Diversity space of multicarrier continuous‐variable quantum key distribution

The diversity space of multicarrier continuous‐variable quantum key distribution (CVQKD) is defined. The diversity space utilizes the resources that are injected into the transmission by the additional degrees of freedom of the multicarrier modulation. 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 extend the manifold extraction for the multiple‐access AMQD‐MQA (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.

The pre-unitary F 1 (U 1 ) transforms such that the input will be sent through the λ i eigenchannels of the Gaussian link, whereas U −1 2 performs its inverse. Note that the pre-F 1 (U 1 ) and post-U −1 2 unitaries are the not inverse of F and U but F −1 1 (U −1 1 ) and U 2 , respectively. In particular, these unitaries define the set S 1 of singular operators, as follows: Specifically, if each transmit user sends a single-carrier Gaussian CV signal to an encoder E, 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 −1 2 , defining the set S 2 of singular operators as The subindices of the operators F 1 ; U −1 2 È É and U 1 ; U −1 2 È É are different in each S i , i=1,2 because these operators are not the inverse of each other.
These operators are determined by the SVD of F T ð Þ, which is evaluated as where F −1 1 ; F 1 ∈ C K in ×K in and U 2 ; U −1 2 ∈ C K out ×K out , K in , and K out refer to the number of sender and receiver users such that and The term Γ∈R is a diagonal matrix with nonnegative real diagonal elements λ 1 ≥ λ 2 ≥ …λ n min ; (6) which are called the eigenchannels of F T ð Þ ¼ U 2 ΓF −1 1 , where which equals to the rank of F T ð Þ, where by an initial assumption. (Note that the eigenchannels are also called the ordered singular values of F T ð Þ.) In terms of the λ i eigenchannels, F T ð Þ can be precisely rewritten as where λ i U 2;i F −1 1;i are rank-one matrices. In fact, the n min squared eigenchannels λ 2 i are the eigenvalues of the matrix where Γ T is the transpose of Γ.

| Rate formulas of multicarrier CVQKD
The complete derivation of the secret key rate formulas can be found in Gyongyosi and Imre 2 ; here, we give a brief overview on the transmission rates of multicarrier CVQKD. In particular, the (real domain) classical capacity of a Gaussian subchannel N i in the multicarrier setting is while in the SVD-assisted AMQD, where σ 2 ω ′′ ¼ σ 2 ω 1 þ c ð Þ> σ 2 ω . Specifically, the SNR (signal to noise ratio) of N i is expressed as while the SNR of N at a constant modulation variance σ 2 ω is SNR ¼ Particularly, in the SVD-assisted AMQD, it referred to as and respectively. From (11) and (12), the (real domain) classical information transmission rates R k N ð Þ and R ′ k N ð Þ of user U k through the l N i Gaussian subchannels in AMQD and SVD-assisted AMQD are precisely as follows: and Precisely, the SNR * i (signal to noise ratio) of the i-th Gaussian subchannel N i for the transmission of private classical information (ie, for the derivation of the secret key rate) under an optimal Gaussian attack is expressed as SNR is precisely evaluated as follows 2 : where and where σ 2 0 is the vacuum noise and N i is the excess noise of the Gaussian subchannel N i defined as where W i is the variance of Eve's EPR state used for the attacking of N i , while is the transmittance of Eve's beam splitter (BS), and T i j j 2 is the transmittance of N i . Precisely, in the SVD-assisted multicarrier CVQKD, and for N i and N, respectively. Particularly, from (18) the P N i ð Þ private classical capacity (real domain) is expressed as The SVD-assisted P ′ N i ð Þ from (22) is then yielded precisely as Assuming l Gaussian subchannels, the (real domain) secret key rate S N ð Þ of AMQD and S ′ N ð Þ of SVD-assisted AMQD are as follows:

