Order statistics and random matrix theory of multicarrier continuous‐variable quantum key distribution

In a multicarrier continuous‐variable quantum key distribution (CVQKD) protocol, the information is granulated into Gaussian subcarrier CVs and the physical Gaussian link is divided into Gaussian sub‐channels. Here, we propose a combined mathematical framework of order statistics and random matrix theory for multicarrier continuous‐variable quantum key distribution. The 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 framework to evaluate the statistical properties of the information flowing processes in multicarrier CVQKD protocols. Using random matrix theory, we express the achievable secret key rates and study the efficiency of the adaptive multicarrier quadrature division‐multiuser quadrature allocation (AMQD‐MQA) multiple‐access multicarrier CVQKD. The proposed combined framework is particularly convenient for the characterization of the physical processes of experimental multicarrier CVQKD.


INTRODUCTION
The continuous-variable quantum key distribution (CVQKD) protocols 1 allow the establishment of an unconditional secure communication 2 over standard, currently established telecommunication networks.  Another motivation behind the use of CVQKD is the development of the quantum Internet 10,44,58 and quantum computers [59][60][61][62][63][64] . 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. [16][17][18][19][20][21][22][23][27][28][29][30][31][32][33] 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 34,[65][66][67] into the transmission because the optimal attack against a CVQKD protocol is provably also Gaussian. [16][17][18][19][20][21][22][23][27][28][29][30][31][32][33]68 Besides the attractive properties of CVQKD, the relevant performance attributes, such as secret key rates and transmission distances, [16][17][18][19][20][21][22][23][27][28][29][30][31][32][33][69][70][71][72][73][74][75][76] still require significant improvements. [37][38][39][40][41] For this purpose, the multicarrier CVQKD has been recently introduced through the adaptive quadrature division modulation (AMQD) scheme. 77 The multicarrier CVQKD injects several additional degrees of freedom onto the transmission which is not available for a standard, single-carrier CVQKD setting. 15,78,86 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, 77,78 but also make possible the introduction and defining of several new phenomena for CVQKD, such as singular layer transmission, 82 enhanced security thresholds, 78 multidimensional manifold extraction, 79 subcarrier domain analysis, 85 secret key rate adapting, 86 or the utilization of inference methods. 80,81 The benefits of multicarrier CVQKD has also been proposed for multiple-access multicarrier CVQKD via the AMQD-MQA (multiuser quadrature allocation). 15 An adaptive quadrature detection technique has also been defined for multicarrier CVQKD, which uses a channel transmittance estimation to decode the continuous variables. 69 The secret key rates of multicarrier CVQKD confirms the multimode bounds of a previous study 47 (see the results on fundamental rate-loss scaling in quantum optical communications in a previous study 47 ). For an additional information on the bounds of private quantum communications, we suggest another work. 48 Here, 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.
First, we provide a distribution statistics framework for multicarrier CVQKD. The distribution statistics of multicarrier CVQKD utilizes the results of order statistics, 87 which is an important subfield of statistical theory, with several applications-from mathematical statistics and engineering to the analysis of traditional communication systems. [88][89][90][91] 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 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. 34,92,93 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.
The novel contributions of our manuscript are as follows: 1. We propose a mathematical framework of order statistics and random matrix theory for multicarrier continuous-variable quantum key distribution. 2. The order 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. 3. We reveal the mathematical formalism of sub-channel selection and show a reduced complexity progressive sub-channel scanning method. 4. We define a random matrix formalism for multicarrier CVQKD to evaluate the statistical properties of the information flowing process. 5. Using random matrix theory, we express the achievable secret key rates and study the efficiency of the adaptive multicarrier quadrature division-multiuser quadrature allocation (AMQD-MQA) multiple-access multicarrier CVQKD.
6. The proposed framework is particularly convenient for the characterization of the physical processes of experimental multicarrier CVQKD.
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.

