Distribution Statistics and Random Matrix Formalism of Multicarrier Continuous-Variable Quantum Key Distribution Laszlo Gyongyosi 1
Quantum Technologies Laboratory, Department of Telecommunications Budapest University of Technology and Economics 2 Magyar tudosok krt, Budapest, H-1117, Hungary 2 MTA-BME Information Systems Research Group Hungarian Academy of Sciences 7 Nador st., Budapest, H-1051, Hungary
[email protected]
Abstract We propose a combined mathematical framework of order statistics and random matrix theory for multicarrier continuous-variable (CV) quantum key distribution (QKD). In a multicarrier CVQKD scheme, the information is granulated into Gaussian subcarrier CVs, and the physical Gaussian link is divided into Gaussian sub-channels. The sub-channels are dedicated to the conveying of the subcarrier CVs. The distribution statistics analysis covers the study of the distribution of the sub-channel transmittance coefficients in the presence of a Gaussian noise and the utilization of the moment generation function (MGF) in the error analysis. We reveal the mathematical formalism of sub-channel selection and formulation of the transmittance coefficients, and show a reduced complexity progressive sub-channel scanning method. We define a random matrix formalism for multicarrier CVQKD to evaluate the statistical properties of the information flowing process. Using random matrix theory, we express the achievable secret key rates and study the efficiency of the AMQD-MQA (adaptive multicarrier quadrature division–multiuser quadrature allocation) multiple-access multicarrier CVQKD. The proposed combined framework is particularly convenient for the characterization of the physical processes of experimental multicarrier CVQKD. Keywords: quantum key distribution, continuous-variables, CVQKD, AMQD, AMQDMQA, random matrix theory, order statistics, quantum Shannon theory.
1
1 Introduction The continuous-variable quantum key distribution (CVQKD) protocols allow the establishment of an unconditional secure communication over standard, currently established telecommunication networks [10–22]. In comparison to discrete-variable (DV) QKD protocols, CVQKD does not require single-photon sources and detectors and can be implemented by standard optical telecommunication devices [1], [9–26], [30–37]. In a CVQKD system, the information is carried by a continuous-variable quantum state that is defined in the phase space via the position and momentum quadratures. Since a Gaussian modulation is a reasonable modulation technique in an experiment, these CV quantum states have a Gaussian random distribution. Precisely, the presence of an eavesdropper on the quantum channel adds a white Gaussian noise into the transmission because the optimal attack against a CVQKD protocol is provably also Gaussian. Besides the attractive properties of CVQKD, the relevant performance attributes, such as secret key rates and transmission distances, still require significant improvements. For this purpose, the multicarrier CVQKD has been recently introduced through the adaptive quadrature division modulation (AMQD) scheme [2]. The multicarrier CVQKD injects several additional degrees of freedom onto the transmission which is not available for a standard, single-carrier CVQKD setting. In particular, these extra benefits and resources not just allow the realization of higher secret key rates and higher amount of tolerable losses with unconditional security but also make possible the introduction and defining of several new phenomena for CVQKD, such as singular layer transmission [4], enhanced security thresholds [5], multidimensional manifold extraction [6], and the characterization of the subcarrier domain [7]. The benefits of multicarrier CVQKD has also been proposed for multiple-access multicarrier CVQKD via the AMQD-MQA (multiuser quadrature allocation) [3]. An adaptive quadrature detection technique has also been defined for multicarrier CVQKD, which uses a channel transmittance estimation to decode the continuous variables [8]. In this work, we particularly focus on the characterization of the transmittance coefficients of the sub-channels, through the statistical analysis of their distribution. We also define a random matrix formalism to describe the process of information flow via the Gaussian subcarrier CVs. We develop a combined framework that utilizes and integrates the results of distribution statistics and random matrix theory. The proposed combined framework provides a tool to characterize the physical distribution and transmission processes of information flowing in experimental multicarrier CVQKD scenarios. In the first part, we provide a distribution statistics framework for multicarrier CVQKD. The distribution statistics of multicarrier CVQKD utilizes the results of order statistics [28–29], which is an important subfield of statistical theory, with several applications—from mathematical statistics and engineering to the analysis of traditional communication systems. We reveal the statistical properties of the multicarrier transmission and define the statistical operators of sub-channel ordering, selection, and formulation of the single-carrier level transmittance coefficients. We also define the conditions that are required for the simultaneous achievement of a maximal secret key rate and an unconditional security at diverse channel parameters. The proposed distribution statistics analysis covers the study of the distribution of the sub-channel transmittance coefficient in
2
the presence of a Gaussian noise and the utilization of the moment generation function (MGF) in the error analysis, such as the distribution of the received Gaussian subcarriers. Random matrix theory represents a useful mathematical tool with a widespread application— from physics, statistics, to engineering problems and communication theory [27]. The popularity of random matrix theory is rooted in the fact that several problems can be directly interpreted and solved via the mathematical framework of random matrix formalism. In the second part, using the results of distributions statics, we define a random matrix formalism for multicarrier CVQKD. Using the framework provided by random matrix formalism, we characterize the statistical properties of the information transmission process, derive the achievable secret key rates, and study the multiuser efficiency of the AMQD-MQA multiple-access multicarrier CVQKD scheme. This paper is organized as follows. In Section 2, some preliminary findings are summarized. Section 3 provides the distribution statistics of multicarrier CVQKD. Section 4 discusses the random matrix formalism of multiple-access multicarrier CVQKD via the analysis of AMQD-MQA. Finally, Section 5 concludes the results. Supplementary information is included in the Appendix.
2 Preliminaries In Section 2, we briefly summarize the notations and basic terms. For further information, see the detailed descriptions of [2–8].
2.1 Basic Terms and Definitions 2.1.1 Multicarrier CVQKD The following description assumes a single user, and the use of n Gaussian sub-channels i for the transmission of the subcarriers, from which only l sub-channels will carry valuable information. In the single-carrier modulation scheme, the j-th input single-carrier state j j = x j +ip j is a Gaussian state in the phase space , with i.i.d. Gaussian random position and momentum quad-
(
ratures x j Î 0, sw2
0
),
(
p j Î 0, sw2
0
) , where
sw2 is the modulation variance of the quadra0
tures. In the multicarrier scenario, the information is carried by Gaussian subcarrier CVs,
fi = x i +ipi , x i Î ( 0, sw2 ) , pi Î ( 0, sw2 ) , where sw2 is the modulation variance of the
subcarrier quadratures, which are transmitted through a noisy Gaussian sub-channel i . Precisely, each i Gaussian sub-channel is dedicated for the transmission of one Gaussian subcarrier CV from the n subcarrier CVs. (Note: index i refers to a subcarrier CV, index j to a single-carrier CV, respectively.) The single-carrier CV state jj in the phase space can be modeled as a zero-mean, circular
3
æ ö symmetric complex Gaussian random variable z j Î ççç 0, sw2 ÷÷÷ , with a variance zj ø è 2ù é sw2 = ê z j ú = 2sw2 , zj 0 ë û and with i.i.d. real and imaginary zero-mean Gaussian random components
(
Re ( z j ) Î 0, sw2
0
) , Im ( z
j
) Î ( 0, sw2
0
).
(1)
(2)
In the multicarrier CVQKD scenario, let n be the number of Alice’s input single-carrier Gaussian states. Precisely, the n input coherent states are modeled by an n-dimensional, zero-mean, circular symmetric complex random Gaussian vector T
z = x + ip = ( z 0 , , z n -1 ) Î ( 0, Kz ) ,
(3)
where each z j is a zero-mean, circular symmetric complex Gaussian random variable
æ z j Î ççç 0, sw2 zj è
÷ö÷ , z = x + ip . j j ÷ø j
(4)
In the first step of AMQD, Alice applies the inverse FFT (fast Fourier transform) operation to vector z (see (3)), which results in an n-dimensional zero-mean, circular symmetric complex T
Gaussian random vector d , d Î ( 0, Kd ) , d = ( d0 , , dn -1 ) , precisely as
d=F
-1
(z) = e
where
dT AAT d 2
(
=e
s2 d02 ++dn2 -1 w0 2
(
) ,
)
di = xd + ipd , di Î 0, sd2 , i
i
i
é 2ù where sw2 = ê di ú = 2sw2 , thus the position and momentum quadratures of fi di ë û
(5)
(6) are i.i.d.
Gaussian random variables with a constant variance sw2 for all i , i = 0, , l - 1 sub-channels:
Re ( di ) = xd Î ( 0, sw2 ) , Im ( di ) = pd Î ( 0, sw2 ) , i
i
(7)
T ù é where Kd = éê dd† ùú , éë d ùû = éê e ig d ùú = e ig éë d ùû , and éê ddT ùú = ê e ig d (e ig d ) ú = e i2 g éê ddT ùú ë û ë û ë û ë û êë ûú for any g Î éë 0, 2p ùû .
The T (
)
transmittance vector of in the multicarrier transmission is T T ( ) = éëT0 ( 0 ) , ,Tn -1 ( n -1 ) ûù Î n ,
(8)
Ti ( i ) = Re (Ti ( i ) ) + i Im (Ti ( i ) ) Î ,
(9)
where
is a complex variable, which quantifies the position and momentum quadrature transmission (i.e., gain) of the i-th Gaussian sub-channel i , in the phase space , with real and imaginary parts 4
0 £ Re Ti ( i ) £ 1
2 , and 0 £ Im Ti ( i ) £ 1
2.
(10)
Particularly, the Ti ( i ) variable has the squared magnitude of
Ti ( i )
2
2
2
= ReTi ( i ) + Im Ti ( i ) Î ,
(11)
ReTi ( i ) = Im Ti ( i ) .
(12)
where The Fourier-transformed transmittance of the i-th sub-channel i (resulted from CVQFT operation at Bob) is denoted by 2
F (Ti ( i ) ) . The
n-dimensional
D Î (
)
2 0, sD n
zero-mean,
circular
symmetric
(13) complex
Gaussian
noise
vector
, of the quantum channel , is evaluated as T
D = ( D0 , , Dn -1 ) Î ( 0, KD ) ,
(14)
where
KD = éê DD† ùú , û ë with independent, zero-mean Gaussian random components
(
)
(15)
(
Dx Î 0, s 2 , and Dp Î 0, s 2 i
i
i
i
),
(16)
with variance s 2 , for each Di of a Gaussian sub-channel i , which identifies the Gaussian i
noise of the i-th sub-channel i on the quadrature components x i , pi in the phase space .
(
2 Thus F ( D ) Î 0, sD
i
) , where
2 sD = 2s 2 . i
(17)
i
The CVQFT-transformed noise vector can be rewritten as T
F ( D ) = ( F ( D0 ) , , F ( Dn -1 ) ) , with independent components F ( Dx
i
) Î ( 0, s2 ) i
and F ( Dp
i
(18)
) Î ( 0, s2 )
on the quadra-
i
tures, for each F ( Di ) . Precisely, it also defines an n-dimensional zero-mean, circular symmetric
(
)
complex Gaussian random vector F ( D ) Î 0, KF ( D ) with a covariance matrix †ù é (19) KF ( D ) = ê F ( D ) F ( D ) ú . ë û The complex Aj ( j ) Î single-carrier channel coefficient is derived from the l Gaussian sub-
channel coefficients as
Aj ( j ) =
1 l
l -1
å i =0 F (Ti ( i )) .
