. 5; 5235 (2014)MT, Guha, Wilde, IEEE Trans. Info. Theory 60, 4987 (2014)
![Page 39: Fundamental rate-loss tradeoff for optical quantum key](https://reader030.vdocuments.mx/reader030/viewer/2022012723/61b50baa9a96407cb73d7f8b/html5/thumbnails/39.jpg)
Finite n analysis
‐ Our upper bound is a weak converseFor a tight upper bound on finite block length, a strong converse or a second order analysis should be established.
‐ However, we can estimate the effect of finite block lengthfrom our result.
![Page 40: Fundamental rate-loss tradeoff for optical quantum key](https://reader030.vdocuments.mx/reader030/viewer/2022012723/61b50baa9a96407cb73d7f8b/html5/thumbnails/40.jpg)
Finite n analysis
: secrecy (based on the trace distance criteria)
n: code length
200 km fiber (0.2dB/km loss)Example in a pure‐loss optical channel:
MT, Guha, Wilde, Nat. Commun. 5; 5235 (2014)
![Page 41: Fundamental rate-loss tradeoff for optical quantum key](https://reader030.vdocuments.mx/reader030/viewer/2022012723/61b50baa9a96407cb73d7f8b/html5/thumbnails/41.jpg)
Few more slides…
Extension of the results to
‐ Quantum repeaters‐Multipartite secret key sharing
Ads..
‐ QCrypt2015‐ Summer internship at NICT
![Page 42: Fundamental rate-loss tradeoff for optical quantum key](https://reader030.vdocuments.mx/reader030/viewer/2022012723/61b50baa9a96407cb73d7f8b/html5/thumbnails/42.jpg)
Upper bound on quantum repeaters
Quantum communication with N repeater stations:
Alice Bob
![Page 43: Fundamental rate-loss tradeoff for optical quantum key](https://reader030.vdocuments.mx/reader030/viewer/2022012723/61b50baa9a96407cb73d7f8b/html5/thumbnails/43.jpg)
Upper bound on quantum repeaters
Quantum communication with N repeater stations:
Alice Bob
N=0
N=1
N=2
N=3
0 50 100 150 2000.01
0.050.10
0.501.00
5.00
Channel loss in dB =-10log10h
Secr
etke
yra
teb
itspe
rmod
e
Compare with the existing repeater protocols?
![Page 44: Fundamental rate-loss tradeoff for optical quantum key](https://reader030.vdocuments.mx/reader030/viewer/2022012723/61b50baa9a96407cb73d7f8b/html5/thumbnails/44.jpg)
Multi-party key distribution
Alice
Bob
Charlie…K
K
K
Broadcast quantum channel
Squashed entanglement of a quantum broadcast channel
Key rate upper bound
‐ Channel examples?‐ How tight?
![Page 45: Fundamental rate-loss tradeoff for optical quantum key](https://reader030.vdocuments.mx/reader030/viewer/2022012723/61b50baa9a96407cb73d7f8b/html5/thumbnails/45.jpg)
Few more slides…
Extension of the results to
‐ Quantum repeaters‐Multipartite secret key sharing
Ads..
‐ QCrypt2015‐ Summer internship at NICT
![Page 46: Fundamental rate-loss tradeoff for optical quantum key](https://reader030.vdocuments.mx/reader030/viewer/2022012723/61b50baa9a96407cb73d7f8b/html5/thumbnails/46.jpg)
QCrypt2015 & UQCC2015
QCrypt2015 (4.5days)
Tokyo, Japan September 28 ‐ October 2, 2015
UQCC2015 (0.5days, Sep 28)
http://2015.uqcc.org/
http://2015.qcrypt.net/
Discuss the latest theoryand experiments on quantum cryptography and related fields
Invites engineers, potential users, media etc, to show the SOTA QKD performance & applications and discuss the possible business solutions
QKD platform
![Page 47: Fundamental rate-loss tradeoff for optical quantum key](https://reader030.vdocuments.mx/reader030/viewer/2022012723/61b50baa9a96407cb73d7f8b/html5/thumbnails/47.jpg)
Where? Tokyo, Japan
Baton Rouge
Tokyo
Central area of Tokyo
QCrypt2015 & UQCC2015September 28 ‐ October 2, 2015
![Page 48: Fundamental rate-loss tradeoff for optical quantum key](https://reader030.vdocuments.mx/reader030/viewer/2022012723/61b50baa9a96407cb73d7f8b/html5/thumbnails/48.jpg)
Central are of Tokyo
Imperial Palace
Akihabara
Shinjuku
Asakusa Temple
QCrypt2015 & UQCC2015
Oral presentation submission deadline: April 27 http://2015.qcrypt.net/
![Page 49: Fundamental rate-loss tradeoff for optical quantum key](https://reader030.vdocuments.mx/reader030/viewer/2022012723/61b50baa9a96407cb73d7f8b/html5/thumbnails/49.jpg)
Summer internship at Quantum ICT Lab in NICT
http://www.nict.go.jp/en/
QKD network, architecture, applications
Quantum optics: Entangled sources, PNRD, QIP protocols
Collaborative projects
‐Optical space communication G‐Nano ICT G‐Frequency standard G
Theory: quantum information theory, quantum optics, physical layer security…
LeaderMasahideSasaki