PRELIMINARIES
This section briefly summarizes the notations and basic terms. For further information, see previous work. 77

Multicarrier CVQKD
First, we summarize the basic notations of AMQD. 77 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 th input single-carrier state | | , where 2 0 is the modulation variance of the quadratures. (For simplicity, 2 0 is referred to as the single-carrier modulation variance, throughout.) Particularly, this Gaussian single-carrier is transmitted through a Gaussian quantum channel  . In the multicarrier scenario, the information is carried by Gaussian subcarrier CVs, , where 2 is the modulation variance of the subcarrier quadratures, which are transmitted through a noisy Gaussian sub-channel  i . 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 the subcarriers, while index to the single-carriers throughout the manuscript.) The single-carrier state | | ⟩ in the phase space  can be modeled as a zero-mean, circular symmetric complex Gaussian random variable z ∈  , and with i.i.d. real and imaginary zero-mean Gaussian random components Re . In the multicarrier CVQKD scenario, let n be the number of Alice's input single-carrier Gaussian states. The n input coherent states are modeled by an n-dimensional, zero-mean, circular symmetric complex random Gaussian vector where each z can be modeled as a zero-mean, circular symmetric complex Gaussian random variable Specifically, the real and imaginary variables (ie, the position and momentum quadratures) formulate n-dimensional real Gaussian random vectors, x = (x 1 , … , x n ) T and p = (p 1 , … , p n ) T , with zero-mean Gaussian random variables with densities (x ) and (p ) as where K z is the n × n Hermitian covariance matrix of z as follows: while z † is the adjoint of z.
For vector z, for any ∈ [0, 2 ]. The density of z is as follows (if K z is invertible): A n-dimensional Gaussian random vector is expressed as x = As, where A is an (invertible) linear transform from R n to R n , and s is an n-dimensional standard Gaussian random vector  (0, 1) n . This vector is characterized by its covariance The Fourier transformation F (·) of the n-dimensional Gaussian random vector v = (v 1 , … , v n ) T results in the n-dimensional Gaussian random vector m = (m 1 , … , m n ) T , as follows: 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 Gaussian random vector d, d ∈  (0, where and the position and momentum quadratures of | i ⟩ are i.i.d. Gaussian random variables where The T (  ) transmittance vector of  in the multicarrier transmission is where is a complex variable, which quantifies the position and momentum quadrature transmission (ie, gain) of the ith Gaussian sub-channel  i , in the phase space , with real and imaginary parts and Particularly, the T i (  i ) variable has the squared magnitude of where The Fourier-transformed transmittance of the ith sub-channel  i (resulted from CVQFT operation at Bob) is denoted by The n-dimensional zero-mean, circular symmetric complex Gaussian noise vector Δ ∈  ( 0, 2 Δ ) n of the quantum channel  , is evaluated as where with independent, zero-mean Gaussian random components and with variance 2  i , for each Δ i of a Gaussian sub-channel  i , which identifies the Gaussian noise of the ith sub-channel  i on the quadrature components in the phase space .
The CVQFT-transformed noise vector can be rewritten as with independent components F ( on the quadratures, for each F (Δ i ).
It also defines an n-dimensional zero-mean, circular symmetric complex Gaussian random vector The complex A (  ) ∈ C single-carrier channel coefficient is derived from the l Gaussian sub-channel coefficients as

DISTRIBUTION STATISTICS FOR MULTICARRIER CVQKD
First, we summarize some preliminary findings from order statistics from a previous work 87 then evaluate the theorems and proofs. Note the proofs throughout Section 3 follow the notations of the previous work. 87

Moment-generating function
The M moment-generating function (MGF)-function 87 of a nonnegative random variable x, where c is a complex dummy variable, P (·) is the probability density function (PDF) function, and 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 where ∈ [0, 2 ] and Assuming an error rate R err (x) where a and b are constants, theR err average error rate is yielded as where P SNR (·) is the PDF of the signal to noise ratio (SNR), which can be rewritten as where M SNR (·) is the M-function of the SNR.
Precisely, from (59), it follows that the P |A ( )| 2 PDF function of can be evaluated via two random correlated variables as Specifically, exploiting the Bayesian formula, it follows that where P Γ 2 (x) is the PDF of the lth (ordered) sub-channel transmittance coefficient while P Γ 1| Γ 2 =x ( ) is the conditional PDF of the sum of the first l − 1 ordered sub-channel coefficients, at 1 Then, let us assume that there are l − 1 available sub-channels, and let the ith sub-channel's quantity be referred to as 1

