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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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