Multidimensional Manifold Extraction for 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 introduce the multidimensional manifold extraction for multicarrier continuousvariable (CV) quantum key distribution (QKD). The manifold extraction utilizes the resources that are injected into the transmission by the additional degrees of freedom of the multicarrier modulation. We demonstrate the results through the AMQD (adaptive multicarrier quadrature division) scheme, which granulates the information into Gaussian subcarrier CVs and divides the physical link into several Gaussian sub-channels for the transmission. We prove that the exploitable extra degree of freedom in a multicarrier CVQKD scenario significantly extends the possibilities of single-carrier CVQKD. The manifold extraction allows for the parties to reach decreased error probabilities by utilizing those extra resources of a multicarrier transmission that are not available in a single-carrier CVQKD setting. We define the multidimensional manifold space of multicarrier CVQKD and the optimal tradeoff between the available degrees of freedom of the multicarrier transmission. We also extend the manifold extraction for the multiple-access AMQDMQA (multiuser quadrature allocation) multicarrier protocol. The additional resources of multicarrier CVQKD allow the achievement of significant performance improvements that are particularly crucial in an experimental scenario. Keywords: quantum key distribution, continuous-variables, CVQKD, AMQD, AMQDMQA, quantum Shannon theory.
1
1 Introduction By utilizing the fundamental laws of quantum mechanics, the continuous-variable quantum key distribution (CVQKD) systems allow to realize an unconditionally secure communication through the currently established communication networks. The CVQKD protocols do not require single photon devices in contrast to the first developed discrete-variable (DV) QKD protocols [1–17]. This significant benefit has immediately made possible to achieve the practical implementation of QKD by the standard devices of traditional telecommunications [18–19], [23–25]. In a CVQKD setting, the information is carried by Gaussian random distributed position and momentum quadratures, which identify a quantum state in the phase space. The quantum states are sent through a noisy link (e.g., an optical fiber or a wireless optical channel [18–19], [26-30]), which adds a white Gaussian noise to the phase space transmission. Despite the fact that the noise characteristic of a CVQKD transmission is plausible and well exploitable in the security proofs, the performance of CVQKD, particularly the currently available secret key rates, is still below the rates of the protocols of traditional telecommunications. This issue brings up a potential requirement on the delivery of an intensive performance enhancement for CVQKD. In particular, for this purpose the multicarrier CVQKD modulation has been recently proposed through the multicarrier transmission scheme of AMQD (adaptive multicarrier quadrature division) [2]. The AMQD allows improved secret key rates and higher tolerable excess noise in comparison with standard (referred to as single-carrier throughout) CVQKD. The multicarrier transmission granulates the information into several Gaussian subcarrier CVs, which are then transmitted through the Gaussian subchannels. Particularly, the AMQD divides the physical Gaussian channel into several Gaussian sub-channels; each sub-channel is dedicated for the conveying of a given Gaussian subcarrier CV. Similar to single-carrier CVQKD, a multicarrier CVQKD also provides an unconditional security against the most powerful attacks [2]. Specifically, the benefits of AMQD have been extended by SVD-assistance (singular value decomposition) through the SVD-assisted AMQD [4]. Precisely, the SVD-assistance further injects an additional degree of freedom into the multicarrier transmission. The multicarrier CVQKD has been also proposed for a multiple-access scenario through the AMQD-MQA (multiuser quadrature allocation) scheme [3]. The AMQD-MQA allows for several legal parties to perform reliable simultaneous secret communication over a shared physical Gaussian link through the combination of a sophisticated allocation mechanism of the Gaussian subcarriers and the careful utilization of the Gaussian sub-channels. The secret key rates and security thresholds of multicarrier transmission have been proven in [5], leading to enhanced secret key rates in both one- and two-way CVQKD. The common root of these improvements is that the additional degrees of freedom injected by the multicarrier transmission act as a resource, allowing for the parties to exceed significantly the possibilities of singe-carrier CVQKD. We also further confirm this statement in this work through the utilization of these extra resources. In traditional communications, the diversity is an effective technique to improve the performance of communication over a noisy channel. The diversity can be obtained through several different tools—most of these solutions are based on sophisticated information coding approaches. The diversity of classical communication channels represents an extra resource from which several
2
benefits can be extracted to improve the performance of the transmission. The diversity in traditional telecommunications can be obtained via time, frequency, space, and coding [20–22]. Basically, in a communication scenario, the diversity provides a useful tool to improve the rates and the reliability. Here we show that similar benefits can be obtained for a multicarrier CVQKD scenario. The proposed solution is called manifold extraction. We propose the manifold extraction for multicarrier CVQKD, achieving an improved transmission by utilizing those available additional degrees of freedom in the Gaussian quantum channel that are obviously not available in a single-carrier CVQKD setting. In particular, the extractable manifold provided by the additional degrees of freedom of a multicarrier CVQKD transmission also allows for the parties in a multiple-access scenario to decrease much more significantly the error probabilities than it does presently in a single-carrier scheme. Specifically, the origin of these benefits is that the additional degrees of freedom of the multicarrier CVQKD provide an exploitable resource for the legal parties. The proposed manifold extraction uses a sophisticated phase space constellation for the Gaussian sub-channels [4] which provides a natural framework to exploit the manifold patterns of the sub-channel transmittance coefficients. The manifold extraction can be applied for an arbitrary distribution of the sub-channel transmittance coefficients and, by exploiting some properties of the phase space constellation it does not require the use of a statistical model. The proposed phase space constellation offers an analogous criterion to an averaging over the statistics of the sub-channel transmittance coefficients. We compare the achievable performance of manifold extraction of multicarrier and single-carrier CVQKD. We determine the optimal manifold-degree of freedom ratio tradeoff curve and define its attributes in a single and multicarrier CVQKD setting. We prove that the manifold extraction in a multicarrier scenario offers significantly decreased error probabilities, and through the sophisticated allocation of the Gaussian subcarrier CVs this benefit can be extended to all legal users of a multiple-access multicarrier CVQKD. We characterize the multidimensional manifold space of multicarrier CVQKD and define the multidimensional optimal tradeoff function in a high-dimensional manifold space. We then study the manifold extraction for multicarrier CVQKD through AMQD, and multiple-access multicarrier CVQKD through AMQD-MQA, respectively. This paper is organized as follows. Section 2 summarizes some preliminary findings. Section 3 defines the multidimensional manifold space for CVQKD. Section 4 proposes the manifold extraction of multicarrier CVQKD and multiple-access multicarrier CVQKD. Finally, Section 5 concludes the results. Supplemental 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 [2–5].
3
2.1 Basic Terms and Definitions 2.1.1 Multicarrier CVQKD In this section we very briefly summarize the basic notations of AMQD from [2]. 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 l refers to the subcarriers, while index j, to the single-carriers, throughout the manuscript.) The single-carrier state j j in the phase space can be
modeled as a zero-mean, circular symmetric complex Gaussian random variable æ ö 2ù é z j Î ççç 0, sw2 ÷÷÷ , with variance sw2 = ê z j ú , and with i.i.d. real and imaginary zero-mean z z è ë û j ø j
(
Gaussian random components Re ( z j ) Î 0, sw2
) , Im ( z
0
j
) Î ( 0, sw2
).
0
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 = ( z1, , z n ) Î ( 0, Kz ) ,
(1)
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 j + ip j . ø÷ j
(2)
Specifically, the real and imaginary variables (i.e., the position and momentum quadratures) forT
mulate n-dimensional real Gaussian random vectors, x = ( x1, , x n )
T
and p = ( p1, , pn ) ,
with zero-mean Gaussian random variables
f (x j ) =
1 sw
0
2p
e
-x j 2
-pj2
2 s2 w0
2 s2 w0
, f ( pj ) =
1 sw
0
2p
e
,
(3)
where Kz is the n ´ n Hermitian covariance matrix of z :
Kz = éê zz† ùú , ë û 4
(4)
while z† is the adjoint of z . For vector z , éë z ùû = éê e ig z ùú = e ig éë z ùû holds, and ë û T é éê zzT ùú = ê e ig z (e ig z ) ë û êë
ù ú = e i2 g éëê zzT ùúû , ûú
(5)
for any g Î éë 0, 2p ùû . The density of z is as follows (assuming that Kz is invertible): f (z) =
1 p det Kz n
-1
†
e -z Kz
z
.
(6)
A n-dimensional Gaussian random vector is expressed as x = As , where A is an (invertible) linear transform from n to n , and s is an n-dimensional standard Gaussian random vector ( 0,1 )n . This vector is characterized by its covariance matrix Kx = éê xxT ùú = AAT , as ë û -
x=
1
(
2p
n
)
(
T
det AA
)
xT x 2 AAT
e (
).
(7) T
The Fourier transformation F (⋅ ) of the n-dimensional Gaussian random vector v = ( v1, , vn ) T
results in the n-dimensional Gaussian random vector m = ( m1, , mn ) , precisely: m = F (v) = e
- mT AAT m 2
(
=e
2 -s 2 m12 ++mn w0
) .
2
(8)
In the first step of AMQD, Alice applies the inverse FFT (fast Fourier transform) operation to vector z (see (1)), which results in an n-dimensional zero-mean, circular symmetric complex T
Gaussian random vector d , d Î ( 0, Kd ) , d = (d1, , dn ) , precisely as d=F
(z) = e
-1
where
dT AAT d 2
(
=e
s2 d12 ++dn2 w0
(
) ,
2
(9)
)
di = xd + ipd , di Î 0, sd2 , i
i
i
(10)
é 2ù where sw2 = ê di ú and the position and momentum quadratures of fi di ë û random variables
(
)
(
are i.i.d. Gaussian
)
Re ( di ) = xd Î 0, sw2 , Im ( di ) = pd Î 0, sw2 , i
i
i
i
T 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 ) ë û ë û ë û êë for any g Î éë 0, 2p ùû .
