1. Channel Coding for Quantum Key Distribution
Gottfried Lechner
gottfried.lechner@unisa.edu.au
Institute for Telecommunications Research
University of South Australia
July 7, 2011
HiPANQ Workshop, Vienna
1 / 26

2. Outline
Basics
Channel Coding
Slepian-Wolf Coding
Binning and the Dual Channel
Linear Block Codes, Syndrome and Rates
QKD Reconciliation
System Setup
Example
Optimisation
Conclusions
2 / 26

3. Shannon: Commiunication in the Presence of Noise
Channel Coding1949 the channel capacity may be defined as
ony, this operation consists of merely changing sound
pressure into a proportional electrical current. In teleg-
,= T-oo
C9g2 M
T
T--a T
A precise meaning will be given later to
of reliable resolution of the M signals.
II. THE SAMPLING THEOR
Let us suppose that the channel has
width W in cps starting at zero frequen
are allowed to use this channel for a c
Fig. 1-General communications system. time T. Without any further restrict
mean that we can use as signal functio
of time whose spectra lie entirely with
raphy, we have an encoding operation which produces a and whose time functions lie within th
transmit data reliably from source to
sequence of dots, dashes, and spaces destination though it is not possible to fulfill both
corresponding to
the letters of the message. To take a more complex tions exactly, it is possible to keep the
maximise the example, speech functionsmultiplex sampled, compressed, outside theW, and PeT.< the time fun
different
in the R of
c must be
PCM telephony the the of
code rate case such that the probability banderror to have we describe interval Can
quantized and encoded, and finally interleaved properly way the functions which satisfy these
the maximum to constructgiven by the capacity of the answer is the following:
rate is the signal. channel C
3. The channel. This is merely the medium used to THEOREM 1: If a function f(t) contai
transmit the signal from the transmitting to the receiv- higher than W cps, it is completely dete
ing point. It may be a pair of wires, a coaxial cable, a its ordinates at a series of points space
band of radio frequencies, etc. During transmission, or apart.
at the receiving terminal, the signal may be perturbed This is a fact which is common knowl
by noise or distortion. Noise and distortion may be dif- munication art. The intuitive justificat
ferentiated on the basis that distortion is a fixed opera- contains no frequencies higher than
tion applied to the signal, while noise involves statistical change to a substantially new value in
and unpredictable perturbations. Distortion can, in one-half cycle of the highest frequency,
principle, be corrected by applying the inverse opera- mathematical proof showing that this
tion, while a perturbation due to noise cannot always be proximately, but exactly, true can be
removed, since the signal does not always undergo the Let F(w) be the spectrum of f(t). Then
same change during transmission.
4. The receiver. This operates on the received signal 1 a00 3 / 26

4. Shannon: Commiunication in the Presence of Noise
Channel Coding1949 the channel capacity may be defined as
ony, this operation consists of merely changing sound
pressure into a proportional electrical current. In teleg-
,= T-oo
C9g2 M
T
T--a T
A precise meaning will be given later to
of reliable resolution of the M signals.
II. THE SAMPLING THEOR
Let us suppose that the channel has
width W in cps starting at zero frequen
are allowed to use this channel for a c
Fig. 1-General communications system. time T. Without any further restrict
mean that we can use as signal functio
of time whose spectra lie entirely with
raphy, we have an encoding operation which produces a and whose time functions lie within th
transmit data reliably from source to
sequence of dots, dashes, and spaces destination though it is not possible to fulfill both
corresponding to
the letters of the message. To take a more complex tions exactly, it is possible to keep the
maximise the example, speech functionsmultiplex sampled, compressed, outside theW, and PeT.< the time fun
different
in the R of
c must be
PCM telephony the the of
code rate case such that the probability banderror to have we describe interval Can
quantized and encoded, and finally interleaved properly way the functions which satisfy these
the maximum to constructgiven by the capacity of the answer is the following:
rate is the signal. channel C
3. The channel. This is merely the medium used to THEOREM 1: If a function f(t) contai
transmit the signal from the transmitting to the receiv- higher than W cps, it is completely dete
ing point. It may be a pair of wires, a coaxial cable, a its ordinates at a series of points space
Channel Coding Theorem [Shannon 1948]
band of radio frequencies, etc. During transmission, or apart.
at the receiving terminal, the signal may be perturbed This is a fact which is common knowl
For any > 0 and by c <orC, for largeand distortion may be dif- munication a code of justificat
Rnoise distortion. Noise enough N, there exists art. The intuitive
ferentiated on the basis that distortion is a fixed opera- contains no frequencies higher than
length N and rate Rc and a decoding algorithm, such that thesubstantially new value in
tion applied to the signal, while noise involves statistical change to a maximal
probability of blockand unpredictable perturbations. Distortion can, in one-half cycle of the highest frequency,
error is less than .
principle, be corrected by applying the inverse opera- mathematical proof showing that this
tion, while a perturbation due to noise cannot always be proximately, but exactly, true can be
removed, since the signal does not always undergo the Let F(w) be the spectrum of f(t). Then
same change during transmission.
4. The receiver. This operates on the received signal 1 a00 3 / 26