FIGURE 1
The Λ U k order-and-sum operator of user U k . Λ U k selects l sub-channels from the total n, via O (the ordered l sub-channels are depicted by the thick frame), then evaluates Without loss of generality, for this quantity, the following relation holds: where After some calculations, the M-function of 1 For l Gaussian sub-channels with 1 In particular, the Pr Λ 0 (·) probability that l sub-channels are selected from the n is as Note that in a worst-case scenario, the condition 1 may not be satisfied for the required number l of sub-channels. In this case, the threshold 1 where 0 < < 1 is a real variable. The condition of (75) satisfies that, at least, the best sub-channel 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, (75) is satisfied, resulting in the ordered coefficients 1 which can be simplified into and the partial function is by some fundamental theory. The probability that k sub-channels are selected from n at condition (75) is expressed as where The proof is concluded here.

Optimized complexity progressive scan
Theorem 2. (Progressive sub-channel selection). The average number of iterations needed for the selection of the l Gaussian sub-channels is minimized via an Λ sub-channel selection operator.
Proof. The complexity of the sub-channel selection operators is discusses via the average number of iterations needed for the procedure. The proof follows the definitions and notations of a previous work. 87 Exploiting the results of Theorem 1, using Λ 0 , the Λ 0 overall average number of the iterations (eg, the number of comparisons of sub-channel transmittance coefficients at a ) .
Operator Λ 0 does require the scan of the total n sub-channels, which is practically inconvenient.
To resolve this problem, we introduce operator Λ 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 sub-channels. Let Λ ′ a slightly modified version of Λ 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 Λ ′ selects the remaining l − k sub-channels from the set  ( of bad sub-channels, and the Lagrange multiplier in (69) at Λ ′ is reevaluated as where is a nonnegative real variable, expressed as where . As follows, (82) allows to the legal parties to preserve the security conditions via a modified security threshold The Λ and Λ ′ average number of iterations of operators Λ and Λ ′ are evaluated as follows. The Λ sub-channel selection operator models the situation if the iteration stops as the 1 Specifically, the Pr Λ (k) probability of the availability of l sub-channels for k = l, and the probability that only k < l, k = 0, … , l − 1 sub-channels are available is 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 are found and the remaining l − k sub-channels are selected set  of the ordered coefficients, the Λ ′ |A ( )| 2 CDF-function is as After some calculations, the Λ ′ overall average number of the iterations at Λ ′ is yielded as which can be rewritten as wherẽis the common mean. while , ≤ x ≤ i is expressed as with an M-function Then, let (  ) be the averaged single-carrier squared magnitude as the sum of i -s, as which can be rewritten as the sum of independent random variables as Specifically, using (103), the M-function of a term kx k is yielded as and the M-function of (  ) is as Then, P ( ) (x) is expressed as Particularly, at l − 1 total sub-channels, let us denote ′ i = 1 the ith normalized Gaussian subcarrier. Then, and the conditional PDF is as where and the M-function of The joint PDF is where P c (x) and c (x) are the common PDF and CDF of the unordered variables, while Using (112), the joint PDF in (111) can be rewritten precisely as Note that by some fundamental theory, and Then, the x and p quadrature-level (single-carrier) error probability at Λ 0 is as where M ŜNR is a scaled SNR quantity,ŜNR= 2 0 ∕ 2 2  , and ∈ [0, 2 ]. In particular, the formula of (116) can be applied to derive the p err ( A ) error probabilities (single-carrier ) quadrature level) of the operators using the M-functions proposed in Sections 3.1 and 3.2, precisely as and ) and by using the M-functions derived in the previous sections, the corresponding p because both operators find l sub-channels; however, the threshold differs that leads to p Λ′ because operator Λ ′ always determines the required number l of the sub-channels at a given threshold. It leads to the final conclusion p

RANDOM MATRIX FORMALISM OF MULTICARRIER CVQKD
First, we summarize the basic functions and random matrix tools in Propositions 3 and 4, to evaluate the results in Theorem 3 and Lemma 1. The proofs throughout Section 4 follow the notations of 92 and. 93 For the detailed description of the AMQD-MQA multiple-access multicarrier CKQKD scheme, see. 15