The T (
)
(11)
ù ú = e i2 g éêë ddT ùúû úû
transmittance vector of in the multicarrier transmission is T T ( ) = ëéT1 ( 1 ) , ,Tn ( n ) ûù Î n ,
5
(12)
where
Ti ( i ) = Re (Ti ( i ) ) + i Im (Ti ( i ) ) Î ,
(13)
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
0 £ Re Ti ( i ) £ 1
2 , and 0 £ Im Ti ( i ) £ 1
2.
(14)
Particularly, the Ti ( i ) variable has the squared magnitude of
Ti ( i )
2
2
2
= Re Ti ( i ) + Im Ti ( i ) Î ,
(15)
where
ReTi ( i ) = Im Ti ( i ) .
(16)
The Fourier-transformed transmittance of the i-th sub-channel i (resulted from CVQFT (continuous-variable quantum Fourier transform) operation at Bob) is denoted by 2
F (Ti ( i ) ) .
(17)
2 The n-dimensional zero-mean, circular symmetric complex Gaussian noise vector D Î ( 0, sD )
n
of the quantum channel , is evaluated as T
D = ( D1, , Dn ) Î ( 0, KD ) ,
(18)
where
KD = éê DD† ùú , û ë with independent, zero-mean Gaussian random components
(
)
(19)
(
Dx Î 0, s 2 , and Dp Î 0, s 2 i
i
i
i
),
(20)
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 in the phase space . The CVQFT-transformed noise vector can be rewritten as T
F ( D ) = ( F ( D1 ) , , F ( Dn ) ) , with independent components F ( Dx
i
) Î ( 0, sF2 ( ) ) i
and F ( Dp
i
(21)
) Î ( 0, sF2 ( ) )
on the quad-
i
ratures, for each F ( Di ) . Precisely, it also defines an n-dimensional zero-mean, circular symmet-
(
)
ric complex Gaussian random vector F ( D ) Î 0, KF ( D ) with a covariance matrix †ù é KF ( D ) = ê F ( D ) F ( D ) ú , ë û
where KF ( D ) = KD , by theory.
6
(22)
2.1.2 SVD-Assisted Multicarrier CVQKD We briefly summarize the SVD-assisted multicarrier CVQKD scheme from [4]. The detailed description of the multiple-access AMQD-MQA scheme is included in [3]. Precisely, the singular layer consists of a pre-unitary F1 (U 1 ) (scaled FFT operation (scaled CVQFT), independent from the IFFT (inverse Fast Fourier transform) operation F (U 1 )) and a post-unitary U 2-1 (CVQFT operation, independent from the U CVQFT† operation) that perform the pre- and post-transform. The pre-unitary F1 (U 1 ) transforms such that the input will be sent through the li eigenchannels of the Gaussian link, whereas U 2-1 performs its inverse. Note that the pre- F1 (U 1 ) and post-
U 2-1 unitaries are the not inverse of F and U but F1-1 (U 1-1 ) and U 2 , respectively. In particular, these unitaries define the set S1 of singular operators, as follows [4]:
S1 = { F1,U 2-1 } .
(23)
Specifically, if each transmit user sends a single-carrier Gaussian CV signal to an encoder , then the pre-operator is the unitary U 1 , the CVQFT operation, whereas the unitary post-operator is achieved by the inverse CVQFT operation U 2-1 , defining the set S2 of singular operators as
S2 = {U 1,U 2-1 } . The subindices of the operators
{F1,U 2-1 }
(24)
and {U 1,U 2-1 } are different in each Si , i = 1, 2 be-
cause these operators are not the inverse of each other. These operators are determined by the SVD of F ( T ) , which is evaluated as
F ( T ) = U 2 GF1-1 ,
(25)
where F1-1, F1 Î Kin ´Kin and U 2 ,U 2-1 Î Kout ´Kout , Kin , and Kout refer to the number of sender and receiver users such that
Kin £ Kout , F1-1F1 = F1F1-1 = I ,
(26)
and U 2U 2-1 = U 2-1U 2 = I .
The term G Î is a diagonal matrix with nonnegative real diagonal elements l1 ³ l2 ³ ln , min
(27) (28)
which are called the eigenchannels of F ( T ) = U 2 GF1-1 , where
n min = min ( Kin , Kout ) ,
(29)
which equals to the rank of F ( T ) , where
Kin £ Kout ,
(30)
by an initial assumption. (Note: the eigenchannels are also called the ordered singular values of F ( T ) .) In terms of the li eigenchannels, F ( T ) can be precisely rewritten as
7
F (T) =
ån
min
liU 2,i F1,-i 1 ,
(31)
where liU 2,i F1,-i 1 are rank-one matrices. In fact, the n min squared eigenchannels li2 are the eigenvalues of the matrix †
F ( T ) F ( T ) = U 2 GGTU 2-1 ,
(32)
where GT is the transpose of G . The complete description of the singular layer of CVQKD can be found in [4].
2.1.3 Rate Formulas of Multicarrier CVQKD The complete derivation of the secret key rate formulas can be found in [5], here we give a brief overview on the transmission rates of multicarrier CVQKD. In particular, the (real domain) classical capacity of a Gaussian sub-channel i in the multicarrier setting is æ C ( i ) = 21 log2 ççç 1 + çè
2
sw2 F (Ti ( i ) ) i
s 2 i
ö÷ ÷÷÷ , ø÷
(33)
÷ö ÷÷÷ , ÷ø
(34)
while in the SVD-assisted AMQD, æ C ¢ ( i ) = 21 log2 ççç 1 + çè
sw2 ¢¢ F (Ti ( i ) )
2
i
s 2 i
where sw2 ¢¢ = sw2 ( 1 + c ) > sw2 . Specifically, the SNR (signal to noise ratio) of i is expressed as
SNRi =
sw2
i
s 2
,
(35)
i
while the SNR of at a constant modulation variance sw2 is SNR =
sw2 s 2
.
Particularly, in the SVD-assisted AMQD, it referred to as
SNRi¢ =
sw2 ¢¢ i
s 2
, and SNR ¢ =
i
sw2 ¢¢ s 2
,
(36)
respectively. From (33) and (34), the (real domain) classical information transmission rates Rk ( ) and Rk¢ ( ) of user U k through the l i Gaussian sub-channels in AMQD and SVDassisted AMQD are precisely as follows:
é æ Rk ( ) £ max êê å l 21 log2 ççç 1 + "i èç êë
sw2 F (Ti ( i ) )
é æ Rk¢ ( ) £ max êê å l 21 log2 ççç 1 + "i çè ëê
sw2 ¢¢ F (Ti ( i ) )
2
i
s 2
÷öù ÷÷÷ úú , ø÷ úû
(37)
÷öù ÷÷÷ úú . ø÷ ûú
(38)
and
8
i
s 2
2
Precisely, the SNRi* (signal to noise ratio) of the i-th Gaussian sub-channel i for the transmission of private classical information (i.e., for the derivation of the secret key rate) under an optimal Gaussian attack [5], [14-16] is expressed as
SNRi* =
sw2
sw2
, and SNR * =
i
s2
where s 2
i*
,
s2
i*
(39)
*
is precisely evaluated as [5]
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
(40)
where
sC2 = s02 + N i ,
(41)
i
and where s2 is the vacuum noise and N i is the excess noise of the Gaussian sub-channel i defined as
( F (T
(Wi -1 )
Ni =
Eve ,i
1- F (TEve ,i )
)
),
2
2
(42)
where Wi is the variance of Eve’s EPR state used for the attacking of i , while
TEve,i
2
= 1 - Ti
is the transmittance of Eve’s beam splitter (BS), and Ti
2
2
(43)
is the transmittance of i .