5. 4 Simplifying Component Decoders
Typical Approach
choose a family of channels with a single parameter (e.g.,
AWGN, BSC, BEC,...)
fix a code rate
optimise the code such that it achieves vanishing error probability
close to capacity
0
10
SPA
MSA
MSA − variable scaling
−1
MSA − fixed scaling 0.60
10 MSA − fixed scaling 0.70
MSA − universal
−2
10
bit error rate
−3
10
−4
10
−5
10
−6
10
0.5 1 1.5 2 2.5
Eb
N0
Figure 4.10: Bit error rates for irregular code with and without post-processing.
4 / 26

6. DAVID SLEPIAN AND JACK K. WOLF
Slepian-Wolf Coding
ation sequences . . .,X- 1,X0,XI,. . . and -x-( ,xg .X,,“’ x “~01101~~~ ..x-:,x,*,x;l-.
ed by repeated independent drawings of - ENCODER RATE RX 0
bles X, Y from a given bivariate distribu- E
C
e minimum number of bits per character CORRELATED
0
SOURCES
se sequences that they can be faithfully
so D
assumptions regarding the encoders and “Ye, ,Y, .Y,;.. Y “‘11000..~ ; ..Y-,*,Yo*8Y,p...
hich are not at all obvious, are presented ENCODER RATE RY
in the Rx-Ry plane. They generalize a
or a single information sequence,namely Fig. 1. Correlated source coding configuration.
uction.
TRODUCTION
tement transmit two correlated sources over two noiseless channels
generalize, to the caseencoding and decoding: H(X, Y)
joint of two
rtain well-known results on the
separate encoding and decoding: H(X) + H(Y) ≥
ngle discrete information source. H(X, Y)
onsideredis that depictedin Fig. 1.
information sequences. * .,X- 1,
,,,Y,, . . . are obtained by repeated
m a discrete bivariate distribution
ch sourceis constrained to operate
e other source, while the decoder HtXIY) H(X) H(X,Y) RX
ded binary messagestreams. We Fig. 2. Admissible rate region W corresponding to Fig. 1
umber of bits per sourcecharacter
oded messagestreams in order to
ensureaccuratereconstruction by the decoderof the outputs
of both information sources.The results are presentedas an
25, 1972; revised December 28, 1972.
sity of Hawaii, Honolulu, Hawaii, and allowed two-dimensional rate region 93for the two encoded
Hill, N.J. 07974. message streamsas shown in Fig. 2. Note that in 93for this
ersity of Hawaii, Honolulu, Hawaii, on
itute of Brooklyn, Brooklyn, N.Y. case we can have both R, < H(X) and R, -c H(Y) al-
5 / 26