Multiuser quadrature allocation for multicarrier CVQKD
of K users.) At l Gaussian sub-channels and K users, the Proof. Let the Z i, entries of Z be independent, arbitrarily distributed complex random variables with identical mean i, = and variance 2 i, = 1 ∕ l such that the Lindeberg condition 92 is satisfied, where  is a probability measure function that returns an event's probability. Let X be an l × l random matrix, while Φ is a K × K random matrix, independent from each other and from Z. The -transform of a nonnegative random variable X is as where is a nonnegative real number and 0 < X ( ) ≤ 1. Without loss of generality, let the logical channel  k of user U k be defined as The single-carrier transmittance coefficient of  k is referred to as A ( k ). Particularly, then, for K, l → ∞ and K ∕ l → , where is a nonnegative variable, for the -transform of where D ∈  {XX † } is an independent random variable with a distribution  {XX † } of the asymptotic spectra of XX † , and where T ∈  {ΦΦ † } is an independent random variable with a distribution of the asymptotic spectra of ΦΦ † . 92 The -transform (Shannon transform) of a nonnegative random variable X is defined as where is a nonnegative real number. Let the -transform of F ( where d and t are random variables such that Note that by denoting d ( ) the solution for (126) and (127), the result in (122) can be rewritten as Throughout the manuscript, let X and Y be defined as independent random variables X ∈  [0,1] and Y ∈  [0,1] drawn from a  uniform distribution on [0, 1].
Specifically, the decomposition of F ( allows us to define a function H (l) (a, ), ( − 1) ∕ K ≤ a < ∕ K, where a is a random variable, precisely as where I is the identity, such that for K, l → ∞, and a ∈ [0, 1] where for t ( ), the result of (126) and (127) holds and H (l) converges to where B (·) is a limiting bounded measureable function, 92 and H (a, ) is evaluated as Without loss of generality, the entries of XZΦ are independent, arbitrarily distributed zero-mean ( i, = 0) complex random variables with variance where Π is an l × K deterministic matrix with uniformly bounded (i, ) entries, with variance profile var (a, b). Let var l (·) be the variance profile function as Then, let us presume that for these entries the condition of , the conditions (133) and (138) are satisfied, then for K, l → ∞, and K ∕ l → , then for independent random variables X ∈  [0,1] and Y ∈  [0,1] drawn from a  uniform distribution on [0, 1], where and ] .
Note that it is precisely a coincidence with the -transform of the empirical (squared) eigenvalue distribution of In particular, if for the entries of F ( T (  )) the conditions (133) and (138)

are satisfied, and W F(T( ))F(T( )) † (·, ·)
and Υ F(T( ))F(T( )) † (·, ·) are given as (140) and (141), respectively, then the -transform of F can be rewritten as Specifically, further note that for the following limit holds: where ∞ depends on ′ . Precisely, ∞ differs for ′ < 1, ′ = 1 and ′ > 1, as where W ∞ (·) is yielded from and where ℘ (·) is yielded from Precisely, the asymptotic theory of singular values of rectangular matrixes assumes the existence of an l × K matrix M, for which the aspect ratio converges to a nonnegative variable 92 , as K, l → ∞.
Proof. Without loss of generality, let the l × K channel matrix of K users be given as F ( . Focusing on the case that that Z has arbitrarily distributed, Z i, zero-mean complex random variables with 2 such that if X ∈  [0,1] , then without loss of generality, the following relation holds for the  distribution of where the nonrandom limits F a (·) , F b (·) for a given a, b ∈ [0, 1) have all bounded moments. Then, it can be shown that, if F ( T (  )) = Z•Ω, where • stands for the Hadamard product, then for K, l → ∞ and K ∕ l → , then for independent random variables X ∈  [0,1] and Y ∈  [0,1] the following coincidence holds:

Theorem 3. (Random matrix formalism of multiple-access multicarrier CVQKD). The random matrix decomposition of the l × K channel matrix F
is user U k 's entry, m is the number of sub-channels of U k , • is the Hadamard product, while Φ is an l × K random matrix, Proof. Let the number of the users be k ≤ K. The transmission of the users is realized through l sub-channels, such that for each U k user, a given number (m) of sub-channels are allocated. The transmittance coefficient statistics of the users are determined via operators Λ 0 , Λ, Λ ′ , see Theorem 1. The derivation also utilizes the random matrix formulas introduced in Propositions 2 and 3, respectively. The proof utilizes the terms and notations of. 52 Let the l × K channel matrix F ( T (  )) of K users be given as where • is the Hadamard (element-wise) product, and is an l × K random matrix with zero-mean and 2 Z = 1 ∕ l variance entries, where is user U k 's entry, X is an l × K random matrix with independent entries for the users, such that where for user U k where is the averaged transmittance coefficients of U k derived via a corresponding distribution statistics operator (see Theorems 1 and 2), m is the number of sub-channels of U k , while Φ is a l × K random matrix, Specifically, for user U k , let us identify  k the logical channel (a set of m sub-channels) of U k of the kth column of where  (i) k identifies the ith logical sub-channel of U k such that Let r be a nonnegative variable that identify the ratio of the sub-channels allocated to the logical channel  k of U k , as where 0 ≤ r ≤ 1. Using (155), the Y output matrix of the K users can be written as where X refers to the input matrix of the K users while F ( Δ (K) ) is the Fourier transformed matrix of the Δ (K) noise matrix of the K users.
Let us utilize the results of Propositions 3 and 4 for K, l → ∞ and K ∕ l → , where is a nonnegative variable. Let U k be transmission efficiency of U k in function of the SNR * U k (evaluated for the transmission of private classical information, see 38,82 ) is as where (X, b) is the two-dimensional profile function of the l × K matrix and where an (r, t) entry of is which identifies the rth logical sub-channel of the tth user, 92 and let where m is the number of Gaussian subcarrier CVs allocated to U k , 2 i is the constant subcarrier modulation variance, while 2 where 2 0 is the vacuum noise and N i is the excess noise of the Gaussian sub-channel  i defined as where W i is the variance of Eve's EPR state used for the attacking of  i , while is the transmittance of Eve's beam splitter (BS), while |T i | 2 is the transmittance of  i , while function  Particularly, the result in (174) can be rewritten as where function is as Then, let us assume that for (a, b), the following relation holds: where m identifies the number of sub-channels of U k used in the transmission, that is, for the remaining l − k sub-channels, where (r, k) identifies the rth subcarrier of user U k . Using (178), an e (·) efficiency function is introduced as where  (·) is derived via (174). Using Proposition 2, let If (181) holds, then without loss of generality, and if SNR * U k → ∞ also holds, then Specifically, using (180), the P s m ( k ) symmetric private classical capacity (the maximum common 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: where ∑ l−1 i=0 SNR * i is the total SNRs of the l Gaussian sub-channels evaluated for the transmission of private classical information expressed as and function Υ (·) is as In particular, defining function where Z ∈  [b,1] is an independent random variable, drawn from a  uniform distribution on , the formula of (184) can be rewritten in a simplified form specifically as . (188)

FIGURE 5
The P s m ( k ) of U k in function of SNR * U k for 0 ≤ r ≤ 1, where SNR * U k is the SNR 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 From (188), the S s m (U k ) symmetric secret key rate (a common rate at which the K users both can reliably transmit private classical information over the l sub-channels of  ) of U k is as The proof is concluded here.
The P s m ( k ) of U k from (188) in function of r is depicted in Figure 5.