Precisely, in the SVD-assisted multicarrier CVQKD, sw2 ¢¢
*
( SNRi¢ ) =
i
s
2
sw2 ¢¢
*
and ( SNR ¢ ) =
,
s2
i*
(44)
*
for i and , respectively. Particularly, from (40) the P ( i ) private classical capacity (real domain) is expressed as æ P ( i ) = log2 ççç 1 + çè 1 2
sw2 F (Ti ( i ) )
2
i
s
2 i*
ö÷ ÷÷ . ÷ ø÷
(45)
÷÷ö ÷÷ . ÷ø
(46)
The SVD-assisted P ¢ ( i ) from (44) is then yielded precisely as æ P ¢ ( i ) = 21 log2 ççç 1 + çè
sw2 ¢¢ F (Ti ( i ) ) i
s2
i*
2
Assuming l Gaussian sub-channels, the (real domain) secret key rate S (
S ¢(
)
)
of AMQD and
öö ÷÷ ÷÷ , ÷÷ ÷÷ ÷÷ øø
(47)
of SVD-assisted AMQD are as follows:
æ æ ç ç S ( ) £ P ( ) = max çç å l log2 çç 1 + "i çè èç
sw2 F (Ti ( i ) )
æ æ ç ç S ¢ ( ) £ P ¢ ( ) = max çç å l log2 çç 1 + "i çè çè
9
2
i
s
2
i*
sw2 ¢¢ F (Ti ( i ) ) i
s
2
i*
2
öö ÷÷ ÷÷ . ÷÷ ÷÷ ÷÷ øø
(48)
2.2 Manifold Extraction In a multicarrier CVQKD scenario, the term manifold is interpreted as follows. Let the i-th comT
ponent p j ,i of a given private random codeword p j = ( p j ,1, , p j ,l )
to be transmitted through
i , where each Gaussian sub-channel is characterized by an independent transmittance coefficient
(Ti ( i ) )
2
. As a first approach, the number l of the Gaussian sub-channels is identified as
the manifold of . Precisely, the information is granulated into subcarriers, which are dispersed by the inverse Fourier transform, and each p j ,i component is identified by independent transmittance coefficients. A more detailed formula will be concluded in the further sections. Specifically, the transmission can be utilized by a permutation phase space constellation P (
)
[4], which was recently proposed for SVD-assisted CVQKD in [4]. Using Pi , i = 2, , l random permutation operators, P (
)
can be defined for the multicarrier transmission as
P ( ) = ( ( 1 ) , , ( l ) )
= ( ( 1 ) , P2 ( 1 ) , , Pl ( 1 ) ) ,
(49)
where d ( ) = d ( ) is the cardinality of ( i ) . Using P ( ) , the available degrees of i j freedom in the Gaussian link can be utilized, and the random permutation operators inject correlation between the i sub-channels via Pi ( 1 ) . In particular, for each Gaussian sub-channel, the distance between the phase space constellation points is evaluated by di , the normalized difference function. Assuming two l-dimensional input T
T
random private codewords pA = ( pA,1, pA,l ) and pB = ( pB,1, pB,l )
and two Gaussian sub-
channels i and j , di is calculated precisely as follows:
di =
1 s2 w ¢¢ s2 *
( pA,i
- pB,i ) ,
(50)
),
(51)
particularly, for the l Gaussian sub-channels
d1l where the term d1l
2
(
> c
1 S¢ l2 ( i )
l
is referred to as the product distance [20-22]. The maximization of this
term ensures the maximization of the extractable manifold, and determines the perr pairwise worst-case error probabilities of pA, pB . As we show in Section 3, by using P (
)
and (51), the perr worst-case pairwise error probabil-
ity can be decreased to the theoretical lower bound. We further reveal that in a multiuser CVQKD scenario, this condition can be extended simultaneously for all uses. Let us assume that the Sk¢ ( ) secret key rate of U k , for "k , is fixed precisely as follows: Sk¢ ( ) =
10
Vk n min
P ¢ ( ),
(52)
where Vk > 0 is referred to as the degree of freedom ratio of U k , and n min has been shown in (29). As one can immediately conclude from (52), Sk¢ ( ) P ¢ ( ) . Without loss of generality, for a given sub-channel i , we redefine Sk¢ ( i ) , V k ,i > 0 precisely as V k ,i
Sk¢ ( i ) =
n min
P ¢ ( i ).
(53)
(Note: From this point, we use the complex domain formulas throughout the manuscript and Sk¢ ( ) and Sk¢ ( i ) are fixed to (52) and (53).) For a given i , an Eerr error event [20–22] is identified as follows:
(
Eerr º log2 1 + F (Ti ( i ) )
2
*
( SNRi¢ )
) < S ¢( ),
(54)
i
and the probability of Eerr at a given S ¢ ( i ) is identified by the perr error probability as follows:
( (
Eerr = perr ( Sk¢ ( i ) ) = Pr log2 1 + F (Ti ( i ) )
2
*
( SNRi¢ )
) < S ¢ ( )) . i
(55)
Particularly, by some fundamental argumentations on the statistical properties of a Gaussian random distribution [20–22], for F (Ti ( i ) )
2
*
( SNRi¢ )
(
perr ( Sk¢ ( i ) ) = Pr F (Ti ( i ) ) while for F (Ti ( i ) )
2
*
( SNRi¢ )
2
0 , perr ( Sk¢ ( i ) ) can be expressed as *
( SNRi¢ )
)
log2 e < S ¢ ( i ) ,
(56)
¥ , the corresponding error probability is as
( (
perr ( Sk¢ ( i ) ) = Pr log2 F (Ti ( i ) )
2
*
( SNRi¢ )
) < S ¢ ( )) . i
Let l = 1 , that is, let’s consider a single-carrier CVQKD, with F (T ( ) )
2
(57)
of , with a secret
key rate S ¢ ( ) . In this setting, perr is expressed precisely as [20]
( (
single perr ( Sk¢ ( ) ) = Pr log2 1 + F (T ( ) )
æ = Pr çç F (T ( çè 1 , = *
))
2
0 nonzero secret key rate is possible, by convention. Particularly, in (51), a given ¶i identifies the minimum distance between the normalS¢ ized 2 k ( i ) points for the phase space constellation ¢ ( i ) of i .
Precisely, by fundamental theory [20-22], it can be proven that for an arbitrary distribution of the F (Ti ( i ) ) Fourier transformed transmittance coefficient, the maximized product distance function of (51) can be derived by an averaging over the following statistic :
(
: F (Ti ( i ) ) Î 0, sF2 where sF2
(Ti ( i ) )
(Ti ( i ) )
),
(61)
2ù é = ê F (Ti ( i ) ) ú , and F (Ti ( i ) ) is a zero-mean, circular symmetric comë û
(
plex Gaussian random variable with i.i.d. 0, 0.5sF2
(Ti ( i ) )
) zero-mean Gaussian random vari-
ables per quadrature components x i and pi , for the i-th Gaussian subcarrier CV. Putting the pieces together, the maximized product distance function d1l
of (51) precisely can
be obtained via an averaging over the statistics of (61); however, (61) is, in fact, strictly provides an analogous criteria of the worst-case perr situation in (59) via a sophisticated phase space constellation , by theory [20–21]. In other words, set , as it is given in (59) together with
represents a universal criteria and provides us an alternative solution to find the worst-case
perr error probability for arbitrary distributed F (Ti ( i ) ) coefficients in a multicarrier CVQKD scenario. Specifically, some of these argumentations can be further exploited in our analysis. First of all, by using (61), the averaged term
(
0, sF2
(Ti ( i ) )
)
1 l
2 ål F (Ti ( i ))
can be modeled as a sum of
distributed random variables, with zero mean and variance of sF2
each i sub-channels. Then, since
1 l
2 ål F (Ti ( i ))
real Gaussian random variables, the distribution of
1 l
for
(Ti ( i ) )
is the averaged sum of 2l independent
2 ål F (Ti ( i ))
precisely can be approxi-
mated by a c22l chi-square distribution with 2l degrees of freedom, by a density function f ( ⋅ ) :
f (x ) =
1
(l -1 ) !
x l -1e -x ,
(62)
x l -1.
(63)
where x ³ 0 . In particular, for x 0 , the density can be written as
f (x ) » AMQD Thus, we arrive at perr as
12
1
(l -1 ) !
æ ö 2 AMQD perr = Pr çç 1l å l F (Ti ( i ) ) < 1 * ÷÷÷ çè ( SNR ¢ ) ø 1 ö çæ ¢ *÷ ççè ( SNR ) ø÷÷÷
ò
= = » where the term
1 l!
1 l!
1
(l -1 ) !
0
x l -1dx
(64)
1
(( SNR¢ ) )
l
*
1
(( SNR ¢ ) ) *
l
,
is negligible.
Specifically, from (58) and (64), the d manifold parameter picks up the following value in the single-carrier CVQKD setting: (65) dsingle = 1 , while in the multicarrier CVQKD setting,
dAMQD = l .
(66)
The result in (66) will be further sharpened in Section 3 since it significantly depends on the properties of the properties of the corresponding phase space constellation ( ) . From (69) it clearly follows that the extractable manifold d determines the error probability of the transmission, and for higher d , the reliability of the transmission improves. Particularly, in a multiple-access CVQKD scenario, there exists another degree of freedom in the channel, the number of information carriers allocated to a given user U . This type of degree of freedom is denoted by V and is referred to as the degree of freedom ratio. Without loss of generality, in the function of V > 0 (65) and (66) precisely can be rewritten as dsingle = 1 - V ,
(67)
while, in the multicarrier CVQKD setting, it refers to the ratio of the subcarriers allocated to a given user, dAMQD = l ( 1 - V ) . (68) Thus, in a multicarrier CVQKD scenario with l Gaussian sub-channels, for a given V > 0 , the overall gain is l. As follows, using (67) and (68), the error probabilities can be rewritten precisely as single perr = AMQD perr =
1
(
*
( SNR ¢ )
)
dsingle
1
(( SNR ¢ ) ) *
dAMQD
= =
1
(
*
( SNR ¢ )
( 1-V )
)
1
l ( 1-V )
(( SNR ¢ ) ) *
,
(69)
.
(70) *
single AMQD The perr and perr error probabilities of (69) and (70) for ( SNR ¢ ) ³ 1 , for l = 5,10 Gaus-
sian sub-channels, and at V = 0.6 are compared in Fig. 1.
13
*
Figure 1. The error probabilities in the single-carrier and multicarrier CVQKD, ( SNR ¢ ) ³ 1 ,
l = 5,10 , and V = 0.6 . In a multicarrier CVQKD protocol run, there exists an optimal tradeoff between d and V ; however, it requires to make some preliminary assumptions, as it is concluded in Lemma 1.
Sk¢ ( ) > 0
Lemma 1. (Manifold extraction in multicarrier CVQKD). For any
Vk =
1 P ¢(
)
Sk¢ ( ) n min , Vk > 0 of user U k , k = 1, , Kout , the dk ( V k ) extractable manifold is the
ratio of perr ( Sk¢ ( 1 n min
and
))
error probability and the n min -normalized private classical capacity *
P ¢ ( ) , derived at the asymptotic limit of ( SNR ¢ ) ¥ .