5
(20)
3 Distribution Statistics for Multicarrier CVQKD First we summarize some preliminary findings from order statistics from [29], then evaluate the theorems and proofs. Note the proofs throughout Section 3 follow the notations of [29].
3.1 Moment-Generating Function The M MGF-function (moment-generating function) [29] of a nonnegative random variable x , x ³ 0 is ¥
Mx ( c ) =
ò P ( x )ecxdx ,
(21)
0
where c is a complex dummy variable, P (⋅ ) is the PDF (probability density function) function, and éê x n ùú = ë û
Mx ( c ) c = 0 .
(22)
Mx ( -c ) = { P ( x ) } ,
(23)
dn dc n
It can be shown that where is the Laplace transform. Particularly, using the M -function, the Q (⋅ ) Gaussian tail function of x > 0 can be expressed precisely as
Q (x ) =
1 p
æ 2 ö p 2 ççç -x2 ÷÷÷ è 2 sin f ø
ò0
df ,
e
(24)
where f Î éë 0, 2p ùû and
Q
2
æ 2 ö p 4 ççç -x2 ÷÷÷ è 2 sin f ø
( x ) = ò0 1 p
df .
e
(25)
Assuming an error rate Rerr ( x ) Rerr ( x ) = aQ
(
)
bx ,
(26)
average error rate is yielded as where a and b are constants, the R err
= R err
¥
ò0
aQ
(
)
bSNR PSNR ( x ) dx ,
(27)
where PSNR (⋅ ) is the PDF of the SNR (signal to noise ratio), which can be rewritten as
= R err
a p
p 2
ò0
MSNR
where MSNR (⋅ ) is the M -function of the SNR.
6
(
-b 2 sin2 f
)df,
(28)
3.2 Sub-channel Distribution Statistics 3.2.1 Marginal, Joint and Conditional Distributions of the Sub-channels Let the normalized independent, ordered transmittance coefficients of the l sub-channels be as 2
F (Ti ( i ) ) , i = 0, , l - 1.
1 l
(29)
given in a descending order such that 1 l
The
(
F Ti ( i )
1 l
)
2
F (T0 ( 0 ) ) ³
1 l
2
F (T1 ( 1 ) ) ³ ³
2
F (Tl -1 ( l -1 ) ) .
1 l
(30)
2
unordered variables are identically distributed with common PDF and CDF
(cumulative distribution function), Pc ( x ) and fc ( x ) , respectively [29]. The Pc ( x 0 , , xl -1 ) common joint PDF of the unordered variables can be expressed as
Pc ( x 0 , , xl -1 ) =
l -1
Pc ( xi ) .
(31)
i =0
Without loss of generality, the ordered variables are not identically distributed and have PDF and CDF functions (32) P1 2 ( x ) and f1 2 (x ). l
Then let P1 l
F (Ti ( i ) )
2
(x ) P1 l
F (Ti ( i ) )
l
F (Ti ( i ) )
the PDF function of the i-th ordered sub-channel coefficient, as l -1
F (Ti ( i ) )
l -1
( x ) = (l -i )l!(!i -1) ! fc ( x ) ( 1 - fc ( x ) )
2
Pc ( x ) ,
(33)
with
P1 l
l -1
F (T0 ( 0 ) )
2
( x ) = lfc ( x )
Pc ( x ) ,
(34)
and
P1 l
The P1 l
2
F (Tk ( k ) ) , 1 F (Tm ( m ) )
l -1
F (Tl -1 ( l -1 ) ) 2
2
( x ) = l ( 1 - fc ( x ) )
( x, y )
joint PDF of
P1
2
l
l
=
1 l
Pc ( x ) . 2
2
( x, y )
l
k -1
l!
( 1 - fc ( x ) )
m -k -1
⋅ ( fc ( x ) - fc ( y ) )
2
F (Tk ( k ) ) , 1l F (Tm ( m ) ) , m > k is as
F (Tk ( k ) ) , 1 F (Tm ( m ) )
( k -1 ) !( m -k -1 ) !(l -m ) !
(35)
Pc ( x ) l -m
Pc ( y ) ( fc ( y ) )
(36)
.
Since the ordered variables are not identically distributed, the Pc ( x 0 , , xl -1 ) common joint PDF of (31) can be rewritten as a joint PDF function P1 2 1 l
F (T0 ( 0 ) ) ,, F (Tl -1 ( l -1 ) ) l -1
2
( x 0, , xl -1 )
l
= l ! Pc ( x i ), x 0 ³ x1 ³ ³ xl -1. i =0
7
(37)
Specifically, assuming that there are l - i available sub-channels, the m-th ordered sub-channel coefficient is referred to as 2
F (Tm¢ ( m ) ) , m = 0, , (l - i ) - 1 .
1 l -i
(38)
Using the variable of (38), and further assuming that 1 l
the P1 l
F (Tm ( m ) )
P1 l
2 1 l
2
F (Ti ( i ) ) =y
F (Tm ( m ) )
2 1 l
(x )
F (Ti ( i ) )
2
= y,
(39)
conditional PDF can be evaluated as P1
(x ) =
2
F (Ti ( i ) ) =y
l
2 2 F (Tm ( m ) ) ,1 F (Ti ( i ) ) l
P1 l
F (Ti ( i ) )
2
(y )
(
f (x ) (l -i ) ! 1- c fc ( y ) ( m -i -1) !(l -m ) !
=
P1 l
( x ,y )
)
m -i -1
F (Ti ( i ) )
2
( )
Pc ( x ) fc ( x ) fc ( y ) fc ( y )
(40)
l -m
, x £ y.
(y )
Without loss of generality, at l - 1 available sub-channels, (38) can be rewritten as 1 l -1
2
F (Tm¢ ( m ) ) , m = 0, , (l - 1 ) - 1 ,
and (39) holds, the conditional PDF is expressed as P1 2 ( x ) = P1 l -1
where the PDF of
F (Tm¢ ( m ) )
F (Tm¢ ( m ) )
1 l -1
2
l
F (Tm ( m ) )
l -1
2
F (Ti ( i ) ) =y
(x ) ,
(42)
is Pc ( x )
P1
2 1 l
(41)
F (Tm¢ ( m ) )
2
( x ) = 1- f ( y ) ,
x ³ y.
(43)
c
In particular, focusing on the case that the l sub-channels are selected via operator L from a set of n variables S =
{
F (Ti ( i ) )
1 l
2
}
n -1 i =0
, the joint PDF function is yielded as
PS ( x 0 , , xl -1 ) =
n! f ( n -l ) ! c
n -l
( xl -1 )
l -1
Pc ( xi ),
i =0
(44)
x 0 ³ x 1 ³ ³ xl -1.
The model of the LU order-and-sum (selection) operator of user U k with single-carrier channel k
U is depicted in Fig. 1. Operator LU is decomposed as LU = OS , where LU first selects l k
k
k
k
sub-channels from the total n via an ordering operation O , which uses the normalized, unordered 1 l
(
F Ti ( i )
)
2
coefficients; then LU
k
evaluates Aj ( U
ordered coefficients via the sum-operator S
) = ( 1 l ) åli-=10 F (Ti ( i )) to yield Aj ( U ) . k
k
8
from the
Sub-channels of
1F l 1 F l
( ) (T ( )) T0 ( 0 ) 1
1F l
(T ( n -1
(Tl -1 ( l -1 ) )
S
Ordering
Sum
1F l
n -1
k
(T0 ( 0 ) )
1
1 F l
LU
O
Uk
Aj ( U
k
)
))
Figure 1. The LU order-and-sum operator of user U k . LU selects l sub-channels from the total k
k
n, via O . (The ordered l sub-channels are depicted by the thick frame); then evaluates Aj ( U
k
)
of U via S . k
3.2.2 Partial Sum Distributions of Ordered Sub-channels Proposition 1 (PDF of the partial sum of sub-channel coefficients). The PDF function of single carrier transmittance Aj ( j ) where G1 = ables 1 l
1 l
l -2
å i =0 F (Ti ( i ))
that
identify 2
F (Tg ( i ) ) ³
F (Tl -1 ( l -1 ) )
2
1 l
the
2
2
=
1 l
l -1
å i =0 F (Ti ( i ) )
and G2 =
1 l
normalized, 2
F (Tg +1 ( g +1 ) ) ³
1 l
2
is P
F (Tl -1 ( l -1 ) )
ordered
Aj ( j )
2
2
( x ) = PG , G ( x - y, y )dy , 1
2
are two random correlated vari-
sub-channel 2
F (Tl -1 ( l -1 ) ) ,
coefficients
g = 0, , l - 2 ,
such
that where
is the minimum transmittance of the l sub-channels.
Proof. Let Aj ( j )
2
be identified as 2
Aj ( j ) =
l -1
1 l
å F (Ti ( i ) )
2
,
(45)
i =0
and let the transmittances to be sorted in a descending order as 1 l
2
F (T0 ( 0 ) ) ³
1 l
2
F (T1 ( 1 ) ) ³
1 l
2
F (Tl -1 ( l -1 ) ) .
(46)
Particularly, the random variable in (45) can be decomposed to the sum of two correlated random variables as 2
Aj ( j ) = G1 + G2 ,
(47)
where l -2
G1 =
1 l
å F (Ti ( i ))
i =0
and 9
2
(48)
1 l
G2 = Precisely, from (47) follows that the P
2
F (Tl -1 ( l -1 ) ) .
Aj ( j )
two random correlated variables as P
Aj ( j )
(49)
PDF function of Aj ( j )
2
can be evaluated via
( x ) = PG ,G ( x - y, y ) dy .
2
1
2
(50)
2
Specifically, exploiting the Bayesian formula, it follows that PG , G ( x , y ) = PG ( x ) ´ PG G 1
2
2
2 =x
1
(y ) ,
(51)
where PG ( x ) is the PDF of the l-th (ordered) sub-channel transmittance coefficient while 2
PG
1
1 l
G2 =x
(y )
is the conditional PDF of the sum of the first l-1 ordered sub-channel coefficients, at
F (Tl -1 ( l -1 ) )
2
=x.
Then let us assume that there are l - 1 available sub-channels, and let the i-th sub-channel’s quantity be referred to as
2
F (Ti ¢( i ) ) , i = 0, , l - 2 .
1 l -1
Without loss of generality, for this quantity the following relation holds: PG , G =y ( x ) = P 1 l -2 2 ( x ), 1 2 F (Ti ¢( i ) ) l -1 å i = 0 where P 1 l -1
F (Ti ¢( i ) )
2
(x )
(52)
is as Pc ( x )
P1 l -1
F (Ti ¢( i ) )
2
( x ) = 1- f ( x ) , x
³y.
