The multiple coding gain with two criteria in an attenuated quantum channel

Abstract

Recently, the capacity of an attenuated quantum channel which is a model of a long distance optical fiber channel was derived. This capacity is in principle achieved by a continuous modulation and a random coding. However, a continuous modulation and a random coding are not practical. Therefore, we need discussion about capacity used for more practical modulation and coding. In this paper, we show characteristics of both information and error probability criteria for an attenuated channel using M-ary PSK modulation and pseudo-cyclic codes.

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International Symposium on Information Theory and its Applications, ISITA2008
Auckland, New Zealand, 7-10, December, 2008
The multiple coding gain with two criteria in an attenuated quantum channel
Ryosuke Sahara†, Shogo Usami†and Tsuyoshi Sasaki Usuda‡
†Postgraduate Course in Science and Technology
Meijo University
Shiogamaguchi, Tenpaku, Nagoya, 468-8502, Japan
‡Department of Applied Information Technology
Aichi Prefectural University
Kumabari, Nagakute-cho, Aichi-gun 480-1198, Japan
Abstract
Recently, the capacity of an attenuated quantum chan-
nel which is a model of a long distance optical fiber
channel was derived.This capacity is in principle
achieved by a continuous modulation and a random
coding. However, a continuous modulation and a ran-
dom coding are not practical. Therefore, we need dis-
cussion about capacity used for more practical modula-
tion and coding. In this paper, we show characteristics
of both information and error probability criteria for
an attenuated channel using M-ary PSK modulation
and pseudo-cyclic codes.
1. Introduction
Superadditivity in capacity is a remarkable fea-
ture when we transmit classical (Shannon) informa-
tion through a quantum channel [1, 2]. In 1997, it was
demonstrated that there are codes with finite codeword
length, that show superadditivity in capacity for binary
pure-state channel [3]. The analytical expression for
the mutual information based on SRM was shown for
any binary linear code [4]. Moreover, various charac-
teristics were shown by using the expression [5].
On the other hand, after the first demonstration
by Peres and Wootters [6], various properties of mu-
tual information for q-ary linearly dependent real sym-
metric state signals were shown [7]. In addition, the
world’s first proof-of-principle experiments on superad-
ditive quantum gain were carried out at the National
Institute on Information and Communication Technol-
ogy (NICT, formerly the Communications Research
Laboratory)[8]. Linearly dependent signals of a single-
photon system were used in the experiment.
Recently, the capacity of an attenuated quantum
channel which is a model of a long distance optical
fiber channel was derived [9]. In the future, the experi-
ments must be extended to linearly independent signals
(coherent-state system) in order to implement ultrafast
and ultraliable quantum communications.
The capacity of an attenuated quantum channel is
in principle achieved by a continuous modulation and
a random coding. However, a continuous modulation
and a random coding are not practical. Therefore, we
need discussion about capacity used for more practical
modulation and coding. For this purpose, some im-
portant result had been shown. The full capacity is
asymptotically achieved by 2-PSK coherent-state sig-
nals when the energy of the signals is very small [10].
Furthermore, there is a range of signal energy, in which
the full capacity is almost achieved not only by 2-PSK
signals but also by multi-ary PSK or QAM signals [11].
Those results show that multiple coding is effective in
order to realize efficient communication with informa-
tion criterion. We consider error probability perfor-
mance in order to carry out reliable communication.
For this purpose, properties of the quantum reliability
function were shown when coherent-states and digital
modulations were used for an attenuated channel [12].
These results with error probability and information
criteria correspond to a random coding.
In this paper, we show characteristics of both infor-
mation and error probability criteria for an attenuated
channel using M-ary PSK modulation and pseudo-
cyclic codes.
Below, base-2 logarithms are assumed unless other-
wise specified.
2. Superadditivity of quantum channel capacity
and mutual information
2.1. Superadditivity in capacity
We consider classical information transmission
through a quantum channel. Let {xi|i = 0,1,··· ,
M − 1} be an input alphabet of (classical) informa-
tion source and {ξi} be their a priori probabilities.
For a classical input information xi, there corresponds
a quantum state (signal) ˆ ρi(in)which is transmitted
through a quantum channel ϵ at a transmitter. A re-
ceiver measures the transmitted state ϵ(ˆ ρi(in))= ˆ ρi(out)
the correspondence between input xiand output yjis
and received classical information is yj.In general,