7. DAVID SLEPIAN AND JACK K. WOLF
Slepian-Wolf Coding
ation sequences . . .,X- 1,X0,XI,. . . and -x-( ,xg .X,,“’ x “~01101~~~ ..x-:,x,*,x;l-.
ed by repeated independent drawings of - ENCODER RATE RX 0
bles X, Y from a given bivariate distribu- E
C
e minimum number of bits per character CORRELATED
0
SOURCES
se sequences that they can be faithfully
so D
assumptions regarding the encoders and “Ye, ,Y, .Y,;.. Y “‘11000..~ ; ..Y-,*,Yo*8Y,p...
hich are not at all obvious, are presented ENCODER RATE RY
in the Rx-Ry plane. They generalize a
or a single information sequence,namely Fig. 1. Correlated source coding configuration.
uction.
TRODUCTION
tement transmit two correlated sources over two noiseless channels
generalize, to the caseencoding and decoding: H(X, Y)
joint of two
rtain well-known results on the
separate encoding and decoding: H(X) + H(Y) ≥ H(X, Y)
ngle discrete information source.
onsideredis that depictedin Fig. 1.
information sequences. * .,X- 1,
Slepian-Wolf Theorem (1973)
,,,Y,, . . . are obtained by repeated
m a discrete bivariate distribution rate region is given by the rate pairs satisfying
The admissible
ch sourceis constrained to operate
e other source, while the decoder HtXIY) H(X) H(X,Y) RX
ded binary messagestreams. We Rx ≥ H(X|Y)
Fig. 2. Admissible rate region W corresponding to Fig. 1
umber of bits per sourcecharacter
oded messagestreams in order to Ry ≥ H(Y|X)
ensureaccuratereconstruction by the decoderof the outputs
of both information + Ry ≥ H(X, are presentedas an
Rx sources.The results Y)
25, 1972; revised December 28, 1972.
sity of Hawaii, Honolulu, Hawaii, and allowed two-dimensional rate region 93for the two encoded
Hill, N.J. 07974. message streamsas shown in Fig. 2. Note that in 93for this
ersity of Hawaii, Honolulu, Hawaii,penalty if X and Y are encoded separately!
There is N.Y. on case we can have both R, < H(X) and R, -c H(Y) al-
itute of Brooklyn, Brooklyn, no
5 / 26

8. nimum number of bits per character CORRELATED
0
SOURCES
quences that they can be faithfully
so D
mptions regarding the encoders and
Slepian-Wolf Coding
are not at all obvious, are presented
“Ye, ,Y, .Y,;.. Y
ENCODER
“‘11000..~
RATE RY
; ..Y-,*,Yo*8Y,p...
he Rx-Ry plane. They generalize a
single information sequence,namely Fig. 1. Correlated source coding configuration.
n.
DUCTION
ent
neralize, to the case of two
n well-known results on the
discrete information source.
deredis that depictedin Fig. 1.
ormation sequences. * .,X- 1,
, . . . are obtained by repeated
discrete bivariate distribution
ourceis constrained to operate
her source, while the decoder HtXIY) H(X) H(X,Y) RX
binary messagestreams. We Fig. 2. Admissible rate region W corresponding to Fig. 1
ber of bits per sourcecharacter
d messagestreams in order to
ensureaccuratereconstruction by the decoderof the outputs
of both information sources.The results are presentedas an
1972; revised December 28, 1972.
of Hawaii, Honolulu, Hawaii, and allowed two-dimensional rate region 93for the two encoded
N.J. 07974. message streamsas shown in Fig. 2. Note that in 93for this
y of Hawaii, Honolulu, Hawaii, on
of Brooklyn, Brooklyn, N.Y. case we can have both R, < H(X) and R, -c H(Y) al-
rsity of South Australia. Downloaded on January 17, 2009 at 20:51 from IEEE Xplore. Restrictions apply.
6 / 26

9. nimum number of bits per character CORRELATED
0
SOURCES
quences that they can be faithfully
so D
mptions regarding the encoders and
Slepian-Wolf Coding
are not at all obvious, are presented
“Ye, ,Y, .Y,;.. Y
ENCODER
“‘11000..~
RATE RY
; ..Y-,*,Yo*8Y,p...
he Rx-Ry plane. They generalize a
single information sequence,namely Fig. 1. Correlated source coding configuration.
n.
DUCTION
ent
neralize, to the case of two
n well-known results on the
discrete information source.
deredis that depictedin Fig. 1.
ormation sequences. * .,X- 1,
, . . . are obtained by repeated
discrete bivariate distribution
ourceis constrained to operate
her source, while the decoder HtXIY) H(X) H(X,Y) RX
binary messagestreams. We Fig. 2. Admissible rate region W corresponding to Fig. 1
ber of bits per sourcecharacter
d messagestreams in order to
ensureaccuratereconstruction by the decoderof the outputs
assume thatbothis transmitted at H(Y) are presentedas an
of Y information sources.The results
1972; revised December 28, 1972.
we operate at a two-dimensional rateFig. 2. Note that two encoded
of Hawaii, Honolulu, Hawaii, and
N.J. 07974.
allowed
message corner point in region Slepian-Wolfthis
streamsas shown of the
93for the
in 93for region
y of Hawaii, Honolulu, Hawaii, on
for this corner we can have can R, < H(X) syndrome of al- channel code
of Brooklyn, Brooklyn, N.Y. case point we both use the and R, -c H(Y) a
as a binning scheme
rsity of South Australia. Downloaded on January 17, 2009 at 20:51 from IEEE Xplore. Restrictions apply.
6 / 26