(53)
c
■
3.2.3 Sub-channel Selection with a Complete Scan Theorem 1 (Sub-channel selection, l from n). The M -function of F (Ti ( i ) )
2
of the Gaussian
sub-channel i selection operator L0 is L
M1 0 l
F (Ti ( i ) )
2
2ö æ f çç 1l F (Ti* ( i ) ) ÷÷÷ è ø
(x ) =
+ ò1
¥
l
where
1 l
2
F (Ti* ( i ) ) £
F
( ( i )) Ti*
2
F (Ti ( i ) )
1 l
P 2
(
1 l
F (Ti ( i ) )
1 l
2
F (Ti ( i ) ) £
1 l
)
e
x 1 F (Ti ( i ) ) l
2
2
d 1l F (Ti ( i ) ) ,
is an threshold for all i , while f ( ⋅ ) and P (⋅ ) are
the CDF and PDF functions of the ordered channels with
2
1 l
F (Ti ( i ) )
2
coefficients. For l Gaussian sub-
2
F (Ti* ( i ) ) , i = 0, , l - 1 , the M -function of Aj ( j )
is
M
æ
L0 Aj ( j )
2
( x ) = f çèç 1l F (Ti* ( i ) ) l
+ ò1 l
¥ F
( ( i )) Ti*
2
P
(
1 l
2ö
÷ ø÷÷
F (Ti ( i ) )
10
2
)
e
x 1 F (Ti ( i ) ) l
2
2
d 1l F (Ti ( i ) ) .
2
Proof. First let us assume that the operation of the selection of the l sub-channels with the best
Ti ( i )
2
coefficient from the total n, is depicted by operator L0 .
Assuming that the l sub-channels are selected via L0 at a F (Ti* ( i ) )
2
threshold, the P
L0 Aj ( j )
2
PDF function can be expressed via the L0
P1 l
l -2
å i =0 F (Ti ( i ) )
2 1 , l
( x - y, y )dy
2
F (Tl -1 ( l -1 ) )
= PG , G 1
2
( x - y, y )dy
(54)
joint PDF function of the sub-channel 1l F (Ti ( i ) ) -s, as
P 2
The apriori fixed F (Ti* ( i ) ) 1 l
¥
L0
Aj ( j )
2
( x ) = ò0
PG , G ( x - y, y ) dy . 1
(55)
2
threshold is determined as
F (Ti ( i ) )
2
1 l
³
2
F (Ti* ( i ) ) , i = 0, , l - 1 ,
(56)
for all l Gaussian sub-channels that carry valuable information. The F (Ti* ( i ) )
2
threshold is derived from the security threshold nEve = 1 l of the optimal
Gaussian attack (see [2]), where l is the Lagrange multiplier as
l = F (T*
)
2
=
2
å i =0 F (Ti* ( i ) ) l -1
1 l
=
1 l
l -1
l -1
å i =0 å k =0Tk*e
- i 2 pik n
2
,
(57)
while T* is the expected transmittance of the l sub-channels under an optimal Gaussian attack. For a detailed description, see [2]. Without loss of generality, let us assume that the sub-channel transmittance coefficients are ordered in a descending order of (46). Then if
1 l
P1¢ l
2
F (Ti ( i ) ) < L0
F (Ti ( i ) )
2
(
1 l
2
F (Ti* ( i ) ) , let
F (Ti ( i ) )
1 l
2
)= f
2ö æ1 ç F (Ti* ( i ) ) ÷÷ dc ç ÷ø 1 F (Ti ( i ) ) è l l L0
2
where dc (⋅ ) is a random function, while for
P1¢ l
L0 F (Ti ( i ) )
2
(
1 l
1 l
F (Ti ( i ) )
F (Ti ( i ) )
2
)=P
2
L0
1 l
F (Ti ( i ) )
Specifically, after some calculations, the M -function of
1 l
l
¥
(
F Ti* ( i )
)
2
L0
P1 l
F (Ti ( i ) )
2
(
1 l
F (Ti ( i ) )
11
2
2
)
e
F (Ti ( i ) )
1 l
2
),
(58)
2
F (Ti* ( i ) ) ,
(
1 l
F (Ti ( i ) )
F (Ti ( i ) )
yielded without loss of generality as 2ö æ1 L L ç F (Ti* ( i ) ) ÷÷ x = f1 0 M1 0 2 ( ) 2 çl ÷ø F (Ti ( i ) ) F (Ti ( i ) ) è l l
+ ò1
1 l
³
(
2
x 1 F (Ti ( i ) ) l
2
2
).
(59)
of sub-channel i is
2
d 1l F (Ti ( i ) ) .
(60)
1 l
For l Gaussian sub-channels with function of Aj ( j )
M
L0
2
2
( x ) = MLF T
( i ( i ))
1 l
l
f
=
+ ò1 l
1 l
³
2
F (Ti* ( i ) ) , i = 0, , l - 1 , the M -
2
(x )
2ö æ1 ç F (Ti* ( i ) ) ÷÷ ÷ø ç F (Ti ( i ) ) è l
L0
(61)
2
1 l
l
2
is 0
Aj ( j )
F (Ti ( i ) )
¥
(
F Ti* ( i )
)
(
L0
P1
2
l
F (Ti ( i ) )
2
F (Ti ( i ) )
1 l
2
)e
x 1 F (Ti ( i ) ) l
2
2
d 1l F (Ti ( i ) ) .
In particular, the PrL ( ⋅ ) probability that l sub-channels are selected from the n is as 0
l n -l æ n ö÷ æ 2 öö æ æ 2 öö æ ÷÷ ç f ç 1 F (T * ( ) ) ÷÷ ÷÷ . PrL ( l ) = ççç ÷÷çç 1 - fc çç 1l F (Ti* ( i ) ) ÷÷ i i ÷÷ çè c çè l ÷÷ 0 è øø øø çè l ø÷ è 2
F (Ti ( i ) ) ³
1 l
Note that in a worst-case scenario, the condition
1 l
F (Ti* ( i ) ) 1 l
satisfied for the required number l of sub-channels. In this case, the threshold
2
(62) may not be
F (Ti* ( i ) )
2
can be redefined as 2
F (Ti* ( i ) ) = m max 1l ( F (Ti ( i ) ) ) ,
1 l
2
(63)
l
where 0 < m < 1 is a real variable. The condition of (63) satisfies that at least, the best subchannel is selected via the sub-channel allocation procedure. It also allows us to reevaluate the M -function of the l sub-channels as follows. Presuming that for l sub-channels, (63) is satisfied, resulting the ordered coefficients 1 l
2
F (Ti ( i ) ) ³
tion, the M
1 l
L0 Aj ( j )
M
2
2
L0
l
2
F (T0 ( 0 ) ) 1 F (Tl -1 ( l -1 ) )
2
(⋅ )
joint PDF func-
l
( x ) -function is precisely as ¥
L0 Aj ( j )
2
F (Ti +1 ( i +1 ) ) . Using the P1
( x ) = ò0 ò 1 l
ò0
F (T0 ( 0 ) )
F (Tl -2 ( l -2 ) )
2
l 1 l
1 l
2
m 1 F (T0 ( 0 ) ) l
F (Tl ( l ) )
2
òm d1 l
d1
0 1 l
(
2
F (T0 ( 0 ) )
2
d1 l
F (T1 ( 1 ) )
2
m 1 F (T0 ( 0 ) )
F (Tl -1 ( l -1 ) )
F (Tl +1 ( l +1 ) )
m 1 F (Tn -1 ( n -1 ) ) l
1 l
F (T0 ( 0 ) )
2
ò0
2
l
d1 l
F (Tl ( l ) )
2
2
l
ò ´P
d1
2
l -1
x1 F (Ti ( i ) ) e l å i =0 2
(64)
2 2
F (T0 ( 0 ) ) 1l F (Tn -1 ( n -1 ) )
which can be simplified into
12
2
)d
1 l
2
F (Tn -1 ( n -1 ) ) ,
M
æn ö
Aj ( j )
¥ x 1 F (T ( ) ) 2 i i
( x ) = l ççç l ÷÷÷÷ ò0
L0
çè ø
2
e
((
l
P
(
F (Ti ( i ) )
1 l
2
)
)) ( M ( x, F (T ( )) ) F (T ( ) ) )) d F (T ( ) ) ,
⋅ f m 1l F (Ti ( i ) )
(
2
n -l
l -1
2
- M x , m 1l
i
2
1 l
1 l
i
i
i
(65)
2
i
i
and the partial function is L
M1 0 l
=
F (Ti ( i ) ) ¥
òx
P
(
1 l
2
( w, x )
F (Ti ( i ) )
)
2
e
w 1 F (Ti ( i ) )
2
l
2
d 1l F (Ti ( i ) ) ,
by some fundamental theory. The probability that k sub-channels are selected from n at condition (63), is expressed as 2 æ 2 2ö PrL ( k ) = Pr çç 1l F (Tk ( k ) ) ³ 1l F (Ti* ( i ) ) ³ 1l F (Tk +1 ( k +1 ) ) ÷÷÷ 0 è ø
=
¥
mx
x
ò0 òmx ò0
P0,k ,k +1 ( x , y, z ) dxdydz ,
(66)
(67)
where
P0,k -1,k ( x , y, z ) =
n!
( k -2 ) !( n -k -1 ) !
Pc ( x ) k -2
⋅ ( fc ( x ) - fc ( y ) )
n -k -1
⋅Pc ( z ) ( fc ( z ) )
Pc ( y )
(68)
. ■
3.3 Optimized Complexity Progressive Scan Theorem 2 (Progressive sub-channel selection). The k average number of iterations needed for the selection of the l Gaussian sub-channels is minimized via an L sub-channel selection operator. Proof. The complexity of the sub-channel selection operators is discusses via the k average number of iterations needed for the procedure. The proof follows the definitions and notations of [29]. Exploiting the results of Theorem 1, using L0 , the kL overall average number of the iterations 0
(e.g., the number of comparisons of sub-channel transmittance coefficients at a F (Ti* ( i ) ) threshold per sub-channels) needed to derive Aj l
kL = 0
æn öæ
2
æ
is yielded as
* 1 ÷÷ çè 1 - fc ççè l F (Ti ( i ) ) å ççççè i ø÷÷ç i =1
l -i æ ÷ö ç ⋅ çç n + å ( n - i - k ) ÷÷÷ . çè ÷ø k =1
13
i 2 öö
÷÷ ÷÷ ÷÷ øø
n -i 2 öö æ æ ç fc ç 1 F (Ti* ( i ) ) ÷÷ ÷÷ ÷÷ çè èç l øø
(69)
Operator L0 does require the scan of the total n sub-channels, which is practically inconvenient. To resolve this problem we introduce operator L which performs a progressive scan: it stops the iteration as the l sub-channels have found and does not require to scan through all the n subchannels. Let L ¢ a slightly modified version of L that also handles the situation when only k < l sub-channels are found; but l was required by the legal parties for the transmission. In this case, the progressive scan operator L ¢ selects the remaining l - k sub-channels from the set ( k , , l -1 ) of bad sub-channels, and the Lagrange multiplier in (57) at L ¢ is reevaluated as
( )
lL¢ = F T
2
2
= F (Ti* ( i ) ) + v ,
(70)
where v is a nonnegative real variable, expressed as
(
2
v = F (Ti* ( i ) ) - min F (T ( k ) ) , , F (T ( l -1 ) ) 2
where F (T ( k ) ) , , F (T ( l -1 ) )
2
2
2
),
(71).
are the corresponding coefficients of ( k , , l -1 ) .