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not one-to-one, and therefore the quantum measure-
ment process can be regarded as a noisy channel. The
channel is characterized by the following conditional
probability
P(yj|xi) = Trˆ ρi(out)ˆΠj,
where {ˆΠj} represent a quantum measurement process
which is a positive operator-valued measure (POM)
satisfying the following relations:
(1)
ˆΠj≥ 0,
ˆΠj=ˆI,
(2)
∑
j
(3)
whereˆI is the identity operator. EachˆΠj is called a
detection operator. P(yj|xi) is the probability that the
signal yjis determined when xiis the input signal. Mu-
tual information is defined with a priori probabilities
{ξi} and conditional probabilities {P(yj|xi)} as
∑
The maximum value of mutual information with re-
spect to detection operators and a priori probabilities
is called the capacity of codeword length 1 (or maxi-
mum mutual information without coding) C1:
I(X;Y ) =
i
ξi
∑
j
log
[
P(yj|xi)
lξlP(yj|xl)
∑
]
.
(4)
C1= max
{ξi}max
{ˆΠj}I(X;Y ).
(5)
For n-th extension of the signals, Cn can be defined
in a similar way in Eq.(5) and called the capacity of
codeword length n.
It is known that the capacities are superadditive
Cn+ Cm≤ Cn+m,
(6)
which is in sharp contrast to a classical channel where
the capacity is additive and the sign of equality is al-
ways attained in Eq.(6). According to the quantum
channel coding theorem, the maximum transmission
rate C is the following limit:
C = lim
n→∞
1
nCn,
(7)
that is called the quantum channel capacity.
In order to give an example of strict superadditivity,
it is sufficient to show the inequality Cn+Cm< Cn+m
(or nC1< Cn).
2.2. Mutual information by Square-root mea-
surement
Let wi = xi,n−1xi,n−2···xi,1xi,0 be a classical
codeword in a code with codeword length n and
|wi⟩ = |ψi,n−1⟩|ψi,n−2⟩···|ψi,1⟩|ψi,0⟩ be the corre-
sponding codeword-state. At this time, it is a collec-
tive decoding that detects each codeword state |wi⟩ as
a single state rather than decoding the individual let-
ter states {|ψl⟩} separately. The square-root measure-
ment(SRM) is known as a typical collective decoding.
Moreover, the SRM is known to be the minimum error
decoding for M-ary pseudo-cyclic code [7]. The SRM
{ˆΠ(SRM)
ˆΠ(SRM)
j
= |µj⟩⟨µj|, (j = 0,1,··· ,N − 1),
(N−1
i=0
j
} for the code is defined as follows [13].
(8)
|µj⟩ =
∑
|wi⟩⟨wi|
)1
2
|wj⟩,
(9)
where N is the number of codewords.
According to Ref.[13], the inner product ⟨wi|µj⟩ be-
tween the codeword-state |wi⟩ and the measurement-
state |µj⟩ is equal to the (i,j) component of the square-
root of the gram matrix Γ = [⟨wi|wj⟩]. That is
⟨wi|µj⟩ = (Γ)
1
2
i,j.
(10)
Therefore, the conditional probability P(wj|wi) by ap-
plying the SRM becomes
P(wj|wi) = |⟨wi|µj⟩|2=
???(Γ)
1
2
i,j
???
2
.
(11)
Mutual information of codeword length n is represented
with conditional probabilities P(wj|wi) as
1
N
ij
In(Xn;Yn) =
∑∑
log
[
P(wj|wi)
∑
1
N
lP(wj|wl)
]
. (12)
2.2.1. Pseudo-cyclic code
In this paper, M-ary pseudo-cyclic code is used as
coding. M-ary (n,k) pseudo-cyclic code of codeword
length n and the number of information symbol k is
considered. Let G(x) be a generator polynomial and
A(x) be an M-ary information polynomial. Then the
codeword polynomial W(x) becomes
W(x) =A(x)G(x)
=wn−1xn−1+ wn−2xn−2+ ··· + w1x + w0,
where wl ∈ {0,1,··· ,M − 1}. Note that for a code-
word polynomial W(x) = wn−1xn−1+ wn−2xn−2+
···+w1x+w0, corresponding codeword and codeword-
state are wi
=
wn−1wn−2···w1w0 and |wi⟩
|ψwn−1⟩|ψwn−2⟩···|ψw1⟩|ψw0⟩, respectively. Here |ψwj⟩
is a letter-state. When the degree of generator poly-
nomial G(x) is m, the number of information digits
(13)
=