Proof. In particular, at a given Vk and Sk¢ ( ) (see (52)), the dk ( V k ) manifold parameter of user
U k , k = 1, , Kout is as follows: dk ( V k ) = where perr ( Sk¢ (
))
lim *
( SNR ¢ )
¥
- log2 perr ( Sk¢ ( 1
n min
P ¢(
)
))
,
(71) *
is the error probability of U k at Sk¢ ( ) , while ( SNR ¢ ) is the SNR of
in an SVD-assisted AMQD modulation for private information transmission (see (44)). Specifically, assuming that the condition of 1 l
* ål F (Ti ( i ) ) ( SNR ¢ ) ¶k ³ c,
(72)
S¢ holds, where c > 0 is a constant and ¶k is the minimum distance of the 2 k ( ) normalized con-
stellation points ji , j j , j ¹ i in a phase space constellation ¢ ( ) , ¢ ( ) Í ( ) , S¢ ¢ ( ) = 2 k ( ) , evaluated precisely as
14
¶k =
1
Sk¢ (
2
)
then at a given secret key rate Sk¢ ( ) , the perr ( Sk¢ (
,
(73)
))
error probability of the transmission of
2
U k decays as perr ( Sk¢ (
æ
) ) = Q çççç è
=
*
2 ( SNR ) ål F (Ti ( i ) ) 2 ¢
1 l
S¢ 2 k ( ) -1 *
( SNR ¢ )
ö ¶k2 ÷÷÷ ø÷
(74)
,
where Q ( ⋅ ) is the Gaussian tail function. Note that the condition of (72) follows from the fact that the separation (i.e., ¶k ) of the constellation points of (
)
has to be significantly larger
than s ; otherwise, the Q ( ⋅ ) Gaussian tail function yields in high error probabilities [20]. Without loss of generality, for an i dedicated to U k with Sk¢ ( i ) , the phase space constellaS¢ tion is referred to as ¢ ( i ) , ¢ ( i ) = 2 k ( i ) , and ¶k ,i =
1
Sk¢ ( i ) 2
2
, and
* æ ö 2 ( SNRi¢ ) 2÷ ÷÷ perr ( Sk¢ ( i ) ) = Q ççç F (Ti ( i ) ) ¶ k i , 2 ÷ø çè Sk¢ ( i ) -1 =2 . *
(75)
( SNRi¢ )
Exploiting the argumentation of (61) on the averaging over the statistics of the channel transmittance coefficients, and the related result in (59), the perr ( Sk¢ ( i ) ) of a given Gaussian subchannel i at Sk¢ ( i ) can be determined precisely as æ S¢ ö -çç 2 k ( i ) -1 ÷÷÷÷ çè ø
perr ( Sk¢ ( i ) ) = 1 - e
( SNRi¢ )*
,
(76)
*
which at ( SNRi¢ ) ¥ coincidences with (75). Thus, using Pi Î , i = 2, , l drawn from a uniform distribution, the private permutation phase space constellation ¢P ( ) , ¢P( ) Í P ( ) , ¢P(
)
S¢ = 2 k ( ) can be defined for
the private multicarrier transmission as ¢P ( ) = ( ¢ ( 1 ) , , ¢ ( l ) )
= ( ¢ ( 1 ) , P2 ¢ ( 1 ) , , Pl ¢ ( 1 ) ) ,
(77)
where d ¢ ( ) = d ¢ ( ) is the cardinality of ¢ ( i ) . The private permutation phase space con i j stellation of (77) can be used as a corresponding ¢ ( i ) , for each i sub-channels. In particular, assuming the use of ( F (Ti ( i ) ) ) in (61), the perr ( Sk¢ ( i ) ) of a given subchannel i with secret key rate Sk¢ ( ) =
Vk n min
P ¢ ( ) , can be rewritten precisely as
15
( (
perr ( Sk¢ ( i ) ) = Pr log2 1 + F (Ti ( i ) )
2
*
( SNRi¢ )
æ ç (( SNRi¢ )* ) 2 = Pr ççç F (Ti ( i ) ) < * ( SNRi¢ ) ççè 1 1 , » = 1-Vk ,i d (V )
V k ,i
(( SNR ¢ ) )
(( SNR¢ ) )
*
*
i
(
k
i
-1 ÷
(78)
k ,i
i
)
2
Pr F (Ti ( i ) ) < x » x , by theory at the distribution of (61) [20–22], and
where
(
dk ,i ( V k ,i ) = 1 - V k ,i . Specifically, if F (Ti ( i ) ) Î 0, sF2
S ¢(
k ,i
ö ÷÷ ÷÷ ø÷
) < S ¢ ( ))
)
(Ti ( i ) )
)
for all i , then for the
secret key rate in the low SNR regimes the following result yields, precisely: 2 æ æ öö sw2 ¢¢ max F (Ti ( i ) ) ÷ ç ç i ÷÷ ÷÷÷ S ¢ ( ) £ çç log2 çç 1 + ÷÷ ÷÷ s2 * çè çè øø æ sw2 ¢¢ max F (Ti ( i ) ) 2 ÷ö 2 i ÷÷ log e » sw ¢¢ log e, » ççç 2 2 2 2 ÷÷ s * s * çè ø
(79)
while in the high SNR regimes
æ æ sw2 ¢¢ max F (Ti ( i ) ) 2 ÷ö ö÷ i ÷÷ ÷÷ S ¢ ( ) £ ççç log2 ççç s2 * ÷÷ø ÷÷ø çè çè » log2
sw2 ¢¢ s2
*
( (
+ log2 max F (Ti ( i ) ) i
and from the law of large numbers [20]: 2 æ æ öö æ æ s 2 F (Ti ( i ) ) ÷÷ çç log çç 1 + 1ç ÷÷ çç å log2 ççç 1 + wi¢¢ 2 = ÷÷ 2 ç ç l lç ÷÷ s * çè ÷÷ çè è øø èç i
(
At F (Ti ( i ) ) Î 0, sF2
(Ti ( i ) )
2
))
(80)
,
sw2 ¢¢ max F (Ti ( i ) ) i
s
2 *
2
÷ö÷ ö÷÷ ÷÷ . ÷ø÷ ø÷÷
(81)
) , the density of (76) is depicted in Fig. 2.
Figure 2. At the distribution of the statistics of (61), the shaded area under the curve gives the perr error probability at a given Sk¢ ( i ) for a Gaussian sub-channel i . 16
Note that for the transmission of classical (i.e., non-private) information, has a cardinality of
= 2Rk at a given Rk¢ ( ) , with a corresponding SNR ¢ and ¶k = ¢
1
Rk¢ ( ) 2
2
; for the details, see
the description of [4]. An error event Eerr of (54) for a sub-channel i can be rewritten as
Eerr :
1 l
( ) 2 ål F (Ti ( i )) < 2( SNR ¢-) 1 ; Sk¢
(82)
*
thus, introducing Z > 1 which brings up by the use of a corresponding (such as the permutation scheme in [4]), a typical error probability is precisely expressed as follows:
( = Pr (
2 * ål log2 (1 + F (Ti ( i )) ( SNRi¢ ) ) < Sk¢ ( i )) V 2 * 1 log2 ( 1 + F (Ti ( i ) ) ( SNRi¢ ) ) < n P ¢ ( i ) ) l ål æ ö ( SNR ¢ ) ) -1 ÷÷ çç 1 ( 2 ÷÷ = Pr çç l å l F (Ti ( i ) ) < ÷÷ ( SNR ¢ ) ç
perr ( Sk¢ ( i ) ) = Pr
1 l
k ,i
min
*
*
çè
1
=
(( SNR ¢ ) ) æ ( SNR ¢ ) ç( ) =ç *
1-Vk
*
çç çç è
(
where Pr F (Ti ( i ) )
2
Vk
Vk *
( SNR ¢ )
=
1
Z( 1-Vk
(( SNR ¢ ) ) *
ø÷
(83)
)
öZ -1 ÷
÷÷ ÷÷ , ø÷
)
ææ ö öV < x » x Z as ççç ççç( SNR ¢ )* ÷÷÷ k -1 ÷÷÷ è ø è ø
*
( SNR ¢ )
0 , by theory. (Note that in
(83), the Ti transmittance coefficients are arbitrarily distributed, in contrast to (78)). As one immediately can conclude, the phase space constellation provides a further decreased perr in comparison to (70). Putting the pieces together, the optimal manifold-degree of freedom ratio tradeoff curve [20] f for a single-carrier scheme (e.g., if l = 1 we trivially have a single-carrier scheme) can be expressed as (84) f ( i ) : dk ( V k ) = Z ( 1 - V k ) , where 0 < Vk £ 1 . Specifically, some calculations then straightforwardly reveal that any phase space constellation ( ) that satisfies the condition of
¶k2 = q
1 S¢ 2 k( )
(85)
achieves the optimal f tradeoff curve, for any constant q > 0 . The recently proposed permutation phase space constellation P ( ) for SVD-assisted AMQD provably satisfies this condition [4]. Without loss of generality, assuming a constant g > 0 , perr can be rewritten precisely as
17
æ æ 2 * g perr ( Sk¢ ( i ) ) : Pr çççç 1l ål log2 çççç 1 + F (Ti ( i ) ) ( SNRi¢ ) è è V 1-g ö k æ ö æç * çç çç ( ( SNR ¢ ) ) ÷÷÷ -1 ÷ ÷÷ ÷ çè 2 ø çç 1 ÷÷ = Pr ç l å l F (Ti ( i ) ) < 1-g ÷ * çç ¢ (( SNR ) ) ø÷÷÷ çè
(
Vk ( 1-g )
(( SNR¢ ) ) = (( SNR ¢ ) ) *
*
1-g
-1
)
1-g
ö ö ÷÷÷ < Sk¢ ( i ) ÷÷÷ ÷ø ÷ø (86)
.
g Thus, for Eerr , the manifold parameter is dkg ( Vk ) as
dkg ( V k ) = dk ( Vk ) ⋅ ( 1 - g ) ³
lim *
( SNR ¢ )
- log2 perr ( Sk¢ ( 1
¥
nmin
P ¢(
)
))
.