As follows, (70) allows to the legal parties to preserve the security conditions via a modified secu¢ = 1 lL¢ . rity threshold nEve The kL and kL¢ average number of iterations of operators L and L ¢ are evaluated as follows. Without loss generality, the L sub-channel selection operator models the situation if the iteration stops as the
1 l
2
F (Ti ( i ) ) ³
is not needed to determine
1 l
1 l
F (Ti* ( i ) )
F (Ti ( i ) )
2
2
condition is satisfied for l sub-channels, that is, it
for the remaining i = l, , n - 1 sub-channels.
Specifically, the PrL ( k ) probability of the availability of l sub-channels for k = l ,
PrL ( k ) =
n
æn öæ
æ
÷ * ç1 å ççèçç h ø÷÷÷ç çè 1 - fc çè l F (Th ( h ) )
h =l
h 2 öö
÷÷ ÷÷ ÷÷ øø
n -h 2 öö æ æ ⋅ çç fc çç 1l F (Th* ( h ) ) ÷÷ , ÷÷ ÷÷ è è øø
(72)
and the probability that only k < l , k = 0, , l - 1 sub-channels are available is k æ n ö÷ æ 2 öö æ1 * ÷÷ ç PrL ( k ) = ççç ÷ç 1 f F T ( ) ( ) ÷÷ ÷ cç i i ÷÷ èl øø çè k ø÷ çè n -k 2 öö æ æ ⋅ çç fc çç 1l F (Ti* ( i ) ) ÷÷ , ÷÷ ÷÷ è è øø
(73)
and
ML
Aj ( j )
where M
L( h ) Aj ( j )
2
(x )
l
2
( x ) = å PrL ( h )M A( ) ( h =0
L h j
j
)
2
(x ) ,
is the conditional MGF at i available Gaussian sub-channels.
Particularly, (74) can be rewritten precisely as
14
(74)
h æ n ö÷ æ 2 öö æ1 ç * ÷÷ ÷ç ç x = å çç ÷ ç 1 - P ç l F (Th ( h ) ) ÷÷ M ÷÷ 2 ( ) è øø Aj ( j ) ç h ø÷ è h =0 è n -h 2 öö æ æ L( h ) ⋅ çç P çç 1l F (Th* ( h ) ) ÷÷ M x ÷÷ 2 ( ) ÷÷ è è øø Aj ( j ) n æ ö g n ÷æ 2 öö æ1 * ÷÷ ç +å ççç ÷ç 1 P F T ( ) ( ) ÷÷ ÷ç g i ÷÷ çè l øø è g ø÷ è g =l ç l -1
L
n -g 2 öö æ æ1 * ÷÷ ç ç ⋅ ç P ç l F (Tg ( i ) ) ÷÷ ÷÷ è è øø
M
L( h )
(
Aj j
)
2
(75)
(x )
æç ö÷l -i çç M L( 2 ) ÷÷ çç A 2 ÷÷÷ è j( j ) ø
,
where
M
æ L( 1 ) öh æ L( 2 ) öl -h ÷ ÷ ç ç = M M x x x ÷ ÷ çç 2 ( ) 2 ( )÷ ç 2 ( )÷ Aj ( j ) è Aj ( j ) ø çè Aj ( j ) ø
L( h )
(76)
which can be rewritten as
æ
ML
Aj ( j )
2
æ
( x ) = ççè 1 - fc ççè 1l F (Ti* ( i ) )
2 öö
L( 1 ) ÷÷ M (x ) ÷÷ ÷÷ øø Aj ( j ) 2
2ö æ L( 2 ) + fc çç 1l F (Ti* ( i ) ) ÷÷÷ M ( x ). è ø Aj ( j ) 2
(77)
After some calculations, the kL overall average number of the iterations to determine Aj is as l i - 1ö 2 öö æ æ æ1 ÷÷ çç * ÷÷ ç ç i 1 f F T ( ) ( ) ÷÷ å çè c çè l i i çç i - l ÷÷ ÷÷ øø è ø i =l n
kL =
i -l 2 öö æ æ ⋅ çç fc çç 1l F (Ti* ( i ) ) ÷÷ ÷÷ ÷÷ è è øø n æ ö i n ÷æ 2 öö æ +n å ççç ÷÷çç 1 - fc çç 1l F (Ti* ( i ) ) ÷÷ ÷÷ ÷÷ è øø è i ø÷ è n -l +1 ç
(78)
n -i 2 öö æ æ1 * ÷÷ ç ç ⋅ ç fc ç l F (Ti ( i ) ) ÷÷ ÷÷ . è è øø
Assuming the situation that the k available number of sub-channels is lower than the expected
l , the best l - k “bad” sub-channels with
1 l
2
F (Ti ( i ) )
( n ) for all values of Ti* ( i ) 0
ing Ti* ( i )
2
2
2
Aj ( j ) . Increas-
2
Aj ( j ) , kL converges from (l ) to ( n ) , while kL¢ increases from (l )
to kL . The reason is that at L0 all n sub-channels are scanned (non-progressive operator). Using 0
L , the progressive operator selects the first l sub-channels that requires a lower number of iterations. At L ¢ , for low Ti* ( i )
2
2
Aj ( j )
in kL¢ » kL » (l ) , while as Ti* ( i )
2
the first l sub-channels were selected, which results
Aj ( j )
2
increases, the iteration is extended to all n
sub-channels that yields kL¢ » ( n ) . ■
3.4 Distribution of Gaussian Subcarriers Proposition 2 (Distribution of the Gaussian subcarriers). The (normalized) ¶i =
1 l
di ¢
2
squared
subcarrier magnitude is exponentially distributed, with single-carrier squared magnitude
aj ( j ) =
l -1
å i = 0 ¶i
=
1 l
l -1
å i =0 di ¢
2
(
, where di ¢ Î 0, sw2
0
) is the noisy Gaussian subcarrier.
Proof.
(
Let the output of i be the noisy Gaussian subcarrier di ¢ Î 0, sw2
0
) . Then the normalized
random variable 1 l
¶i =
di ¢
2
(84)
is exponentially distributed with density f ( ¶i ) =
1 s¶2
e
-¶i s2 ¶i
, ¶i ³ 0 ,
(85)
i
which can be rewritten as -x
f ( x ) = 1 e ¶ , x ³ 0 ,
(86)
¶
is the common mean. where ¶ The PDF of Pt ( x ) = P¶ ( x ) f¶ ( ¶i ) , 0 < x < ¶i , is evaluated as i
i
Pt ( x ) = while Pz ( x ) = P¶ ( x ) i
P¶ ( x ) i
f¶ ( ¶i )
=
i
-x 1 ¶ e ¶ -¶i ¶
,
(87)
1-e
( f¶ ( ¶i ) - f¶ ( d ) ), d £ x £ ¶i , is expressed as i
i
Pz ( x ) =
P¶ ( x ) i
f¶ ( ¶i )- f¶ ( d ) i
i
17
=
e
-x ¶
æ -d -¶i çççe ¶ -e ¶ ¶ çç çè
÷÷ö ÷÷ ÷÷ ÷ø
,
(88)
with an M -function ¶i
Mz ( w ) =
òd
=
e
æç çèe
Pz ( x )e wx dx
w d -d ¶
-d ¶
-e
w ¶i -¶i ¶
-¶ ¶ -e i
÷ö÷ 1-w ¶ ÷ø
(
)
(89)
.
Then let a j ( j ) be the averaged single-carrier squared magnitude as the sum of ¶i -s, as l -1
aj ( j ) =
å i = 0 ¶i 2 l -1 = 1l å i = 0 di ¢ ,
(90)
which can be rewritten as the sum of independent random variables as l
aj ( j ) =
n
å å xk i =1 k =i
= x1 + 2x 2 + + lxl + lxl +1 + lx n .
(91)
Specifically, using (91), the M -function of a term kx k is yielded as
Mkx
¥
( w ) = ò0
h
Pkx
(
= 1-
h
k w¶ h
( x )ewx dx -1
)
(92)
,
and the M -function of a j ( j ) is as
(
Ma ( ) ( w ) = 1 - w ¶ j j
n
-l
-1
) (1 - wk¶l )
.
(93)
k =l +1
Then Pa ( ) ( x ) is expressed as j j
Pa ( ) ( x ) j j
n -l æ l -1 l +k -1 çç x 1 + çç ¶l l -1 ! ¶ å ( -1 ) è i( ) k =1 l -2 æ -kx m öö 1 l ! n l ÷÷ ( ) ç ÷÷ ⋅ n -l -k ! k ! ( kl ) ⋅ ççe l ¶ - å m1 ! -kx ÷÷ ÷÷ . ( ) l¶ çè ÷÷ øø m =0
=
n! e ( n -l ) ! l !
-x ¶
(94)
( )
Particularly, at l - 1 total sub-channels, let us denote ¶i¢ =
1 l -1
di ¢
2
the i-th normalized Gaus-
sian subcarrier. Then
P¶ ¢ ( y ) = 1 e i
¶
-x -y ¶
, x ³y,
and the conditional PDF is as PG G =y ( x ) = P l -2 ¢ ( x ) 1 2 å i = 0 ¶i
= where G1 =
l -2
å i = 0 ¶i ,
1
(l -2 )
x - (l - 1 ) y ) l -1 (
(l -2 ) ! ¶
e
-
(95)
x -( l -1 )y ¶
, x ³ ( l - 1 ) y,
(96)
G2 = ¶l -1 , i.e., ¶i -s are ordered such that ¶ 0 ³ ³ ¶l -2 ³ ¶l -1 , while
18
M¶ ¢ ( w ) = i
= and the M -function of
l -2
å i =0 ¶i¢
+¥
ò0
P¶ ¢ ( y )e wx dx
(97)
i
1 e wy , 1-w ¶
is
M
l -1
å
( w ) = ( M¶ ¢ )
l -2 ¶¢ i =0 i
i
=
(
1 1-w ¶
e(
l -1
)
l -1 )wx
(98)
.
The joint PDF is
PG , G ( x , y ) = 2
1
n! ( n -l ) !(l -1 ) !
n -l
( fc ( x ) ) l -1
⋅ ( 1 - fc ( x ) )
Pc ( x ) P l -2 ¢ ( y ) , å i = 0 ¶i
(99)
where Pc ( x ) and fc ( x ) are the common PDF and CDF of the unordered variables, while n -l
PG ( x ) =
n! l -1 ! ¶ ( )
2
h
( -1 )
å (n -l -h )
h =0
e !h !
-
(l +h )x ¶
.
(100)
, x ³ 0, y ³ (l - 1 ) x .
(101)
Using (100), the joint PDF in (99) can be rewritten precisely as PG , G ( x , y ) 2
1
n -l
=
h
( -1 )
(l -2 )
n!
å (n -l -h )!(l -1)!(l -2 )! h ! ¶ ( y - (l - 1) x ) l
e
-y +( h +1 )x ¶
h =0
Note that by some fundamental theory 1- fc ( x )
lim
x +¥ Pc ( x )
= W > 0,
(102)
and
lim f¶
l +¥
( x ) = e-e
0
-x -log l
.