10. Binning with Syndrome
Bin 1 Bin 2 Bin 3
encoding of X can be done by random binning
the syndrome of a linear code is used for binning
7 / 26

11. Dual Channel
Correlated sources:
assume sources X and Y with P(X, Y) = P(X)P(Y|X)
generate X according to P(X)
transmit X over the channel P(Y|X) to obtain Y
8 / 26

12. Dual Channel
Correlated sources:
assume sources X and Y with P(X, Y) = P(X)P(Y|X)
generate X according to P(X)
transmit X over the channel P(Y|X) to obtain Y
What is the channel that is seen by the channel decoder?
in general it is the dual channel which is not equal to P(Y|X) nor
P(X|Y)
the channel seen by the decoder is always a symmetric channel
with uniform input
therefore, linear codes can be used
for the simple case of two binary sources correlated via a BSC all
these channels are the same
8 / 26

13. Linear Block Codes, Syndrome and Rates
x
N
C= x ∈ {0, 1} xHT = 0
N M
Rc = N−M
N =1− M
N
9 / 26

14. Linear Block Codes, Syndrome and Rates
x
N
C= x ∈ {0, 1} xHT = 0
N M
Rc = N−M
N =1− M
N
x
s
N
Cs = x ∈ {0, 1} xHT = s
N M
Rs = M
N = 1 − Rc
efficiency parameter f = M
H(X|Y) = M
NH(X|Y) = Rs
H(X|Y)
9 / 26

15. LDPC Codes
N
 
0 0 0 0 0 0 0 0 1 0 1 0 1 0 1 0
 0 0 0 0 0 0 1 1 0 1 0 1 0 0 0 0 
 
 0 1 1 0 0 0 0 1 0 0 1 0 0 0 0 0 
 
 1 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 
H=


 M
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 
 0 0 0 0 1 0 0 0 0 1 0 0 1 1 0 0 
 
 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0 1  dc
0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 1
dv
10 / 26

16. Outline
Basics
Channel Coding
Slepian-Wolf Coding
Binning and the Dual Channel
Linear Block Codes, Syndrome and Rates
QKD Reconciliation
System Setup
Example
Optimisation
Conclusions
11 / 26

17. Quantum Key Distribution
Public
Channel
Alice Bob
X Y
Quantum
Channel
Alice and Bob generate a common key
they communicate via a quantum channel and a public channel
Eve attempts to gain knowledge of the key
12 / 26

18. System Setup
X Public
Encoder
(Alice) Channel
Bob
Y
the quantum channel creates a correlated source
Alice observes X and Bob observes Y
Alice has to communicate at least H(X|Y) over the public channel
this corresponds to the corner point of the Slepian-Wolf region
13 / 26

19. Aims
Aims of QKD:
Alice and Bob want to create a common key
the goal is to maximise the key generation rate
this does not necessarily require error free communication (as
long as the errors are detectable)
the key generation rate can be limited by
the quantum channel (quantum source)
the data rate over the public channel
the processing capabilities of Bob
14 / 26

20. Example
Algorithm 1 Algorithm 2
WER
1 1
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
0 0
0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.15 0.2 0.25 0.3 0.35 0.4 0.45
rs rs
15 / 26

21. Example
Algorithm 1 Algorithm 2
WER
1 1
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
0 0
0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.15 0.2 0.25 0.3 0.35 0.4 0.45
rs rs
15 / 26

22. Example
Algorithm 1 Algorithm 2
WER Keyrate
1 1
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
0 0
0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.15 0.2 0.25 0.3 0.35 0.4 0.45
rs rs
15 / 26