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k = n − m and Mkcodewords is generated. More-
over, when G(x) = 1, selection of codewords is not
performed.
Since M-ary pseudo-cyclic code is group covariant
[7], mutual information of codeword length n by using
M-ary pseudo-cyclic code becomes
∑
where w0= 00···0.
In(Xn;Yn) = logN +
j
P(wj|w0)logP(wj|w0),
(14)
3. An attenuated quantum channel
We consider classical information transmission
through an attenuated quantum channel with energy
transmissivity η (0 ≤ η ≤ 1). And a coherent-state is
used as a signal quantum state.
3.1. Coherent state
In general, when the quantum state is transmitted
through a noisy channel, the output quantum state be-
comes a mixed state even if the input quantum state
is pure. However, when coherent-states are transmit-
ted through an attenuated channel, nature of coherent-
states is kept. That is, when coherent-states are used
as transmitted signals, the attenuated channel ϵ(·) with
transmissivity η can be described as the map which
transforms an input coherent-state |α⟩ into the output
state ϵ(|α⟩)= |√ηα⟩.
3.1.1. M-ary PSK signals
We consider PSK (Phase-shift keying) as a digital
modulation scheme. When coherent-states are used,
M-ary PSK signals are represented as follows.
????αexp
where α is amplitude of signals and i =
average photon number NSof input signals is given by
|αl⟩ =
[
i2l
M
]⟩
, l = 0,1,··· ,M − 1,
(15)
√−1. The
NS=
M−1
∑
M−1
∑
l=0
ξl|αl|2,
=
l=0
ξl|α|2,
= |α|2,
(16)
where ξlare a priori probabilities of signal |αl⟩. Below,
we transcribe an M-ary PSK coherent state signal into
M-PSK.
0:0000
7:0100
5:0111
3:0010
4:0110
6:0101
1:0001
2:0011
8:1100
9:1101
10:1111
15:1000
13:1011
14:1001
11:1110
12:1010
Figure 1: Gray mapping for 16-PSK.
4. The bit error rate in multiple coding
We consider gain of the multiple coding with error
probability criterion. We compared BER of no-coding
with that of the collective decoding. The output of the
collective decoding becomes one of codewords. There-
fore, the collective decoding has an error correction
function. We use gray mapping in order to transform a
multiple codeword into a binary codeword. Gray map-
ping is the mapping of transforming M-ary signals into
the gray code. As an example, gray mapping with 16-
PSK signals are shown in Fig.1.
By applying gray mapping to the M-ary code of
the codeword length n, it becomes the code of the
codeword length µ = nlogM and a codeword is ci=
bµ−1bµ−2···b1b0 (b ∈ {0,1}). BER by applying gray
mapping becomes
BER =
N−1
∑
j=1
dH(c0,cj)
µ
P(wj|w0),
(17)
where dH(ci,cj) is the Hamming distance between the
codewords ci and cj, c0 = 00···0, and P(wj|wi) is
the conditional probability in Eq.(11).
5. The multiple coding gain with two criteria
We consider error probability and information cri-
teria when we use pseudo-cyclic codes as actual cod-
ings. However, the capacity of M-PSK with codeword
length 1 C1has not been derived. Therefore, it is un-
verifiable whether mutual information for M-PSK has
superadditivity. In this paper, we compute the maxi-
mum value of mutual information without coding when
the SRM is used as the decoding. The maximization
is perfomed with respect to the number of signals, i.e.
from 2 to M-PSK. If mutual information of codeword