(103)
Then, the x and p quadrature-level (single-carrier) error probability at L0 is as L0 perr ( Aj ) =
where M
L0
s2 SNR= w
Aj ( j )
0
2
⋅SNR
1 p
p 2
ò0
M
æ sin2 ( p2 ) ÷ö ç 2 ÷df , 2 ç çè sin f ÷ø Aj ( j ) ⋅SNR
L0
(104)
2 is a scaled SNR quantity, is the M -function of Aj ( j ) ⋅ SNR ; SNR
2s 2 , and f Î éë 0, 2p ùû . In particular, the formula of (104) can be applied to derive the
perr ( Aj ) error probabilities (single-carrier quadrature level) of the operators using the M -
functions proposed in sub-sections 3.1-3.2, precisely as L perr ( Aj ) =
1 p
p 2
ò0
ML
Aj ( j )
2
⋅SNR
æ sin2 ( p2 ) ÷ö çç 2 ÷ d f , çè sin f ÷ø
(105)
and L¢ perr ( Aj ) =
1 p
p 2
ò0
M L¢
Aj ( j )
19
2
⋅SNR
æ sin2 ( p2 ) ÷ö çç 2 ÷ d f . çè sin f ÷ø
(106)
(
Specifically, assuming Ti ( i ) Î 0, sT2
( i )
i
) , and by using the
M -functions derived in the
L0 previous sections, the corresponding perr ( Aj ) single-carrier quadrature level error probability at 2 L0 in function of Aj ( j ) ⋅ SNR , are yielded as shown in Fig. 3. The results can be extended
for arbitrarily distributed Ti ( i ) coefficients.
L0 perr ( Aj )
Figure 3. The
(
Ti ( i ) Î 0, sT2
i
( i )
at
L0
M
via
L0 Aj ( j )
2
⋅SNR
, in function of
2 Aj ( j ) ⋅ SNR ,
).
Using operators L and L ¢ with M L
Aj ( j ) ⋅SNR 2
(⋅ )
and M L¢
Aj ( j ) ⋅SNR 2
, the complexity can be re-
L duced, however, the yielding single-carrier quadrature level error probabilities perr ( Aj ) and L¢ perr ( Aj ) are slightly increased, as depicted in Fig. 4.
L L¢ Figure 4. The perr ( Aj ) and perr ( Aj ) at L and L¢ via ML
Aj ( j ) ⋅SNR
(
2 , Ti ( i ) Î 0, sT2 function of Aj ( j ) ⋅ SNR
i
20
( i )
).
2
(⋅ )
and M L¢
Aj ( j ) ⋅SNR 2
, in
L0 L¢ The perr ( Aj ) is approximately coincidences with perr ( Aj ) because both operators find l subL0 L¢ channels however the threshold differs that leads to perr ( Aj ) ³ perr ( Aj ) . In comparison to L , L L¢ perr ( Aj ) ³ perr ( Aj ) because operator L¢ always determines the required number l of the subL0 L¢ L channels at a given threshold. It leads to the final conclusion perr ( Aj ) £ perr ( Aj ) £ perr ( Aj ) .
■
4 Random Matrix Formalism of Multicarrier CVQKD First, we summarize the basic functions and random matrix tools in Propositions 3 and 4, to the evaluate the results in Theorem 3 and Lemma 1. The proofs throughout Section 4 follow the notations of [27] and [28]. For the detailed description of the AMQD-MQA multiple-access multicarrier CKQKD scheme, see [3].
4.1 Multiuser Quadrature Allocation for Multicarrier CVQKD Proposition 3 (Random matrix decomposition of F ( T ( ) ) of K users.) At l Gaussian subchannels and K users, the l ´ K channel matrix F ( T ( ) ) of K users can be decomposed as F (T(
) ) = CZF ,
where Z is an l ´ K random matrix, C is an l ´ l random matrix, while
F is a K ´ K random matrix. Proof. Let the Zi, j entries of Z be independent, arbitrarily distributed complex random variables with identical mean mi, j = m and variance si2, j = 1 l such that the Lindeberg condition [27] 1 K
å ò i, j
{
2
Zi , j ³d }
Zi, j d = 0 ,
(107)
is satisfied, where is a probability measure function that returns an event’s probability. Let C be an l ´ l random matrix, while F is a K ´ K random matrix, independent from each other and from Z . The h -transform of a nonnegative random variable X is as hX ( g ) = éê 1+1g X ùú , (108) ë û where g is a nonnegative real number, and 0 < hX ( g ) £ 1 . Without loss of generality, let the logical channel k of user U k , defined as k = { 0 , m-1 } .
(109)
The single-carrier transmittance coefficient of k is referred to as Aj ( k ) . Particularly, then, for K , l ¥ and K l c, where c is a nonnegative variable, for the h transform of F ( T (
†
)) F ( T ( ))
21
é ù = êV † ( D, g ) ú , êë F ( T( ) )F ( T( ) ) úû is an independent random variable with a distribution CC†
h
where D Î
{ CC† }
F ( T(
where T Î FF†
{
) )F ( T( ) )
{
totic spectra of CC† , and V
(110)
†
F ( T(
†
) )F ( T( ) )
( D, g ) =
} of the asymp-
1 , é ù ê ú ê ú T 1+ gcD ê ùú ê 1+ gT éê DV úú , D g ( ) ê ú † ê êë úû úû F ( T( ) )F ( T( ) ) ë
(111)
} is an independent random variable with a distribution of the asymptotic spec-
tra of FF† [27]. The n -transform (Shannon transform) of a nonnegative random variable X is defined as nX ( g ) = éë log2 ( 1 + g X ) ùû ,
(112)
where g is a nonnegative real number. †
Let the n -transform of F ( T ( ) ) F ( T ( ) ) is as
n
F ( T(
†
) )F ( T( ) )
( g ) = nCC ( cgd ) + cnFF ( gt ) - c g gg †
†
d t
log2 e ,
(113)
where gd and gt are random variables such that [27] gd gt g
= 1 - hFF† ( gt )
(114)
= 1 - hCC† ( cgd ) .
(115)
and
c
gd gt g
Note that by denoting gd ( g ) the solution for (114) and (115), the result in (110) can be rewritten as
h
F ( T(
†
) )F ( T( ) )
= hCC† ( cgd ( g ) ) .
(116)
Throughout the manuscript, let X and Y be defined as independent random variables X Î é 0,1 ù and Y Î é 0,1 ù drawn from a uniform distribution on éë 0,1 ùû . ë
û
ë
û
Specifically, the decomposition of F ( T ( l to define a function H( ) ( a, g ) ,
( j - 1)
) ) = CZF
with a j-th column F ( T ( ) )j allows us
K £ a < j K , where a is a random variable, precisely
as l H( ) (a, g ) =
-1 æ †ö ÷ ç F ( T ( ) )j çç I + g å F ( T ( ) )k F ( T ( ) )k ÷÷ F ( T ( ) )j ; 2 çè F ( T( ) )j ø÷ k¹j
1
†
(117)
where I is the identity, such that for K , l ¥ , and a Î éë 0,1 ùû l H( ) (a, g )
gt ( g ) g éë D ùû
;
l where for gt ( g ) the result of (114) and (115) holds, and H( ) converges to
22
(118)
H( a , g ) é ë B ( X ,a ) ùû
,
(119)
where B (⋅ ) is a limiting bounded measureable function [27], and H ( a, g ) is evaluated as
é ù ú. 1 H ( a, g ) = êê B ( X , a ) é ùú 1 1 , gc B X Y X + ( ) ê ú 1+ gH(Y , g ) ëê ëê ûú ûú
(120)
Without loss of generality, the entries of CZF are independent, arbitrarily distributed zero-mean ( mi, j = 0 ) complex random variables with variance
si2, j =
Pi , j l
,
(121)
where P is an l ´ K deterministic matrix with uniformly bounded ( i, j ) entries, with variance profile var ( a, b ) [27]. Let varl ( ⋅ ) be the variance profile function as
varl : éë 0,1 ) ´ éë 0,1 ) such that for
i -1 l
i l
£a
0 ) ( éë rJ ( X ,Y ) X ùû > 0 )
c
< 1.
If (169) holds, then without loss of generality, feff ( c, SNRU* ) = c éê log2 ( 1 + SNRU* (Y , SNRU* k k k ë and if SNRU* ¥ also holds, then k
28
(169)
)) ùûú ,
(170)
(b, SNRU* Specifically, using (168), the Psym ( k )
é
ù ú. rJ X ,Y ) ùú Xúú ëê (Y ) ûú û
) êê 1+c éê ( (
rJ X ,b )
(171) ëê symmetric private classical capacity (the maximum comk
mon rate at which the K users both can reliably transmit private classical information over the l sub-channels of ) of the k logical channel of U k is expressed as follows
Psym ( k ) =
1 K
( feff ( c, å i =0 SNRi* ) l -1
(
(
) )
l -1 l -1 é é ù ù + ê log2 1 + å i = 0 SNRi*c ê rJ ( X ,Y ) ¡ Y , å i = 0 SNRi* X ú ú ë ë û û l -1 l -1 l -1 * é * * ù -cå i = 0 SNRi ê Y , å i = 0 SNRi ¡ Y , å i = 0 SNRi ú log2 e ) ë û é æ öù l -1 1 ÷÷÷ ú = K1 ( f eff c, å i = 0 SNRi* + êê log2 ççç l -1 * ÷ú , SNR X y ÷ ç i ) øú è ( å i =0 êë û l -1 é * ù + ê y X , å i = 0 SNRi ú - 1 log2 e ) , ë û l -1
å i =0 SNRi*
) (
( (
(
where
(
)
(172)
) ) )
is the total SNRs of the l Gaussian sub-channels evaluated for the transmis-
sion of private classical information expressed as
l -1
l -1
å i =0 SNRi* = å i =0 sw2
s2
i*
i
, where s 2
i*
is
shown in (158), X Î é 0,1 ù and Y Î é 0,1 ù , and ë
ë
û
û
é ù rJ ( X ,b ) l -1 ú, b, å i = 0 SNRi* = êê l -1 l 1 1+ c å i = 0 SNRi* éê rJ ( X ,Y )¡ (Y , å i =0 SNRi* ) X ùú ú ëê ë û ûú
(
)
(173)
and function ¡ ( ⋅ ) is as
(
)
l -1
¡ b, å i = 0 SNRi* =
(
l -1
In particular, defining function b, å i = 0 SNRi*
1
(
l -1
l -1
1+ å i =0 SNRi* b, å i = 0 SNRi*
)
.
(174)
) precisely as
é ù ê ú rJ ( X ,b ) l -1 ú, b, å i = 0 SNRi* = êê (175) é ùú l 1 ú 1 ê 1+ å SNRi*c( 1-b ) êê rJ ( X ,Z ) ú Xú i =0 * * 1 l -1 ê 1+åli êë úû úû = 0 SNRi ( Z , å i = 0 SNRi ) ë where Z Î éb,1 ù is an independent random variable, drawn from a uniform distribution on
(
ë
)
û
é b,1 ù , the formula of (172) can be rewritten in a simplified form specifically as ë û l -1 l -1 é Psym ( k ) = K1 c ê log2 1 + å i = 0 SNRi* Y , å i = 0 SNRi* ë From (176), the Ssym (U k ) symmetric secret key rate (a common rate at which
(
(
)) ùúû .