23. Example
Algorithm 1 Algorithm 2
WER Keyrate Time
1 1
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
0 0
0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.15 0.2 0.25 0.3 0.35 0.4 0.45
rs rs
15 / 26

24. Optimisation Problem
maximum achievable key rate
rk,max = fk (rs , pX,Y )
word error probability
pe = fe (rs , pX,Y , A)
decoding complexity
td = ft (rs , pX,Y , A)
16 / 26

25. Optimisation Problem
maximum achievable key rate
rk,max = fk (rs , pX,Y )
word error probability
pe = fe (rs , pX,Y , A)
decoding complexity
td = ft (rs , pX,Y , A)
Optimisation Problem
rk = max {rk,max (1 − pe )}
subject to td < td,max
where the maximisation is taken over
0 < rs < 0.5
all decoding algorithms A
16 / 26

26. Optimisation
The decoding algorithm can either be
fixed
chosen from a fixed set of algorithms
adaptively changed during the decoding process (e.g., gear-shift
decoding)
The coding rate can either be
fixed
chosen from a fixed set of rates (rate-compatible codes)
adaptively changed during the decoding process (rateless codes)
17 / 26

27. Message-Passing Decoders
Lch Lcv,j Lvc,j
Lvc,i Lcv,i
Sum-Product Algorithm (SPA)
dv dc
Lvc,j
Lvc,i = Lch + Lcv,j Lcv,i = 2 tanh−1 tanh
2
j=1 j=i j=1 j=i
Min-Sum Algorithm (MSA)
dv dc
Lvc,i = Lch + Lcv,j Lcv,i = α · min |Lvc,j | · sign(Lvc,j )
j=i
j=1 j=i j=1 j=i
Binary Message-Passing (BMP)
mvc,i = majority(mch , mcv,j ) mcv,i = xor mvc,j
j=i
18 / 26

28. Gear-Shift Decoding
1238 IEEE
labeled
rithms in
.T
gear-shif
available
quence c
panded g
E. Conv
For eq
gence th
Fig. 2. Simple gear-shifting trellis with of size six and three algorithms. than the
Notice that some vertices have fewer than three outgoing edges; this happens
when some algorithms have a closed EXIT chart at this message-error rate, or
chooses
when two algorithms result in a parallel edge (in which case, only the lower sulting E
complexity algorithm is retained). and henc
In the
from Ardakani and Kschischang, “Gear-shift decoding,” IEEE Trans. Com. 2006 timum ge
Clearly, every gear-shifting sequence corresponds to a path complex
19 / 26

29. Fixed Rate vs Rateless
error rate on the quantum channel known and large block length
transmit syndrome over public channel and discard key if
decoding is not successful (one bit feedback)
error rate on the quantum channel varies
not enough data on the public channel leads to high error rate
too much data on the public channel reduces the keyrate
20 / 26

30. Literature
Information Theory
David Slepian and Jack K Wolf.
Noiseless coding of correlated information sources.
IEEE Transactions on Information Theory, 19(4):471 – 480, 1973.
Aaron D Wyner.
Recent results in the Shannon theory.
IEEE Transactions on Information Theory, 20(1):2 – 10, 1974.
Jun Chen, Da ke He, and Ashish Jagmohan.
On the duality between Slepian–Wolf coding and channel coding under mismatched
decoding.
IEEE Transactions on Information Theory, 55(9):4006 – 4018, 2009.
21 / 26

31. Literature
Coding
Robert G Gallager.
Low-density parity-check codes.
IEEE Transactions on Information Theory, 8(1):21 – 28, 1962.
Michael Luby.
LT codes.
In IEEE Symposium on Foundations of Computer Science, 2002, pages 271 – 280, 2002.
Amin Shokrollahi.
Raptor codes.
IEEE Transactions on Information Theory, 52(6):2551 – 2567, 2006.
T. Richardson and R. Urbanke.
Modern Coding Theory.
Cambridge University Press, 2008.
22 / 26