Multiuser transmission over identical Gaussian sub-channels Lemma 1. (Random matrix formalism of multiple-access multicarrier CVQKD for identically allocated sub-channels). Let us set
Proof. The proof utilizes Proposition 3. Let us identify set the sub-channels allocated to the 1 ≤ k ≤ K ≤ l users. The transmittance coefficients are determined via operators Λ 0 , Λ , and Λ ′ (see Theorem 1). The proof utilizes the terms and notations of. 92 The random matrix decomposition of the h × K channel matrix F ( where is an h × K isotropic unitary matrix, independent of X and Φ, UU † = I, with arbitrarily distributed random variables such that the Lindeberg condition (see (119)) is satisfied, with mean i, = and variance 2 i, = 1 ∕ l, X is an h × h random matrix where and is the averaged transmittance coefficients of U k taken over , such that Precisely, if the users transmit through the same subcarriers, then for all U i , i = 0, … , k − 1 users, an identical  U i =  k logical channel is allocated, see (197); thus, (194) can be rewritten as with the averaged coefficient of (196) for the h sub-channels of user U k . Particularly, since all users use the same Gaussian sub-channels, the empirical distribution of XX † converges to a nonrandom limit F |C| 2 , thus where |C| is a random variable having a distribution of the asymptotic singular value distribution of the singular values of X, ie, the distribution of |C| 2 is determined by the asymptotic spectra of XX † , thus without loss of generality where A ( k ) is the A coefficient of the  k channel of U k , see (196). Let |Ω| be a random variable having a distribution of the asymptotic singular value distribution of the singular values of Φ, the distribution of |Ω| 2 is determined by the asymptotic spectra of ΦΦ † . Specifically, the e (·) efficiency function in (180) can be rewritten as where U k is the multiuser efficiency, evaluated as Without loss of generality, the term U k Precisely, the P s m ( k ) symmetric private classical capacity of the  k logical channel of U k is particularly yielded as where 1] , and F(T( ))F(T( )) † is the -transform of F specifically as while function Using (208), the following relation holds for the K users: . (210)

FIGURE 6
The density function (x) for = 0.2, 0.5, 1 Precisely, the S s m (U k ) symmetric secret key rate of U k over identical sub-channels is expressed as In particular, using the density function (x) (Marchenko-Pastur density function 92 ), where (·) is a related random function, and u = ( 1 − √ ) 2 and v = ( 1 + √ ) 2 , (210) can be rewritten precisely as The proof is concluded here.

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 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.

ETHICS STATEMENT
This work did not involve any active collection of human data.

DATA ACCESSIBILITY STATEMENT
This work does not have any experimental data.    var (a, b) Variance profile function.
Nonnegative variable, such that K ∕ l → holds, where K is the number of users, l is the number of available sub-channels.
Logical channel of user U k , a set formulated from m subcarriers such that  k = ( Symmetric private classical capacity of the  k logical 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  .
Symmetric secret key rate of the  k logical channel of U k , a common rate at which the K users both can reliably transmit private classical information over the l sub-channels of  . b Ratio of the k actual number of users and the total users The random matrix decomposition at arbitrarily allocated sub-channels. Random matrix decomposition of F ( is the averaged transmittance coefficient of U k derived by operator Λ 0 , Λ or Λ ′ , m is the number of sub-channels of U k , • is the Hadamard product operator, while Φ is an l × K random matrix, Φ = diag (Φ 0 , … , Φ K−1 ).

XUΦ
The random matrix decomposition at identically allocated sub-channels.
is an h × K isotropic unitary matrix with arbitrarily distributed random variables with mean i, = and variance 2 i, = 1 ∕ l, u k = [ q k,0 , … , q k,h−1 ] T , X is an h × h random matrix the averaged transmittance coefficients of U k in , such that  i =  k for i = 0, … , K −1, while Φ is a K × K random matrix, Φ = (Φ 0 , … , Φ K−1 ).

U K out
The unitary CVQFT operation, U K out = 1 √ K out e −i2 ik K out , i, k = 0, … , K out − 1, K out × K out unitary matrix.
where IFFT stands for the Inverse Fast Fourier Transform, zero-mean Gaussian random quadrature components, and 2 F is the variance of the Fourier transformed Gaussian state.
The decoded single-carrier CV of user U k from the subcarrier CV, | | k,i  Gaussian quantum channel.  i An i-th Gaussian sub-channel, i = 0, … , n − 1.
∈ C. The real part identifies the position quadrature transmission, the imaginary part identifies the transmittance of the position quadrature.
∈ C, quantifies the position and momentum quadrature transmission, with (normalized) real and imaginary parts 0 ≤ ReT i ( Eve's transmittance for the i-th subcarrier CV. z A d-dimensional, zero-mean, circular symmetric complex random Gaussian vector that models d Gaussian CV input states, z = x + ip = (z 0 , … , z d−1 ) T ,  (0, K z ), The m-th element of the k-th user's vector y k , expressed as k,m = ∑ l F ] T ∈ C l , the complex transmittance vector.