(176)
the K users both
can reliably transmit private classical information over the l sub-channels of ) of U k is as l -1 l -1 é ù Ssym (U k ) £ K1 c ê log2 1 + å i = 0 SNRi* Y , å i = 0 SNRi* ú . (177) ë û The Psym ( k ) of U k from (176) in function of r is depicted in Fig. 5.
(
(
29
))
Figure 5. The Psym ( k ) of U k in function of SNRU* for 0 £ r £ 1 , where SNRU* is the SNR k
k
for the transmission of private classical information over the k logical channel of U k , via m Gaussian sub-channels; r = m l identifies the ratio of the sub-channels allocated to U k .
4.2 Multiuser Transmission over Identical Gaussian Sub-channels Lemma 1 (Random matrix formalism of multiple-access multicarrier CVQKD for identically allocated sub-channels). Let set = { i } , i = 0, , h - 1 , h £ l , refer to the sub-channels of the 1 £ k £ K £ l users, obtained via L0 , L , L ¢ . The random matrix decomposition of the
h ´K
channel matrix F ( T (
))
is F ( T (
) ) = CUF ,
where U = ( u 0 , , uK -1 ) is an
h ´K
T isotropic unitary matrix, uk = éë qk ,0 , , qk ,h -1 ùû , C
is an h ´ h
random matrix
h -1 (k ) (k ) (k ) T C = diag ( 0 , , h -1 ) , with k = éê Aj ,0 , , Aj ,h -1 ùú , where Aj = h1 å i = 0 F (Ti ( i ) ) is the ë û averaged transmittance coefficients of U k in , such that i = k for i = 0, , K - 1 , while
F is a K ´ K random matrix, F = ( F0 , , FK -1 ) . Proof. The proof utilizes Proposition 3. Let us identify set = { i } , i = 0, , h - 1 , h £ l ,
(178)
the sub-channels allocated to the 1 £ k £ K £ l users. The transmittance coefficients are determined via operators L0 , L , L ¢ (see Theorem 1). The proof utilizes the terms and notations of [27]. The random matrix decomposition of the h ´ K channel matrix F ( T ( F (T(
) ) = CUF ,
))
is (179)
where U = ( u 0 , , u K -1 )
30
(180)
is an h ´ K isotropic unitary matrix, independent of C and F , UU † = I , with arbitrarily distributed random variables such that the Lindeberg condition (see (107)) is satisfied, with mean mi, j = m and variance si2, j = 1 l , T uk = éë qk ,0 , , qk ,h -1 ùû ,
(181)
C = diag ( 0 , , h -1 ) ,
(182)
(k ) (k ) T k = éê Aj ,0 , , Aj ,h -1 ùú , ë û
(183)
C is an h ´ h random matrix where
and (k )
Aj
=
1 h
h -1
å i =0 F (Ti ( i ))
(184)
is the averaged transmittance coefficients of U k taken over , such that
i = k for i = 0, , K - 1 ,
(185)
while F is a K ´ K random matrix, F = ( F0 , , FK -1 ) .
(186)
Precisely, if the users transmit through the same subcarriers, then for all U i , i = 0, , k - 1 users, an identical U = k logical channel is allocated, see (185); thus (182) can be rewritten as [27] i
C = diag ( k ) ,
(187)
with the averaged coefficient of (184) for the h sub-channels of user U k . Particularly, since all users use the same Gaussian sub-channels, the empirical distribution of
CC† converges to a nonrandom limit F
C
2
, thus
CC† F where C
C
2
,
(188)
is a random variable having a distribution of the asymptotic singular value distribu-
tion of the singular values of C , i.e., the distribution of C
2
is determined by the asymptotic
spectra of CC† , thus without loss of generality (k )
C = Aj ( k ) = Aj
,
(189)
where Aj ( k ) is the Aj coefficient of the k channel of U k , see (184). Let be a random variable having a distribution of the asymptotic singular value distribution of the singular values of F , the distribution of
2
is determined by the asymptotic spectra of FF† .
Specifically, the feff ( ⋅ ) efficiency function in (168) can be rewritten as
feff ( c, SNRU*
k
) = c éêë log2 (1 + éêë Aj ( k )
where hU is the multiuser efficiency, evaluated as k
31
2
ù * * ú SNRU k hU k ( SNRU k û
)) ùúû ,
(190)
hU
é ù ê ú 2 ê ú C ú = é 1 2ù ê é ù ê 2 ê C ú úú ê 1+SNRU* k c C 2 êê é 2 ù ë û úú ê 1+ ê C ú SNR* 2 h ú ú ê Uk Uk ú ê ê ú ë ë ûû ë û é ê 2 ê Aj ( k ) = é 1 2 ù êê é 2 ê Aj ( k ) ú ê 1+SNR* c A 2 êê êë úû ( ) Uk j k ê é ù ê 2 ê 1+ êê Aj ( k ) úú SNRU* k êë úû ëê ë
k
such that
h ( SNRU* where
(
)
2ù é h SNRU* ê Aj ( k ) ú = k ë û
(
1+
k
hU
SNRU* k
) h ( SNRU* éêë Aj ( k ) k
k
2
(191)
ù ú ú ú, ùú úú ú úú 2 hU ú ú k û û
)
ù ú , û
(192)
)
2ù é ê Aj ( k ) ú hU êë úû k
2 2ù é é ù Aj ( k ) ê Aj ( k ) ú ê ê ú. ë ûú = ê 2ù 2 2ù 2ù é é é * * c c h + + A A A A SNR 1 1 SNR ( ) ( ) ( ) ( ) ( ) ê ú ê ú ê ú ) ( Uk ëê j k ûú ) Uk úúû j k êë ( Uk ëê j k ûú )( j k ëê ûú (193)
Without loss of generality, the term hU
hU
k
k
( SNRU* ) in (190) for ( SNRU* ) = (1 - c ) .
0 £ c £ 1 is expressed as
k
(194)
k
Precisely, the Psym ( k ) symmetric private classical capacity of the k logical channel of U k is particularly yielded as
where
Psym ( k ) =
1 K
=
1 K
l -1
å i =0 SNRi* = ål sw2
s2
i
i*
(
(
))
l -1 é ù c ê log2 1 + Y , å i = 0 SNRi* ú ë û l -1 å i =0 SNRi* 1 æ ÷ö ç1 - h † ( x ) ÷ dx , ò0 xç F ( T( ) )F ( T( ) ) è ø
, Y Î é 0,1 ù , and h ë
F ( T(
û
(195)
is the h -transform of
†
) )F ( T( ) )
†
F ( T ( ) ) F ( T ( ) ) , specifically as æ çç l -1 * SNR h SNRi* = † † çå å i CC i =0 F ( T( ) )F ( T( ) ) çç i = 0 è
(
h
(
l -1
while function Y , å i = 0 SNRi*
(
l -1
÷÷ö ÷, l -1 * ÷ † ( å i = 0 SNRi ) ÷ ÷ø
c-1+ h
F ( T( ) )F ( T(
h
F ( T( ) )F ( T(
) )†
(196)
))
) [27] is as
Y , å i = 0 SNRi*
(
)
l -1
)
l -1
1+ Y , å i = 0 SNRi*
)
2 l -1 é ù å i =0 SNRi* F (Ti ( i ) ) ú. = êê 2 l -1 l -1 * * ú c c SNR 1 1 , SNR Y F T Y Y + + ( ( ) ) ( ) å å ( i =0 i ) ûú i i i i =0 ëê
Using (196), the following relation holds for the K users æ çç I + l -1 SNR *F ( T ( 1 Tr å i =0 i ççè K
(
= 1-
1æ ç1 cç è
h
F ( T(
†
) )F ( T( ) )
32
†
))
(å
F (T(
l -1
SNRi* i =0
(197)
-1 ö
)))
)
÷ö . ÷ø
÷÷ ÷ ø÷
(198)
Precisely, the Ssym (U k ) symmetric secret key rate of U k over identical sub-channels is expressed as
Ssym (U k ) £
l -1
1 K
å i =0 SNRi* 1 æ ÷ö ç1 - h † ( x ) ÷ dx . ò0 xç F ( T( ) )F ( T( ) ) è ø
(199)
In particular, using the density function fc ( x ) (Marchenko-Pastur density function [27]),
(
fc ( x ) = 1 -
1 c
+
)
+
+
( x -u ) ( v -x )
x (x ) +
2pcx
(
where x (⋅ ) is a related random function, and u = 1 - c
2
)
,
(200)
(
and v = 1 +
2
)
c , (198) can be
rewritten precisely as 1 Tr K
=
b
òa
æ çç I + l -1 SNR *F ( T ( å i =0 i ççè
(
1 l -1
1+ å i =0 SNRi*x
†
))
F (T(
-1 ö
)))
÷÷ ÷ ø÷
(201)
fc ( x ) dx .
The density function fc ( x ) for c = 0.2, 0, 5 and c = 1 is depicted in Fig. 6.
Figure 6. The density function fc ( x ) for c = 0.2, 0.5,1 . ■
5 Conclusions The multicarrier CVQKD systems are aimed to avoid the main drawbacks of CVQKD, such as low secret key rates, low tolerable losses and short communication distances, allowing the legal parties to establish an unconditional secure communication over the standard networks. We provided a distribution statistics and random matrix framework for multicarrier CVQKD. Exploiting order statistics, we defined different statistics methods to derive the averaged transmittance coefficients from the Gaussian sub-channel transmittances. The provided operators order, select, and sum the sub-channel coefficients for the information transmission at a given threshold. We analyzed the complexity and the error probabilities of the sub-channel selection operators and defined an optimized progressive sub-channel scanning scheme. We characterized a random matrix formalism for multicarrier CVQKD that utilizes the mathematical background of random matrix 33
theory. The framework is formulated for multiple-access multicarrier CVQKD, through the multiuser quadrature allocation multiuser transmission scheme. Using the random matrix formalism, we studied the private classical information capabilities of the users at different subcarrier allocation mechanisms. The proposed combined framework of order statistics and random matrix theory is particularly convenient for the firm portrayal of the information flowing in experimental multicarrier CVQKD scenarios.
Acknowledgements The author would like to thank Professor Sandor Imre for useful discussions. This work was partially supported by the GOP-1.1.1-11-2012-0092 (Secure quantum key distribution between two units on optical fiber network) project sponsored by the EU and European Structural Fund, and by the COST Action MP1006.