32. Literature
QKD Basics
Gilles Brassard and Louis Salvail.
Secret-key reconciliation by public discussion.
In Advances in Cryptology EUROCRYPT’93, pages 410–423, 1994.
Tomohiro Sugimoto and Kouichi Yamazaki.
A study on secret key reconciliation protocol ”cascade”.
In IEICE Transactions on Fundamentals of Electronics, Communications and Computer
Sciences, volume E83-A, pages 1987–1991, 2000.
W T Buttler, S K Lamoreaux, J R Torgerson, G H Nickel, C H Donahue, and C G Peterson.
Fast, efficient error reconciliation for quantum cryptography.
arXiv, quant-ph, 2002.
Hao Yan, Xiang Peng, Xiaxiang Lin, Wei Jiang, Tian Liu, and Hong Guo.
Efficiency of Winnow protocol in secret key reconciliation.
In Computer Science and Information Engineering, 2009 WRI World Congress on, volume 3,
pages 238 – 242, 2009.
23 / 26

33. Literature
Coding for QKD (non-exhaustive)
David Elkouss, Anthony Leverrier, Romain Alleaume, and Joseph J Boutros.
Efficient reconciliation protocol for discrete-variable quantum key distribution.
In International Symposium on Information Theory, pages 1879–1883, 2009.
David Elkouss, Jesus Martinez-Mateo, Daniel Lancho, and Vicente Martin.
Rate compatible protocol for information reconciliation: An application to QKD.
In Information Theory Workshop (ITW), 2010 IEEE, pages 1 – 5, 2010.
David Elkouss, Jesus Martinez-Mateo, and Vicente Martin.
Efficient reconciliation with rate adaptive codes in quantum key distribution.
arXiv, quant-ph, 2010.
David Elkouss, Jesus Martinez-Mateo, and Vicente Martin.
Secure rate-adaptive reconciliation.
In Information Theory and its Applications (ISITA), 2010 International Symposium on, pages
179 – 184, 2010.
Kenta Kasai, Ryutaroh Matsumoto, and Kohichi Sakaniwa.
Information reconciliation for QKD with rate-compatible non-binary LDPC codes.
In Information Theory and its Applications (ISITA), 2010 International Symposium on, pages
922 – 927, 2010.
Jesus Martinez-Mateo, David Elkouss, and Vicente Martin.
Interactive reconciliation with low-density parity-check codes.
In Turbo Codes and Iterative Information Processing (ISTC), 2010 6th International
Symposium on, pages 270 – 274, 2010.
24 / 26

34. Literature
Implementation (non-exhaustive)
Chip Elliott, Alexander Colvin, David Pearson, Oleksiy Pikalo, John Schlafer, and Henry Yeh.
Current status of the DARPA quantum network.
arXiv, 2005.
Jerome Lodewyck, Matthieu Bloch, Raul Garcia-Patron, Simon Fossier, Evgueni Karpov,
Eleni Diamanti, Thierry Debuisschert, Nicolas J Cerf, Rosa Tualle-Brouri, Steven W
McLaughlin, and Philippe Grangier.
Quantum key distribution over 25 km with an all-fiber continuous-variable system.
arXiv, quant-ph, 2007.
´
Simon Fossier, Eleni Diamanti, Thierry Debuisschert, Andre Villing, Rosa Tualle-Brouri, and
Philippe Grangier.
Field test of a continuous-variable quantum key distribution prototype.
arXiv, quant-ph, 2008.
Simon Fossier, J Lodewyck, Eleni Diamanti, Matthieu Bloch, Thierry Debuisschert, Rosa
Tualle-Brouri, and Philippe Grangier.
Quantum key distribution over 25 km, using a fiber setup based on continuous variables.
In Lasers and Electro-Optics, 2008 and 2008 Conference on Quantum Electronics and Laser
Science. CLEO/QELS 2008, pages 1 – 2, 2008.
A Dixon, Z Yuan, J Dynes, A Sharpe, and Andrew Shields.
Megabit per second quantum key distribution using practical InGaAs APDs.
In Lasers and Electro-Optics, 2009 and 2009 Conference on Quantum Electronics and Laser
Science. CLEO/QELS 2009, pages 1 – 2, 2009.
25 / 26

35. Conclusions
Reconciliation for QKD is a Slepian-Wolf coding problem (in a
corner point)
linear codes are sufficient for the optimal solution
maximising the key rate is not necessarily equivalent to
minimising the error rate
complexity constraints may lead to a non-trivial optimisation
problem to find the best codes and decoding algorithms
rate adaptive or rateless schemes might be necessary in case
where the error rate on the quantum channel is unknown
26 / 26