(87) ■
In Section 3, we give a proof on the multicarrier CVQKD scenario and show that there exists an optimal tradeoff between Vk and dk for any U k . We further reveal that in a multicarrier setting, the manifold extraction significantly exceeds the possibilities of a single-carrier CVQKD scenario.
3 Multidimensional Manifold Space Theorem 1 (Multidimensional manifold space for multicarrier CVQKD). For any Kin , Kout , and
V k > 0 , the manifold is a dim ( ) = K in Vk + ( Kout - V k ) V k dimensional space. The number N dim^ of dimensions orthogonal to in the dim ( S ( F ( T ( ) ) ) ) = K in Kout dimensional space S ( F ( T ( with perr ( V k ) =
)))
is N dim^ = Kin Kout - ( K in V k + ( Kout - V k ) V k ) = ( Kin - V k )( Kout - V k ) , 1
(( SNR ¢ ) ) *
N
dim^
, and optimal tradeoff curve h ( ) : dk ( V k ) = N dim^ .
Proof. The proof assumes a Kin , Kout multiuser scenario. First we express perr as follows:
æ ç perr = Pr çç Eerr = çè
n min
* æ ( SNR ¢ ) 2 ÷ö ç log 1 li ÷÷ < + å 2 ççè Kin ø i =1
Vk ( r +Kin -1 ) r
÷ö P ¢ ( ) ÷÷÷ , ÷ø
(88)
where li2 are the squared random singular values of F ( T ( ) ) [20–22]. Assuming Vk > 0 , the li singular values can be decomposed into subsets s 0 and s1 such that set
s 0 = { l1, , lV contains the largest Vk singular values of F ( T (
)) ,
max {l1, , lV "i
k
k
}
(89)
li < li +1, i = 1, , Vk , and where
} £ 1.
(90)
The remaining n min - Vk singular values formulate the subset s1 , as
s1 = { lV
k
+1, , ln min
18
},
(91)
where
max { lV "i
k
} £ ( SNR1 ¢ )
+1, , ln min
*
In particular, from (89) and (91), for the rank of F ( T (
.
(92)
) ) , the following relation identifies an
error event Eerr : Eerr : rank ( F ( T (
) ) ) £ Vk .
(93)
Thus, the perr at a given Vk is precisely referred to as perr ( V k ) = Pr ( rank ( F ( T (
) ) ) £ Vk ) .
(94)
Specifically, at Vk = 0 , perr ( Vk ) is evaluated as
æ ö perr ( Vk = 0 ) = Pr çç F ( T ( ) ) : max { l1, , ln } £ 1 * ÷÷÷ min "i èç ( SNR ¢ ) ø 1 = . Kin Kout
(95)
(( SNR ¢ ) ) *
At Vk 0 , in (88) the corresponding relation is
li £
1 *
( SNR ¢ )
,
(96)
thus for the sum of the n min squared eigenvalues li , n min
å li2
=
i =1
å F (T ( U )) k
k
2
,
(97)
where U refers to the logical channel of U k , which consists of the allocated Gaussian subcarrik
ers of that user. As follows, (96) holds if only
F (T ( U
k
))
2
£
1 *
( SNR ¢ )
,
(98)
thus,
æ perr ( Vk 0 ) = Pr çç F (T ( U k çè 1 » Kin Kout
(( SNR¢ )
*
=
(
2
0 ,
19
(101)
perr ( Vk > 0 ) = Pr ( rank ( F ( T ( ) ) ) £ rank ( M ) = Vk )
= where
N dim^
refers
to
the
1
(
*
( SNR ¢ )
dim^
)
N
dim^
(102)
,
dimensions
orthogonal
to
in
the
dim ( S ( F ( T ( ) ) ) ) = K in Kout dimensional space S of F ( T ( ) ) , and M is a matrix with rank Vk . Thus, for Vk > 0 , perr ( Vk ) is related to as the difference between matrix F ( T ( ) ) and a rank-
Vk matrix M, since F ( T ( ) ) is a Kin ´ Kout matrix with rank = Vk ; that is, it has Vk > 0 linearly independent row vectors from the Kout rows [20–22]. Without loss of generality, M can be characterized by Kin Vk + ( Kout - V k ) V k parameters by theory, from which (101) straightforwardly follows. Putting the pieces together, the N dim^ number of dim^ dimensions orthogonal to manifold space
in the dim ( S ( F ( T ( ) ) ) ) dimensional space of F ( T ( ) ) is precisely N dim^ = Kin Kout - dim ( )
= Kin Kout - ( Kin Vk + ( Kout - Vk ) Vk ) = ( Kin - V k )( Kout - V k ) .
Particularly, from (102) the multidimensional optimal tradeoff function is yielded as h ( ) : d ( Vk ) = N dim^ .
(103)
(104)
The N dim^ in function of dim ( ) , at Kin = Kout - 1 , Vk = 0, 3; 0.6; 0.9 is depicted in Fig. 3.
Figure 3. The values of N dim^ in function of dim ( ) , at Kin = Kout - 1 , Vk = 0, 3; 0.6; 0.9 . ■
20
4 Manifold Extraction for Multicarrier CVQKD Theorem 2 (Manifold extraction for multicarrier CVQKD). For user U k , the manifold extraction at l Gaussian sub-channels leads to an optimal h tradeoff curve h : dk ( Vk ) = l Z ( 1 - Vk ) = lf for any multicarrier scheme, where f is the optimal tradeoff curve of a single-carrier CVQKD protocol, f : dk ( V k ) = Z ( 1 - V k ) , and Z > 1 . Proof. In the first part of the proof, we assume the case l = 1 , which is analogous to a single-carrier transmission. In the second part of the proof, we study the multicarrier case for l Gaussian subchannels and reveal that a multicarrier case allows significantly improved manifold extraction. In the single-carrier scenario, the phase space constellation ¢ ( ) , ¢ ( ) Í ( ) , S¢ ¢ ( ) = 2 k ( ) , leads to ¶k =
1
S¢ ( 2k
)
2
, perr ( Sk¢ (
Sk¢ ( ) -1
)) = 2
*
( SNR ¢ )
, and f : dk ( V k ) = Z ( 1 - Vk ) ,
as it has been already shown in the proof of Lemma 1. This is precisely the situation for l = 1 . Specifically, for the multicarrier CVQKD case, let us assume that there are l Gaussian subchannels dedicated to the transmission of the di , i = 1, , l Gaussian subcarriers, with *
( SNRi¢ ) =
sw2 ¢¢ i
s2
i*
per i , and secret key rate Sk¢ ( i ) as
Sk¢ ( i ) =
V k ,i n min
P ¢ ( i ).
(105)
Without loss of generality, for the total constraint of SVD-assisted AMQD [4], one has precisely l
2 * å log2 (1 + F (Ti ( i )) ( SNRi¢ ) ) ³ lSk¢ ( i ) .
(106)
i =1
In particular, by further exploiting the results of SVD-assisted AMQD [4] and following the derivations of Section 5 in [4], here we determine the private random codeword difference for two lT
T
dimensional input codewords pA = ( pA,1, pA,l ) and pB = ( pB,1, pB,l ) . The probability that
pA is distorted onto pB conditioned on F ( T (
))
is evaluated precisely as follows:
æ Pr ( pA pB F ( T ( ) ) ) = Q çç çè
sw2 ¢¢
2s
2 *
å F (Ti ( i ))
2
l
¶i
2
÷÷ö , ÷÷ ø
(107)
where ¶i is the normalized difference of pA,i and pB,i , calculated as follows: ¶i =
1 s2 w ¢¢ s2 *
( pA,i - pB,i ) .
(108)
Assuming the case that in (107), the condition sw2 ¢¢
2s 2
*
å F (Ti ( i )) l
holds, one obtains
21
2
¶i
2
cl
1 S¢ l ll 2 k ( i )
,
(110)
for any constant c > 0 and for an arbitrary pair of pA and pB [4]. In particular, at a secret key rate Sk¢ per i , the cardinality of ¢ ( i ) is as follows: S¢ ¢ ( i ) = 2 k ( i ) .
(111)
S¢ Thus, in the private transmission each ¢ ( i ) is precisely defined with 2 k ( i ) CV states fi
for each i , at an averaged Sk¢ ( i ) as S¢ lS¢ ¢ ( ) = 2ål k ( i ) » 2 k ( i ) .
(112)
Specifically, evaluating the Q (⋅ ) Gaussian tail function at
min
"F (Ti ( i ) )
F (Ti ( i ) )
2
1
= min
s2 w ¢¢ s2 *
"i
æ çç u èç Eve
1 ¶i
2
ö - 1 ÷÷÷ , ø
(113)
where uEve is Eve’s corresponding security parameter in an optimal Gaussian attack (for an exact derivation of this parameter, see the description of the AMQD modulation in [2]). The result in (113) leads to a worst-case scenario precisely as æ 2 sw2 ¢¢ perr = Pr ( pA pB F ( T ( ) ) ) = Q çç min F (Ti ( i ) ) ¶i å 2 çè "F (Ti ( i ) ) 2s * l
such that for the l i Gaussian sub-channels, the following constraint is satisfied: 2 ö 2 æ æ s 2 F ( T( ) ) ÷ s 2 F (T ( ) ) ö ÷÷ = å log2 çç 1 + w ¢¢ 2i i ÷÷÷ ³ lSk¢ ( i ) . log2 ççç 1 + w ¢¢ 2 s * s * ÷ø è èç ø÷ l
2
ö÷ ÷÷ , ø÷
(114)
(115)
In particular, the optimal manifold extraction dk ( V k ) requires the maximization of the product distance ¶1l
2l
at (114); thus without loss of generality, the optimizing condition at l Gaussian
sub-channels is a maximization as: 2l
dk ( V k ) : max ¶1l
>c
"i
1 S¢ l2 k ( i )
.