References [1] [2] [3] [4] [5] [6] [7] [8] [9]
[10] [11] [12] [13]
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[14] C. Weedbrook, S. Pirandola, R. Garcia-Patron, N. J. Cerf, T. Ralph, J. Shapiro, and S. Lloyd. Rev. Mod. Phys. 84, 621 (2012). [15] Wi. Shieh and I. Djordjevic. OFDM for Optical Communications. Elsevier (2010). [16] L. Gyongyosi, Scalar Reconciliation for Gaussian Modulation of Two-Way Continuousvariable Quantum Key Distribution, arXiv:1308.1391 (2013). [17] P. Jouguet, S. Kunz-Jacques, A. Leverrier, P. Grangier, E. Diamanti, Experimental demonstration of long-distance continuous-variable quantum key distribution, arXiv:1210.6216v1 (2012). [18] M. Navascues, F. Grosshans, and A. Acin. Optimality of Gaussian Attacks in Continuous-variable Quantum Cryptography, Phys. Rev. Lett. 97, 190502 (2006). [19] R. Garcia-Patron and N. J. Cerf. Unconditional Optimality of Gaussian Attacks against Continuous-Variable Quantum Key Distribution. Phys. Rev. Lett. 97, 190503 (2006). [20] F. Grosshans, Collective attacks and unconditional security in continuous variable quantum key distribution. Phys. Rev. Lett. 94, 020504 (2005). [21] M R A Adcock, P Høyer, and B C Sanders, Limitations on continuous-variable quantum algorithms with Fourier transforms, New Journal of Physics 11 103035 (2009) [22] S. Pirandola, S. Mancini, S. Lloyd, and S. L. Braunstein, Continuous-variable Quantum Cryptography using Two-Way Quantum Communication, arXiv:quantph/0611167v3 (2008). [23] L. Hanzo, H. Haas, S. Imre, D. O'Brien, M. Rupp, L. Gyongyosi. Wireless Myths, Realities, and Futures: From 3G/4G to Optical and Quantum Wireless, Proceedings of the IEEE, Volume: 100, Issue: Special Centennial Issue, pp. 1853-1888. (2012). [24] D. Tse and P. Viswanath. Fundamentals of Wireless Communication, Cambridge University Press, (2005). [25] D. Middlet, An Introduction to Statistical Communication Theory: An IEEE Press Classic Reissue, Hardcover, IEEE, ISBN-10: 0780311787, ISBN-13: 978-0780311787 (1960) [26] S. Kay, Fundamentals of Statistical Signal Processing, Volumes I-III, Prentice Hall, (2013) [27] A. M. Tulino, Sergio Verdu, Random Matrix Theory and Wireless Communications, now Publishers, Hanover, USA, ISBN: 1-933019-00-X; ISSNs: Paper version 1567-2190; Electronic version 1567-2328 (2004). [28] A. Edelman, N. Raj Rao, Random matrix theory, Acta Numerica, pp- 1-65, DOI: 10.1017/S0962492904000236 (2005). [29] H-C. Yang, M-S. Alouini, Order Statistics in Wireless Communications, Cambridge University Press, Cambridge, UK (2011). [30] S. Imre, F. Balazs: Quantum Computing and Communications – An Engineering Approach, John Wiley and Sons Ltd, ISBN 0-470-86902-X, 283 pages (2005). [31] D. Petz, Quantum Information Theory and Quantum Statistics, Springer-Verlag, Heidelberg, Hiv: 6. (2008).
35
[32] R. V. Meter, Quantum Networking, John Wiley and Sons Ltd, ISBN 1118648927, 9781118648926 (2014). [33] L. Gyongyosi, S. Imre: Properties of the Quantum Channel, arXiv:1208.1270 (2012). [34] K Wang, XT Yu, SL Lu, YX Gong, Quantum wireless multihop communication based on arbitrary Bell pairs and teleportation, Phys. Rev A, (2014). [35] Babar, Zunaira, Ng, Soon Xin and Hanzo, Lajos, EXIT-Chart Aided Near-Capacity Quantum Turbo Code Design. IEEE Transactions on Vehicular Technology (submitted) (2014). [36] Botsinis, Panagiotis, Alanis, Dimitrios, Ng, Soon Xin and Hanzo, Lajos LowComplexity Soft-Output Quantum-Assisted Multi-User Detection for Direct-Sequence Spreading and Slow Subcarrier-Hopping Aided SDMA-OFDM Systems. IEEE Access, PP, (99), doi:10.1109/ACCESS.2014.2322013 (2014). [37] Botsinis, Panagiotis, Ng, Soon Xin and Hanzo, Lajos Fixed-complexity quantumassisted multi-user detection for CDMA and SDMA. IEEE Transactions on Communications, vol. 62, (no. 3), pp. 990-1000, doi:10.1109/TCOMM.2014.012514.130615 (2014).
36
Supplemental Information S.1 Notations The notations of the manuscript are summarized in Table S.1. Table S.1. Summary of notations. i
Index for the i-th subcarrier Gaussian CV, fi = x i + ipi .
j
Index for the j j = x j + ip j .
j-th
Gaussian
single-carrier
CV,
Number of Gaussian sub-channels i for the transmission l
of the Gaussian subcarriers. The overall number of the subchannels is n. The remaining n - l sub-channels do not transmit valuable information.
x i , pi
Position and momentum quadratures of the i-th Gaussian subcarrier, fi = x i + ipi .
x i¢, pi¢
Noisy position and momentum quadratures of Bob’s i-th noisy subcarrier Gaussian CV, fi¢ = x i¢ + ipi¢ .
x j , pj
Position and momentum quadratures of the j-th Gaussian single-carrier jj = x j + ip j .
x j¢ , p j¢
Noisy position and momentum quadratures of Bob’s j-th recovered single-carrier Gaussian CV jj¢ = x j¢ + ip j¢ .
x A,i , pA,i
Alice’s quadratures in the transmission of the i-th subcarrier.
fi , fi¢
Transmitted and received Gaussian subcarriers.
Q (⋅ )
Gaussian tail function.
37
z Î ( 0, Kz )
A d-dimensional input CV vector to transmit valuable information. A
d-dimensional
noisy
output
vector,
T
z ¢T = A† z + ( F d ( D ) ) = ( z 0¢ , , zd¢ -1 ) ,
z¢T
SNRU*
z j¢ =
(
1 l
å i =0 F (Tj,i ( j,i ) )) z j l -1
where
( (
+ F ( D ) Î 0, 2 sw2 + s 2 0
)).
The SNR quantity of U k , evaluated for the transmission of k
private classical information. Total SNR of the l Gaussian sub-channels, evaluated for the transmission of private classical information, l -1
l -1
å i =0 SNRi* = å i =0 sw2
i
, where sw2 is the constant
s2
i*
i
subcarrier modulation variance, and
s2 * i
=
sw2 i
æ sw2 F (Ti ( i ) ) 2 +sC2 ö÷-1 çç i i ÷ çç 1+s 2 s2 F (T ( ) ) 2 - 1 ÷÷÷ , i i è Ci wi ø
l -1
where sC2 = s02 + N i , s2 is the vacuum noise, N i is the
i =0
excess noise of the Gaussian sub-channel i as
å SNRi*
i
Ni =
( F (T
(Wi -1 )
Eve ,i
1- F (TEve ,i )
2
)
2
),
where Wi is the variance of Eve’s EPR state used for the attacking of i , while TEve,i
2
= 1 - Ti
2
tance of Eve’s beam splitter (BS), and Ti
2
is the transmitis the transmit-
tance of i . 1 l
F Ti ( i )
(
)
2
Normalized, unordered sub-channel transmittance coefficient, real variable.
1 l
F (Ti ( i ) )
2
Normalized, ordered sub-channel transmittance coefficient, real variable.
Aj ( i )
Single-carrier channel coefficient, complex variable, expressed as Aj ( j ) =
38
1 l
l -1
å i =0 F (Ti ( i )) .
Aj ( j )
Single-carrier channel coefficient, real variable, expressed as
2
2
Aj ( j ) =
1 l
l -1
å i =0 F (Ti ( i ))
2
, real variable.
M (⋅ )
The MGF (Moment-Generating function) function.
P (⋅ )
The PDF (Probability Density Function) function.
Pc ( ⋅ )
The common PDF of unordered random variables.
f (⋅ )
The CDF (Cumulative Distribution Function) function.
fc ( ⋅ )
The common CDF of unordered random variables.
Rerr ( x )
Error rate function.
R err
Average error rate.
LU
Channel selection operator for user U k , LU = OS , where k
O is the sub-channel ordering operator, while S is the sum generating operator.
k
Threshold parameter for the sub-channel selection process. F (Ti
*
( i ) )
2
If
1 l
2
F (Ti* ( i ) ) £
1 l
F (Ti ( i ) )
2
for Gaussian sub-
channel i , the sub-channel is qualified as “good”, otherwise it is put into the “bad” set. Sub-channel selection with complete scan at apriori fixed 2
threshold F (Ti* ( i ) ) , which is determined via the op-
L0
timal Gaussian attack. All sub-channels (n) are scanned through, the best l selected. In a modified version of L0 , the threshold is evaluated via the transmittance coefficient of the best available sub-channel as 1 l
2
F (Ti* ( i ) ) = m max 1l ( F (Ti ( i ) ) ) . 2
l
Progressive sub-channel selection with an apriori fixed 2
L
threshold F (Ti* ( i ) ) , at an optimized complexity. The iteration does not require full scan, as the l sub-channels are determined the iteration stops.
39
Progressive sub-channel selection with apriori fixed thresh2
L¢
old F (Ti* ( i ) ) , at an optimized complexity. If no l subchannels available at the threshold, the operator selects the remaining sub-channels from the bad channel set (bad: the apriori fixed threshold does not hold for the sub-channel). Lagrange multiplier, used to determine the F (Ti* ( i ) )
2
threshold parameter, as
l = F (T*
l
1 l
= while T*
)
2
=
l -1
1 l
å i =0 F (Ti* ( i ) ) l -1
l -1
å i =0 å k =0Tk*e
-i 2 pik n
2
2
,
is the expected transmittance of the l sub-
channels under an optimal Gaussian attack. Lagrange multiplier at L ¢ ,
( )
lL¢ = F T
2
2
= F (Ti* ( i ) ) + v ,
where v is a nonnegative real variable,
lL¢
v = F (Ti* ( i ) ) - min
((
2 2
F (T ( k ) ) , , F (T ( l -1 ) ) 2
and F (T ( k ) ) , , F (T ( l -1 ) )
2
2
))
,
are the correspond-
ing coefficients of the bad set ( k , , l -1 ) . l -2
Correlated random variables, G1 =
G1 , G2
G2 = cients 1 l
1 l
1 l
å F (Ti ( i ))
2
, and
i =0
2
F (Tl -1 ( l -1 ) ) , where the transmittance coeffiare
sorted 2
F (T0 ( 0 ) ) ³
1 l
in
a 2
F (T1 ( 1 ) ) ³
descending 1 l
order 2
F (Tl -1 ( l -1 ) ) ,
2
and Aj ( j ) = G1 + G2 .
kL , kL , kL¢ 0
The average number of the iterations (e.g., the number of comparisons of sub-channel transmittance coefficients at a
40
F (Ti* ( i ) ) threshold per sub-channels). L0 L L¢ perr ( Aj ) , perr ( Aj ) , perr ( Aj )
M
2 at operator L0 . The MGF function of Aj ( j ) ⋅ SNR
L0
Aj ( j ) ⋅SNR 2
ML
(⋅ )
Aj ( j ) ⋅SNR 2
M L¢
Single-carrier level, quadrature error probability at operators L0 , L and L ¢ .
2 at operator L . The MGF function of Aj ( j ) ⋅ SNR 2 at operator L ¢ . The MGF function of Aj ( j ) ⋅ SNR
Aj ( j ) ⋅SNR 2
h -transform. For a nonnegative random variable X is defined as hX ( g ) = éê 1+1g X ùú , where g is a nonnegative real ë û number, and 0 < hX ( g ) £ 1 .
h
n -transform (Shannon transform). For a nonnegative random variable X is defined as, nX ( g ) = éë log2 ( 1 + g X ) ùû ,
n
where g is a nonnegative real number. The n -transform of F ( T (
n
F ( T(
†
) )F ( T( ) )
(g )
n
F ( T(
, gd gt g
(x )
gd gt g
d t
†
gt
and
= 1 - hFF† ( gt ) , c
are
random
log2
variables,
= 1 - hCC† ( cgd ) .