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00.5
1.0
1.5
2.0
0.5
1.0
1.5
2.0
(3,k) coded 4-PSK
(5,k) coded 4-PSK
4-PSK without coding
I (X ;Y )/n
n
n
n
Figure 2: Maximum value of mutual information for
coded 4-PSK with respect to the average number of
received photons. Case that codeword length n = 3
and 5.
length n (Eq.(14)) is greater than that without coding,
we say there is a coding gain. And we compare BER
by multiple coding with BER for 2-PSK without cod-
ing because BER for 2-PSK is the smallest of all BER
for M-PSK without coding.
In an actual communication system, a code with
long codeword length may be used. However to com-
pute square root of a higher order Gram matrix is com-
putationally difficult. Hence we consider a coding gain
for the case that codeword length is short. Moreover,
by changing a generator polynomial, there are many
(n,k) pseudo-cyclic codes. We consider all the possi-
bilities, and the value of the best code is shown.
5.1. Multiple coding gain for 4-PSK with code-
word length n=3 and 5
Fig.2 shows the maximum values of mutual infor-
mation for 4-PSK with codeword length n=3 and 5 in a
number of codes we computed, as a function of the av-
erage number of received photons ηNS. We computed
mutual information for all codes with codeword length
n = 3 and 5.In Fig.2 and the subsequent figures,
“(n,k) code” means the maximum value of mutual in-
formation in all generator polynomials we computed.
And “M-PSK without coding” means the maximum
value of mutual information without coding in 2 to M-
PSK.
From Fig.2, mutual information for coded 4-PSK
with codeword length 3 is larger than that without
coding when ηNS ≤ 0.1. Moreover, mutual informa-
tion for coded 4-PSK with codeword length 5 is larger
than that without coding when ηNS≤ 1. This means
that the transmitted information increases by multi-
ple coding. And when we lengthen codeword length
100
10-5
BER
10-10
0
0.5
1.0
1.52.0
(3,1) coded 4-PSK
(5,2) coded 4-PSK
(5,1) coded 4-PSK
2-PSK without coding
Figure 3: BER for 4-PSK in the same codes used in
Fig.2, with respect to the average number of received
photons.
Table 1: Generator polynomials for pseudo-cyclic codes
used in Figs. 2 and 3.
(n,k)code
(3,2)
(3,1)
(5,4)
(5,3)
(5,2)
(5,1)
Generator polynomial G(x)
x + 2
x2+ x + 2
x + 2
x2+ x + 2
x3+ 2x2+ x + 1
x4+ 2x3+ x2+ 2x + 3
more, it is expected that the transmitted information
increases.
Fig.3 shows BER for coded 4-PSK with codeword
length n=3 and 5 by the same codes used in Fig.2, as
a function of the average number of received photons
ηNS. We show BER whose information transmission
rate is less than 1 bit/symbol. BER for coded 4-PSK
with codeword length 3 is lower than BER for 2-PSK
without coding when ηNS≤ 0.2. And BER for coded
4-PSK with codeword length 5 are lower than BER
for 2-PSK without coding in all ranges. Therefore the
coded 4-PSK with codeword length 5 has quantum cod-
ing gain with both information and error probability
criteria.
We show in Table1 generator polynomials of
pseudo-cyclic codes used in Figs.2 and 3.
5.2. Multiple coding gain with error probability
criterion for M-PSK when M=4, 8, and 16
Here we show detailed properties of BER for M-ary
PSK coherent-state signals. In all cases, information

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100
10-4
BER
10-8
0
0.5
1.0
1.52.0
(2,1) coded 4-PSK
(4,2) coded 4-PSK
(6,3) coded 4-PSK
(8,4) coded 4-PSK
2-PSK without coding
Figure 4: Minimum BER for coded 4-PSK in a number
of codes we computed, with respect to the average num-
ber of received photons. Case that information trans-
mission rate is equal and codeword length n = 2,4,6
and 8.
transmission rate is assumed to be equal (1bit/symbol).
Fig.4 shows minimum BER for coded 4-PSK in a
number of codes we computed, with respect to the av-
erage number of received photons ηNS. We computed
BER for all codes when information transmission rate
is constant and codeword length n = 2,4,6 and 8.
Then we chose the minimum value for each codeword
length and plotted it.
BER for coded 4-PSK with
codeword length 2, 4, 6, and 8 are lower than BER for
2-PSK without coding when ηNS ≥ 0.4. The longer
the codeword length is the lower the BER is. That is,
when we lengthen codeword length more, it is expected
that BER decreases.
Fig.5 shows minimum BER for coded M-PSK in
a number of codes we computed, with respect to the
average number of received photons ηNS (M=4, 8,
and 16). We apply the minimum codeword length on
condition that information transmission rate is equal
(1bit/symbol). BER for coded 4, 8 and 16-PSK are
lower than that without coding when ηNS≥ 0.4. The
higher the number of signals is the lower the BER is.
6. Conclusion
In this paper, we have shown that characteristics
of multiple coding gain with both information and er-
ror probability criteria for an attenuated channel us-
ing M-ary PSK modulations and pseudo-cyclic codes.
We have shown properties of mutual information and
BER for 4-ary PSK coherent-state signals. As a result,
the coded 4-PSK with codeword length 5 has quantum
coding gain with both information and error probabil-
ity criteria. Moreover, we have shown that BER for
100
10-3
BER
10-6
0
0.5
1.0
1.5 2.0
(2,1) coded 4-PSK
(3,1) coded 8-PSK
(4,1) coded 16-PSK
2-PSK without coding
Figure 5: Minimum BER for coded M-PSK in a num-
ber of codes we computed, with respect to the average
number of received photons (M=4, 8, and 16). Case of
minimum codeword length on condition that informa-
tion transmission rate is equal.
coded M-PSK when information transmission rate is
equal. As a result, the longer the codeword length is
the lower the BER is. And the more the number of
signals is the lower the BER is.
Acknowledgements: This work has been supported
in part by MEXT.KAKENHI (No.18360186). One of
the authors (R.S.) thanks REFEC (No.E-20136).
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