(116)
Since for P ( i ) this condition is satisfied, by using the P ( i ) random permutation operators defined in [4] as ( i ) for the Gaussian sub-channels, the optimality of dk ( Vk ) can be satisfied. Using (108), the constraint of (115) can be rewritten as follows [4], [20]: æ Q ( ) ö å log2 ççèç1 + i¶i 2i ø÷÷÷ ³ lSk¢ ( i ) ,
(117)
l
where Qi ( i ) =
2
2
F (Ti ( i ) ) ¶i sw2 ¢¢
(118)
s2
*
and 1 "Qi ³0 2
min
å Qi l
=
min
sw2 ¢¢
2 "F (Ti ( i ) ) 2s
*
where 22
å F (Ti ( i )) l
2
¶i
2
,
(119)
å F (Ti ( i )) "F (T ( ) ) min i
2
å
=
l
i
1 s2 w ¢¢ s2 *
l
æ çç u èç Eve
1 ¶i
2
ö - 1 ÷÷÷. ø
(120)
Without loss of generality, from (119) and (120), the Gaussian tail function in (114) can be precisely rewritten as æ 2 ö Q çç 21 å nEve - ¶i ÷÷÷ (121) ÷ø çè l
(
)
and
æ
å log2 ççèç uEve l
1 ¶i
2
ö÷ ÷= ø÷
æ
ö Qi ( i ) ÷
å log2 çççè1 +
¶i
l
2
÷÷ = lSk¢ ( i ) . ø
(122)
Particularly, from these derivations, the manifold extraction for the multicarrier scenario is yielded as follows. The Eerr error event can be rewritten as l
Eerr :
2 * å log2 ( 1 + F (Ti ( i )) ( SNRi¢ ) ) < lSk¢ ( i ) ;
(123)
i =1
thus for perr ( Sk¢ ( ) ) ,
perr ( Sk¢ (
æ l ö÷ 2 * ) ) = Pr çççç å log2 1 + F (Ti ( i ) ) ( SNRi¢ ) < lSk¢ ( i ) ÷÷÷ è i =1 ø÷ æ l ö÷ V 2 * ç = Pr çç å log2 1 + F (Ti ( i ) ) ( SNRi¢ ) < l n k ,i P ¢ ( i ) ÷÷÷ . çè i =1 min ø÷
(
)
(
)
(124)
Specifically, it can be further evaluated as
perr ( Sk¢ (
) ) = ( Pr ( log2 ( 1 +
F (Ti ( i ) )
2
*
( SNRi¢ )
)
(
S¢ Kin 2 k ( ) -1
Without loss of generality, let l be the smallest eigenvalue of M j , 24
*
( SNR ¢ )
)
.
(130)
l = min ( li ) ,
(131)
"i
where M j stands for the private codeword difference matrix, 1
Mj =
*
( SNR ¢ )
( pA - pB ) .
(132)
In particular, using (131) and (129) can be written precisely as S¢ Pr ( pA pB F ( T ( ) ) ) = Q 21 l2Kin 2 k ( ) - 1 .
(
(
))
(133)
Precisely, the result of (133) follows from the fact that for a Kin ´ Kout matrix M j , the following relation holds for M j and its smallest eigenvalue l , by theory:
l2 =
†
min
" F ( T(
))
F ( T ( ) ) M j M†j F ( T ( ) ) .
Some calculations then straightforwardly reveal that for Q
(
1 2 l Kin 2
(2
Sk¢ (
(134) ) -1
)) > 1,
the condi-
tion on l is as follows:
l2 >
1 S Kin 2 k ( ) -1
(
)
1 S K in 2 k ( )
.
(135)
Introducing a covariance matrix Ko as * I Kin
Ko = ( SNR ¢ ) where I K
in
Kin
,
(136)
is the Kin ´ Kin identity matrix, Eerr can be rewritten as
(
Eerr = log2 det I K
out
+ F (T(
†
) ) Ko F ( T ( ) )
) < S ¢( ) ,
(137)
where without loss of generality,
æ æ ç ç S ¢ ( ) £ P ¢ ( ) = max çç å l log2 çç 1 + "i çè çè
sw2 ¢¢ F (Ti ( i ) )
Let the SNIR (signal-to noise plus interference ratio) of i be
2
i
s2
i*
öö ÷÷ ÷÷ . ÷÷ ÷÷ ÷÷ øø
*
( SNIRi¢ )
(138)
in an SVD-assisted
AMQD setting, then Eerr can be rewritten precisely as * æ †ö ( SNR ¢ ) Eerr = log2 det çç I K + F ( T ( ) ) K F ( T ( ) ) ÷÷÷ out in èç ø
=
Kin
å (1 + ( SNIRi¢ )
*
i =1
) < S ¢ ( ).
(139)
Then, let us assume that r sub-channels are interfering with each other in the SVD-assisted multicarrier transmission. Specifically, at r interfering sub-channels, after some calculations, it can be found that the Sk¢ ( ) secret key rate reduces to precisely
Sk¢ ( ) = rSk¢ ( ) r +K1 -1 . ( ) in Thus, the resulting perr ( Sk¢ (
))
(140)
error probability is [20]
* æ æ †ö ( SNR ¢ ) perr ( Sk¢ ( ) ) = Pr çç log2 det çç I K + F ( T ( ) ) K F ( T ( ) ) ÷÷÷ < çè çè out in ø
25
Vk ( r +Kin -1 ) r
ö P ¢ ( ) ÷÷÷ , (141) ø
where I K
out
is the Kout ´ Kout identity matrix.
Then, by exploiting a union bound averaged over the statistics (see (61)) for each i [20–21], the hK
in >Kout
optimal tradeoff curve is yielded as follows:
hK
in >Kout
: dk ( Vk ) = 2 ( 2 - Vk ) .
(142)
Assuming the situation Kin £ Kout , some further results can also be derived. By using (139) and the properties of the multidimensional manifold space (see Theorem 1), and by averaging over the statistics, the hK £K tradeoff function [20] without loss of generin
out
ality is
hK
in £Kout
: dk ( Vk ) = ( i, ( Kin - i ) ⋅ ( Kout - i ) ) , i = 0, , n min .
Putting the pieces together, for each function hK
and hK
in >Kout
in £Kout
(143)
, the manifold extraction is
optimized via the maximization of the n min smallest singular values as l1n
min
, where for each
0 £ li £ 2 Kin and are determined from M j (see (132)).
Exploiting the argumentation of (110), the corresponding condition on l1n
min
for the optimal
tradeoff curve h is precisely as
h : max l1n "i
1
> c2
min
n min
1 n min
1n 2 min
1S ¢ k( i)
,
(144)
22
for any constant c > 0 . The results for any Kin > Kout and Kin £ Kout at Kin = 2, Kout = 4 are summarized in Fig. 5.