Distribution function of variable x. Probability measure function that returns an event’s probability. The
fc ( x )
†
))
gd
where
, expressed as
( g ) = n CC ( cgd ) + cnFF ( gt ) - c g gg
†
) )F ( T(
†
)) F ( T ( ))
Marcenko-Pastur
(
fc ( x ) = 1 relating
(
b = 1+
1 c
+
)
x (x ) +
random
c
2
)
41
.
density +
+
( x -a ) (b -x ) 2 pcx
function,
function,
, where x (⋅ ) is a
(
a = 1-
c
2
)
and
Laplace transform.
X Î é 0,1 ù ,Y Î é 0,1 ù
Independent, random variables, drawn from a uniform distribution on éë 0,1 ùû .
Z Î éb,1 ù
Independent random variable, drawn from a uniform distribution on éë b,1 ùû .
ë
û
ë
ë
û
C
û
Random variable having a distribution of the asymptotic singular value distribution of the singular values of C . The 2
distribution of C
is determined by the asymptotic spec-
tra of CC† .
Random variable having a distribution of the asymptotic singular value distribution of the singular values of F . The distribution of
2
is determined by the asymptotic spectra
of FF†
var ( a, b ) rF ( T( ) ) ( a, b ) Fa (⋅ ) , Fb ( ⋅ )
Variance profile function. Channel
profile
function
F (T(
of
)) ,
2 rF ( T( ) ) (a, b ) : ëé 0,1 ûù .
Nonrandom limits. Nonnegative variable, such that K l c holds, where K
c
K D( )
is the number of users, l is the number of available subchannels. Noise matrix of K users. Logical channel of user U k , a set formulated from m sub-
k
(
(0)
(l )
carriers such that k = k , , k
)
T
(i )
, where k
identifies the i-th logical sub-channel of U k . Symmetric private classical capacity of the k logical Psym ( k )
channel of U k , the maximum common rate at which the K users both can reliably transmit private classical information over the l sub-channels of .
42
Symmetric secret key rate of the k logical channel of U k ,
Ssym ( k )
b
a common rate at which the K users both can reliably transmit private classical information over the l subchannels of . Ratio of the k actual number of users and the total users K, b = Kk . The random matrix decomposition at arbitrarily allocated sub-channels. Random matrix decomposition of F ( T ( ) ) , where Z = ( Q0 , , QK -1 ) is an l ´ K
random matrix,
T Qk = éë k ,0 , , k ,l -1 ùû is user U k ’s entry, C is an l ´ K C = ( 0 , , K -1 ) , with random matrix
C ZF
(k ) (k ) T i = éê Aj ,0 , , Aj ,l -1 ùú ë û (k )
Aj
=
1 m
å m F (Ti ( i ))
for
Uk ,
user
where
is the averaged transmittance
coefficient of U k derived by operator L0 , L or L ¢ , m is the number of sub-channels of U k , is the Hadamard product operator, while F is F = diag ( F0 , , FK -1 ) .
an
l ´K
random
matrix,
The random matrix decomposition at identically allocated sub-channels. The h ´ K channel matrix F ( T ( ) ) is
F ( T ( ) ) = CUF , where U = ( u 0 , , uK -1 ) is an h ´ K isotropic unitary matrix with arbitrarily distributed random variables with mean mi, j = m and variance si2, j = 1 l ,
CUF
T uk = éë qk ,0 , , qk ,h -1 ùû , C is an h ´ h
with
(k ) (k ) T k = éê Aj ,0 , , Aj ,h -1 ùú , ë û
å h F (Ti ( i ))
is the averaged transmit-
C = diag ( 0 , , h -1 ) , (k )
where Aj
=
1 h
random matrix
tance coefficients of U k in , such that i = k for
i = 0, , K - 1 , while F is a K ´ K random matrix, F = ( F0 , , FK -1 ) . UK
out
The
unitary
43
CVQFT
operation,
UK
out
=
1 Kout
-i 2 pik
e Kout ,
i, k = 0, , Kout - 1 , Kout ´ Kout unitary matrix.
UK
The unitary inverse CVQFT operation, U K in
in
=
1 Kin
i 2 pik
e Kin ,
i, k = 0, , Kin - 1 , Kin ´ Kin unitary matrix. The variable of a single-carrier Gaussian CV state, ji Î . Zero-mean, circular symmetric complex Gaussian
z Î ( 0, sz2 )
é 2ù random variable, sz2 = ê z ú = 2sw2 , with i.i.d. zero 0 ë û mean, Gaussian random quadrature components
(
x , p Î 0, sw2
2 D Î ( 0, sD )
0
) , where s
2 w0
is the variance.
The noise variable of the Gaussian channel , with i.i.d. zero-mean, Gaussian random noise components on the position and momentum quadratures
Dx , Dp Î ( 0, s 2 ) ,
é 2ù 2 sD = ê D ú = 2s 2 . ë û The variable of a Gaussian subcarrier CV state, fi Î .
d Î ( 0, sd2 )
Zero-mean, circular symmetric Gaussian random variable, é 2ù sd2 = ê d ú = 2sw2 , with i.i.d. zero mean, Gaussian ranë û dom quadrature components xd , pd Î ( 0, sw2 ) , where sw2 is the (constant) modulation variance of the Gaussian subcarrier CV state.
F -1 (⋅ ) = CVQFT† ( ⋅ )
The inverse CVQFT transformation, applied by the encoder, continuous-variable unitary operation.
F (⋅ ) = CVQFT ( ⋅ )
The CVQFT transformation, applied by the decoder, continuous-variable unitary operation.
F -1 ( ⋅ ) = IFFT ( ⋅ )
Inverse FFT transform, applied by the encoder.
sw2 sw2 =
1 l
Single-carrier modulation variance. 0
ål sw2
i
Multicarrier modulation variance. Average modulation variance of the l Gaussian sub-channels i .
44
The i-th Gaussian subcarrier CV of user U k , where IFFT stands for the Inverse Fast Fourier Transform, fi Î ,
= F
-1
( z k ,i )
( ) Î ( 0, s ) ,
é 2ù sd2 = ê di ú , i ë û
di Î 0, sd2 ,
fi = IFFT ( zk ,i )
i
= di .
xd
i
2 wF
(
pd Î 0, sw2 i
F
)
di = xd + ipd , i
are
i.i.d.
i
zero-mean
Gaussian random quadrature components, and sw2
is the
F
variance of the Fourier transformed Gaussian state. jk ,i = CVQFT ( fi
The decoded single-carrier CV of user U k from the subcar-
)
rier CV, expressed as F ( di
)=
F ( F -1 ( z k ,i ) ) = z k ,i .
Gaussian quantum channel.
i , i = 0, , n - 1
Gaussian sub-channels. Channel transmittance, normalized complex random variable, T ( ) = Re T ( ) + i Im T ( ) Î . The real part
T ( )
identifies the position quadrature transmission, the imaginary part identifies the transmittance of the position quadrature. Transmittance coefficient of Gaussian sub-channel i , Ti ( i ) = Re (Ti ( i ) ) + i Im (Ti ( i ) ) Î ,
quantifies
the position and momentum quadrature transmission, with (normalized) real and imaginary parts
Ti ( i )
0 £ Re Ti ( i ) £ 1
2,
0 £ Im Ti ( i ) £ 1
2,
where
ReTi ( i ) = Im Ti ( i ) .
TEve
Eve’s transmittance, TEve = 1 - T ( ) .
TEve,i
Eve’s transmittance for the i-th subcarrier CV.
T
z = x + ip = ( z 0 , , zd -1 )
A d-dimensional, zero-mean, circular symmetric complex random Gaussian vector that models d Gaussian CV input states, ( 0, Kz ) , Kz = éê zz† ùú , where z i = x i + ipi , ë û T
x = ( x 0 , , xd -1 ) ,
(
pi Î 0, sw2
0
T
p = ( p0 , , pd -1 ) ,
(
x i Î 0, sw2
0
),
) i.i.d. zero-mean Gaussian random variables.
45
An l-dimensional, zero-mean, circular symmetric complex random Gaussian vector, ( 0, Kd ) , Kd = éê dd† ùú , ë û
(
T
d=F
-1
d = (d0 , , dl -1 ) , di = x i + ipi ,
(z)
x i , pi Î 0, sw2
F
)
are
i.i.d. zero-mean Gaussian random variables, sw2 = 1 sw2 . F
(
0
)
é 2ù The i-th component is di Î 0, sd2 , sd2 = ê di ú . i i ë û
(
yk Î 0, éê yk yk† ùú ë û
)
A d-dimensional zero-mean, circular symmetric complex Gaussian random vector. The m-th element of the k-th user’s vector yk , expressed as
yk , m
yk , m =
ål F (Ti ( i ))F (di ) + F ( Di ) .
Fourier
F ( T ( ))
T
T ( ) = éëT0 ( 0 ) ,Tl -1 ( l -1 ) ûù transmittance vector.
of
Î l ,
Complex vector, expressed as F ( D ) = e
F (D)
the
complex
T -F ( D ) KF ( D )F ( D ) 2
, with
†ù é covariance matrix KF ( D ) = ê F ( D ) F ( D ) ú . ë û
AMQD block, y éë j ùû = F ( T (
y éë j ùû
t = F ( d ) éë j ùû
transform
2
An
exponentially
f ( t ) = ( 1 2sw2n )e
) ) F ( d ) éë j ùû + F ( D ) éë j ùû .
distributed
-t
2sw2
variable,
with
density
, éë t ùû £ n 2sw2 .
Eve’s transmittance on the Gaussian sub-channel i ,
TEve,i
TEve,i = Re TEve,i + i Im TEve,i Î , 0 £ Im TEve,i £ 1
di n min
sw2
2 , 0 £ TEve,i
2
0 £ Re TEve,i £ 1
2,
< 1.
A di subcarrier in an AMQD block. The min { n 0 , , nl -1 } minimum of the ni sub-channel coefficients, where ni = s 2
F (Ti ( i ) )
2
and ni < nEve .
Constant modulation variance, sw2 = nEve - n min ( d )p
(x )
nEve =
where
46
1 l
, ,
l = F (T*
)
2
=
1 n
n -1
å i =0
n -1
å k =0Tk*e
- i 2 pik n
2
and T* is the
expected transmittance of the Gaussian sub-channels under an optimal Gaussian collective attack.
S.2 Abbreviations AMQD BS CDF CV CVQFT CVQKD DV FFT IFFT MGF MQA PDF QKD SNR SVD
Adaptive Multicarrier Quadrature Division Beam Splitter Cumulative Distribution Function Continuous-Variable Continuous-Variable Quantum Fourier Transform Continuous-Variable Quantum Key Distribution Discrete Variable Fast Fourier Transform Inverse Fast Fourier Transform Moment-Generating Function Multiuser Quadrature Allocation Probability Density Function Quantum Key Distribution Signal to Noise Ratio Singular Value Decomposition
47