Figure 5. The optimal tradeoff curves hK
Kin = 2, Kout = 4 . The hK
in >Kout
in >Kout
for any Kin > Kout , and hK
in £Kout
at
curve is maximized in dk ( Vk ) = 4 at Vk = 0 , and picks up the
minimum dk ( Vk ) = 0 at Vk = 2 , for any Kin > Kout . For any Kin £ Kout , the hK
in £Kout
curve
has the max. in dk ( Vk ) = K in Kout , Vk = 0 , and the min. dk ( Vk ) = 0 at Vk = min ( K in , Kout ) . ■
26
5 Conclusions The additional degree of freedom injected by the multicarrier transmission represents a significant resource to achieve performance improvements in CVQKD protocols. The proposed manifold extraction exploits those extra resources brought in by the multicarrier CVQKD modulation and is unavailable in a single-carrier CVQKD scheme. We introduced the term of multidimensional manifold extraction and proved that it can significantly improve the reliability of the phase space transmission. We demonstrated the results through the AMQD multicarrier modulation and extended it to the multiple-access multicarrier scenario through the AMQD-MQA scheme. We studied the potential of a multidimensional manifold space of multicarrier CVQKD and the optimized tradeoff curve between the manifold parameter and the additional degree of freedom ratio. The results confirm that the possibilities in a multicarrier CVQKD significantly exceed the singlecarrier CVQKD scenario. The extra degrees of freedom allow the utilization of sophisticated optimization techniques for the aim of performance improvement. The available and efficiently exploitable extra resources have a crucial significance in experimental CVQKD, particularly in longdistance 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]
S. Pirandola, S. Mancini, S. Lloyd, and S. L. Braunstein, Continuous-variable Quantum Cryptography using Two-Way Quantum Communication, arXiv:quantph/0611167v3 (2008). L. Gyongyosi, Adaptive Multicarrier Quadrature Division Modulation for Continuousvariable Quantum Key Distribution, arXiv:1310.1608 (2013). L. Gyongyosi, Multiuser Quadrature Allocation for Continuous-Variable Quantum Key Distribution, arXiv:1312.3614 (2013). L. Gyongyosi, Singular Layer Transmission for Continuous-Variable Quantum Key Distribution, arXiv:1402.5110 (2014). L. Gyongyosi, Security Thresholds of Multicarrier Continuous-Variable Quantum Key Distribution, arXiv:1404.7109 (2014). S. Pirandola, R. Garcia-Patron, S. L. Braunstein and S. Lloyd. Phys. Rev. Lett. 102 050503. (2009). S. Pirandola, A. Serafini and S. Lloyd. Phys. Rev. A 79 052327. (2009). S. Pirandola, S. L. Braunstein and S. Lloyd. Phys. Rev. Lett. 101 200504 (2008). 27
[9] [10] [11] [12] [13]
[14] [15]
[16] [17] [18]
[19] [20] [21]
[22] [23] [24] [25] [26] [27]
C. Weedbrook, S. Pirandola, S. Lloyd and T. Ralph. Phys. Rev. Lett. 105 110501 (2010). C. Weedbrook, S. Pirandola, R. Garcia-Patron, N. J. Cerf, T. Ralph, J. Shapiro, and S. Lloyd. Rev. Mod. Phys. 84, 621 (2012). William Shieh and Ivan Djordjevic. OFDM for Optical Communications. Elsevier (2010). L. Gyongyosi, Scalar Reconciliation for Gaussian Modulation of Two-Way Continuousvariable Quantum Key Distribution, arXiv:1308.1391 (2013). 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). M. Navascues, F. Grosshans, and A. Acin. Optimality of Gaussian Attacks in Continuous-variable Quantum Cryptography, Phys. Rev. Lett. 97, 190502 (2006). R. Garcia-Patron and N. J. Cerf. Unconditional Optimality of Gaussian Attacks against Continuous-Variable Quantum Key Distribution. Phys. Rev. Lett. 97, 190503 (2006). F. Grosshans, Collective attacks and unconditional security in continuous variable quantum key distribution. Phys. Rev. Lett. 94, 020504 (2005). 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) 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). S. Imre and L. Gyongyosi. Advanced Quantum Communications - An Engineering Approach. Wiley-IEEE Press (New Jersey, USA), (2012). D. Tse and P. Viswanath. Fundamentals of Wireless Communication, Cambridge University Press, (2005). D. Middlet, An Introduction to Statistical Communication Theory: An IEEE Press Classic Reissue, Hardcover, IEEE, ISBN-10: 0780311787, ISBN-13: 978-0780311787 (1960) S. Kay, Fundamentals of Statistical Signal Processing, Volumes I-III, Prentice Hall, (2013) S. Imre, F. Balazs: Quantum Computing and Communications – An Engineering Approach, John Wiley and Sons Ltd, ISBN 0-470-86902-X, 283 pages (2005). D. Petz, Quantum Information Theory and Quantum Statistics, Springer-Verlag, Heidelberg, Hiv: 6. (2008). R. V. Meter, Quantum Networking, John Wiley and Sons Ltd, ISBN 1118648927, 9781118648926 (2014). L. Gyongyosi, S. Imre: Properties of the Quantum Channel, arXiv:1208.1270 (2012). K Wang, XT Yu, SL Lu, YX Gong, Quantum wireless multihop communication based on arbitrary Bell pairs and teleportation, Phys. Rev A, (2014).
28
[28] Babar, Zunaira, Ng, Soon Xin and Hanzo, Lajos, EXIT-Chart Aided Near-Capacity Quantum Turbo Code Design. IEEE Transactions on Vehicular Technology (submitted) (2014). [29] 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). [30] 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).
29
Supplemental Information S.1 Notations The notations of the manuscript are summarized in Table S.1. Table S.1. The summary of the notations.
Q (⋅ ) rank (⋅ )
Gaussian tail function. Rank function.
E
An event.
i
Index for the i-th subcarrier Gaussian CV, fi = x i + ipi . Index
j
for
the
j-th
Gaussian
single-carrier
CV,
j j = x j + ip j . 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.
( xi , pi )
Position and momentum quadratures of the i-th Gaussian
( xi¢, pi¢ )
Noisy position and momentum quadratures of Bob’s i-th
( x j , pj )
Position and momentum quadratures of the j-th Gaussian
( x j¢ , p j¢ )
Noisy position and momentum quadratures of Bob’s j-th
x A,i , pA,i
Alice’s quadratures in the transmission of the i-th subcar-
subcarrier, fi = x i + ipi . noisy subcarrier Gaussian CV, fi¢ = x i¢ + ipi¢ . single-carrier j j = x j + ip j . recovered single-carrier Gaussian CV jj¢ = x j¢ + ip j¢ .
rier. The
SNRi
SNR
SNRi =
of
sw2 i s 2
.
i
30
the
i-th
Gaussian
sub-channel
i ,
The SNR of the i-th Gaussian sub-channel i in the SVD-
SNRi¢
sw2 ¢¢
assisted multicarrier transmission, SNRi¢ =
i
s 2
.
i
The SNR of the Gaussian channel , SNR =
SNR
sw2 s 2
.
The SNR of the Gaussian channel in an SVD-assisted SNR ¢
protocol, SNR ¢ =
sw2 ¢¢ s 2
.
The SNR of the i-th Gaussian sub-channel i in a private SNRi*
transmission, SNRi* =
sw2
i
s2
.
i*
The SNR of the i-th Gaussian sub-channel i in an SVD*
( SNRi¢ )
*
assisted private transmission, ( SNRi¢ ) =
sw2 ¢¢ i
.
s2
i*
The SNR of the Gaussian channel in a private trans-
SNR
*
mission, SNR * =
sw2 s2
.
*
The SNR of the Gaussian channel in an SVD-assisted *
( SNR ¢ )
*
private transmission, ( SNR ¢ ) =
sw2 ¢¢ s2
.
*
P ( i )
The private classical capacity of a Gaussian sub-channel
P ¢ ( i )
The private classical capacity of a Gaussian sub-channel
S ( ) , Sk ( )
The secret key rate in a multicarrier setting, and the secret
i . i at SVD-assistance. key rate of user U k . The secret key rate in an SVD-assisted multicarrier transmission, and the secret key rate of user U k . In the manifold
S ¢ ( ) , Sk¢ ( i )
extraction these are fixed as Sk¢ ( ) =
Sk¢ ( i ) =
V k ,i n min
Vk n min
P ¢ ( ) , and
P ¢ ( i ) , respectively.
Product distance derived for the l Gaussian sub-channels,
d1l
d1l
2
(
> c
1
S ¢( i )
l2
31
) , for any constant c > 0 l
and secret key
rate S ¢ ( i ) > 0 per i .
perr
Error probability. An
pA
l-dimensional
random
private
codeword,
T
pA = ( pA,1, pA,l ) , where the i-th component pi is dedicated to i . Set of transmittance coefficients, such that for "j
: F (Tj ( j ) ) = min { F (Ti ( i ) ) } ,
where
"i
F (Ti ( i ) ) ³
1 *
( SNR ¢ )
of
. It refers to the worst-case scenario at
which a S ¢ ( ) > 0 nonzero secret key rate could exist.
( i )
A statistical averaging over the distribution of the Ti ( i ) transmittance coefficients.
c2l2
Chi-square distribution with 2l degrees of freedom, has a density f ( x ) = Degree
Vk dk
¶k , ¶ k ,i
freedom
ratio
=
in
1
(( SNR¢ )
*
)
dsingle
Uk ,
user
in
1
(( SNR ¢ ) ) *
a
- log2 perr ( Sk¢ (
lim *
( SNR ¢ )
¥
1
n min
single-carrier
P ¢(
)
))
.
transmission,
, where dsingle = 1 - V .
probability
AMQD perr =
of
Sk¢ ( ) n min .
probability
single perr
Error AMQD perr
)
x l -1e -x , where x ³ 0 .
Manifold parameter, dk ( Vk ) = Error
single perr
of
1 P ¢(
Vk =
1
(l -1 ) !
dAMQD
a
multicarrier
transmission,
, dAMQD = l ( 1 - V ) .
Distance function for the phase space constellation ¢ (
)
and ¢ ( i ) of user U k , ¶i =
.
1
Sk¢ ( i ) 2
2
, and ¶k ,i =
1
Sk¢ ( i ) 2
2
The optimal manifold-degree of freedom ratio tradeoff curve f
for a single-carrier transmission, f : dk ( V k ) = Z ( 1 - V k ) , where 0 < Vk £ 1 . The optimal manifold-degree of freedom ratio tradeoff curve
h
for
multicarrier
32
transmission,
f : dk ( Vk ) = l Z ( 1 - Vk ) ,
where 0 < Vk £ 1 , at l sub-channels. Number of interfering sub-channels in an SVD-assisted mul-
r
ticarrier scenario. The multidimensional optimal manifold-degree of freedom
h ( )
ratio tradeoff curve over the multidimensional manifold space . Multidimensional
manifold
space,
has
dimension
dim ( ) = K in Vk + ( Kout - V k ) V k .
The number of dimensions orthogonal to in the space of
N dim^
S ( F ( T ( ) ) ) , N dim^ = ( K in - Vk )( Kout - Vk ) .
S ( F ( T ( )))
The multidimensional space of F ( T ( ) ) , has dimension of dim ( S ( F ( T (
) ) ) ) = Kin Kout .
The squared random singular values of F ( T (
li2
An
optimizing
covariance
* I Kin
Ko = ( SNR ¢ )
Ko
Kin
, where I K
matrix,
)) .
defined
as
is the Kin ´ Kin identity
in
matrix. The
Mj
private 1
Mj =
l = min ( li ) "i
*
( SNR ¢ )
codeword
difference
matrix,
( pA - pB ) .
Smallest eigenvalue of the M j private codeword difference matrix. A
h
maximization
criteria
over
the
distribution
of
F ( T ( )) . Sets of singular operators S1 = { F1,U 2-1 } , S2 = {U 1,U 2-1 } .
S1 , S2
The
SVD
of
F (T) ,
where
F1-1, F1 Î Kin ´Kin
and
U 2 ,U 2-1 Î Kout ´Kout are unitary matrices, Kin and Kout
F ( T ) = U 2 GF1-1 ,
refer to the number of sender and receiver users such that
F ( T ) = U 2 GU 1-1
Kin £ Kout ,
F1-1F1 = F1F1-1 = I , U 2U 2-1 = U 2-1U 2 = I ,
and G Î is a diagonal matrix with non-negative real diagonal elements li , F ( T ) =
l1 ³ l2 ³ ln
ån
min
liU 2,i F1,-i 1 .
The non-negative real diagonal elements of the diagonal min
matrix
GÎ , 33
called
the
eigenchannels
of
F ( T ) = U 2 GF1-1 . The n min squared eigenchannels li2 are the eigenvalues of
li2
†
F ( T ) F ( T ) = U 2 GGTU 2-1 . n min = min ( K in , Kout ) , equals to the rank of F ( T ) , where
n min
Kin £ Kout . Stream matrix, s = ( s1, , sn
s
T
)
min
Î ( 0, Ks ) , defined by
the unitary F1 (U 1 ) applied on z Î ( 0, Kz ) . A stream variable si that identifies the CV state si
si ¢ = liU 2,i F1,-i 1 si , and
phase space . Expressed as
si
s¢ = F ( T ) s = The
ån
min
liU 2,i F1,-i 1 si .
Fourier-transformed
(
eigenchannel
)
U 2-1 ( gi ) = U 2-1
(å
n min j ¹i
interference,
2ù é KU -1 ( g ) = sg2 = ê gi ú , 2 i i ë û
U 2-1 ( gi ) Î 0, KU -1 ( g ) , 2 i
U 2-1 ( gi )
in the
)
ljU 2, j F1,-j1 s j .
The
variance
sg2 0 , in the low-SNR regimes. 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
(
x , p Î 0, sw2
0
random
) , where s
2 w0
quadrature
components
is the variance.
The noise variable of the Gaussian channel , with i.i.d.
D Î (
2 0, sD
)
zero-mean, Gaussian random noise components on the position and momentum quadratures
Dx , Dp Î ( 0, s 2 ) ,
é 2ù 2 = ê D ú = 2s 2 . sD 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 34
is the modulation variance of the Gaussian subcarrier CV state.
F -1 (⋅ ) = CVQFT† ( ⋅ )
F
tinuous-variable unitary operation.
(⋅ ) = IFFT (⋅ ) sw2
sw2 =
1 l
coder, continuous-variable unitary operation. The CVQFT transformation, applied by the decoder, con-
F (⋅ ) = CVQFT ( ⋅ ) -1
The inverse CVQFT transformation, applied by the en-
Inverse FFT transform, applied by the encoder. Single-carrier modulation variance.
0
Multicarrier modulation variance. Average modulation vari-
ål sw2
ance of the l Gaussian sub-channels i .
i
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
i , i = 1, , n
)
The decoded single-carrier CV of user U k from the subcarrier CV, expressed as F ( di
)=
F ( F - 1 ( z k ,i ) ) = z k ,i .
Gaussian quantum channel. 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 , Ti ( i )
quantifies
the position and momentum quadrature transmission, with (normalized)
real
0 £ Re Ti ( i ) £ 1
2,
ReTi ( i ) = Im Ti ( i ) . 35
and
imaginary
0 £ Im Ti ( i ) £ 1
parts
2,
where
TEve
Eve’s transmittance, TEve = 1 - T ( ) .
TEve,i
Eve’s transmittance for the i-th subcarrier CV. The subset of allocated users, Í K . Only the allocated
ÍK
users can transmit information in a given (particularly the j-th) AMQD block. The cardinality of subset is . An allocated user from subset Í K .
U k , k = 1, ,
A d-dimensional, zero-mean, circular symmetric complex
T
z = x + ip = ( z 1, , z d )
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
(
T
x = ( x1, , xd ) , p = ( p1, , pd ) , with x i Î 0, sw2
(
pi Î 0, sw2
0
0
),
) i.i.d. zero-mean Gaussian random variables.
An l-dimensional, zero-mean, circular symmetric complex random Gaussian vector of the l Gaussian subcarrier CVs, T ( 0, Kd ) , Kd = éê dd† ùú , d = (d1, , dl ) , di = x i + ipi , ë û
(
d = F -1 ( z )
x i , pi Î 0, sw2 variables,
F
)
are i.i.d. zero-mean Gaussian random
sw2 = 1 sw2 .
(
The
i-th
component
is
0
F
)
é 2ù 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 ) .
T Fourier transform of T ( ) = ëéT1 ( 1 ) ,Tl ( l ) ûù Î l ,
F ( T ( ))
the complex transmittance vector. Complex vector, expressed as F ( D ) = e
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 ( ) ) F ( d ) éë j ùû + F ( D ) éë j ùû .
y éë j ùû
t = F ( d ) éë j ùû
T -F ( D ) KF ( D )F ( D )
2
An
exponentially
f ( t ) = ( 1 2sw2n )e
36
distributed
-t 2sw2
variable,
, éë t ùû £ n 2sw2 .
with
density
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
Rk
2 , 0 £ TEve,i
2
0 £ Re TEve,i £ 1
2,
< 1.
Transmission rate of user U k . A di subcarrier in an AMQD block. For subset Í K with users and l Gaussian sub-channels for the trans-
di
1 n
mission, di =
n min
sw2
å k =1 zke
-i2 pik n ,i
= 1, , l .
The min { n1, , nl } minimum of the ni sub-channel coeffi2
cients, where ni = s 2
F (Ti ( i ) )
Modulation
sw2 = nEve - n min ( d )p
nEve =
1 l
variance,
, l = F (T*
)
2
=
1 n
and ni < nEve .
n
n
å k =1Tk*e
å i =1
(x )
,
-i 2 pik n
where 2
and
T* is the expected transmittance of the Gaussian sub-
channels under an optimal Gaussian collective attack. Additional sub-channel coefficient for the correction of modulation imperfections. For an ideal Gaussian modula-
nk
tion, nk = 0 , while for an arbitrary p ( x )
(
nk = n min 1 - ( d )p
(x )
) , where k =
distribution
(
1
nEve -n min ( d )p
(x )
-1
)
.
The set of i Gaussian sub-channels from the set of l good T U éë j ùû = éë 1, , s ùû k
sub-channels that transmit the s subcarriers of user U k in the j-th AMQD block. The constant modulation variance sw2 ¢ for eigenchannel li , i
sw2 ¢ i
æ ö evaluated as sw2 ¢ = m - ççç s 2 max li2 ÷÷÷ = i è ø n min total constraint sw2 ¢ =
ån
min
sw2 ¢ = i
1 l
1 nmin
ål sw2
i
sw2 ¢ , with a
= sw2 .
The modulation variance of the AMQD multicarrier trans-
sw2 ¢¢
mission
in
the
SVD
environment.
Expressed
as
æ ö sw2 ¢¢ = nEve - ççç s 2 max li2 ÷÷÷ , where li is the i-th eigenè ø n min
37
channel of F ( T ) , max li2 is the largest eigenvalue of n min
†
F (T)F (T) , 1 l
with
a
total
constraint
ål sw2 ¢¢ = sw2 ¢¢ > sw2 . i
The statistical model of F ( T ) at a partial channel side information, S ( F ( T ) ) = xK-1 GxK , where xK-1 in
out
S ( F ( T ))
and xK
in
are unitaries that formulate the input covariance matrix
Ks = xK ÃxK-1 , in
Ã
while
is
a
diagonal
matrix,
in
{
Ks = Qdiag sw2 ¢ , , sw2 ¢
out
Kin
1
}Q . †
Phase space constellation . Random phase space permutation constellation for the transmission of the Gaussian subcarriers, expressed as æ P ( ) = P ( 1 ) , , P ( l ) = çç f1d , P2 f1d , , Pl f1d çè P ( 1 ) P ( 2 ) P ( l )
P (
)
where fi
are the Gaussian subcarrier CVs, Pi , i = 2, , l
is a random permutation operator, d ( ) = d ( ) is the i j cardinality
( i ) .
of
o ( ( ) ) =
å ( nEve -
The
di
l
d P ( ) i
ö÷ ÷, ø÷
2
optimality
function
is
).
Cardinality of ( i ) . The normalized difference of two Gaussian subcarriers dA,i
di
and dB,i , di =
1 s2 w ¢¢ s2
(dA,i - dB,i ) .
Difference function of Gaussian subcarriers (phase space symbols)
¶
dA,i
and
dA, j
in
constellations
P ( k ) ,
k = 1, , l . For two Gaussian sub-channels 1 and 2 ,
¶ ( P ( 1 ) ) = min (dA,i - dA, j ) , "dA,i
¶ ( P ( 2 ) ) = ⋅ ¶ ( P ( 1 ) ) , where ³ 2 , j ¹ i .
38
S.2 Abbreviations AMQD BS CV CVQFT CVQKD DV FFT IFFT MQA QKD SNR SNIR SVD
Adaptive Multicarrier Quadrature Division Beam Splitter Continuous-Variable Continuous-Variable Quantum Fourier Transform Continuous-Variable Quantum Key Distribution Discrete Variable Fast Fourier Transform Inverse Fast Fourier Transform Multiuser Quadrature Allocation Quantum Key Distribution Signal to Noise Ratio Signal to Noise plus Interference Ratio Singular Value Decomposition
39