| Manifold extraction
In a multicarrier CVQKD scenario, the term manifold is interpreted as follows. Let the i-th component p j,i of a given private random codeword p j ¼ p j;1 ; …; p j;l T to be transmitted through N i , where each Gaussian subchannel is char- As a first approach, the number l of the Gaussian subchannels is identified as the manifold of N. 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 C P S N ð Þ. Using P i , i=2,…,l random permutation operators, C P S N ð Þ can be defined for the multicarrier transmission as the available degrees of freedom in the Gaussian link can be utilized, and the random permutation operators inject correlation between the N i subchannels via P i C S N 1 ð Þ. In particular, for each Gaussian subchannel, the distance between the phase space constellation points is evaluated by δ i , the normalized difference function. Assuming two l-dimensional input random private codewords ; …p B;l T and two Gaussian subchannels N i and N j , δ i is calculated precisely as follows: particularly, for the l Gaussian subchannels where the term δ 1 …l j j is referred to as the product distance. 53-55 The maximization of this term ensures the maximization of the extractable manifold, and determines thep err pairwise worst-case error probabilities of p A ,p B .
As we have shown in Section 3, by using C P S N ð Þ and (30), thep err worst-case pairwise error probability 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 users.
Let us assume that the S ′ k N ð Þ secret key rate of user U k , for ∀k, is fixed precisely as follows: where ς k >0 is referred to as the degree of freedom ratio of U k , while n min has been shown in (7). As one can immediately conclude from (31) Without loss of generality, for a given subchannel Note: From this point, we use the complex domain formulas throughout the manuscript and S ′ k N ð Þ and S ′ k N i ð Þ are fixed to (31) and (32).
For a given N i , an E err error event 53-55 is identified as follows: and the probability of E err at a given S ′ N i ð Þ is identified by the p err error probability as follows: Particularly, by some fundamental argumentations on the statistical properties of a Gaussian random distribution, 53-55 can be expressed as , the corresponding error probability is as Let l=1, that is, let us consider a single-carrier CVQKD, with F T N ð Þ ð Þ j j 2 of N, with a secret key rate S′ N ð Þ. In this setting, p err is expressed precisely as 53 by theory. Specifically, assuming a multicarrier CVQKD scenario with l Gaussian subchannels and secret key rate S ′ N i ð Þ per N i , p AMQD err is derived as follows: Without loss of generality, we construct the set T, such that where for ∀i, i=1,,…l the following condition holds: In particular, the transmission through the Gaussian subchannels is evaluated via set T, which refers to the worstcase scenario at which a S ′ N ð Þ > 0 nonzero secret key rate is possible, by convention. Particularly, in (30), a given ∂ i identifies the minimum distance between the normalized 2 S ′ k N i ð Þ points for the phase space constellation C ′ S N i ð Þ of N i . Precisely, by fundamental theory, 53-55 it can be proven that for an arbitrary distribution of the F T i N i ð Þ ð ÞFourier transformed transmittance coefficient, the maximized product distance function of (30) can be derived by an averaging over the following statistic S: where σ 2 Þis a zero-mean, circular symmetric complex Gaussian random vari- Þ zero-mean Gaussian random variables per quadrature components x i and p i , for the i-th Gaussian subcarrier CV.
As it has been shown in Gyongyosi, 6 the δ manifold parameter picks up the following value in the single-carrier CVQKD setting: while in the multicarrier CVQKD setting, The result in (42) will be further sharpened in Section 3 since it significantly depends on the properties of the corresponding phase space constellation C S N ð Þ. From (45), it clearly follows that the extractable manifold δ determines the error probability of the transmission, and for higher δ, 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 ς and is referred to as the degree of freedom ratio. Without loss of generality, in the function of ς>0 (41) and (42) precisely can be rewritten as while in the multicarrier CVQKD setting, it refers to the ratio of the subcarriers allocated to a given user, Thus, in a multicarrier CVQKD scenario with l Gaussian subchannels, for a given ς>0, the overall gain is l. As it can be verified, using (43) and (44), the error probabilities can be evaluated as (see also Gyongyosi 6 ) The result in (46) is therefore a reduced error probability in comparison to the single-carrier CVQKD scenario (see (45)). Thus, the extra degree of freedom available in a multicarrier CVQKD setting allows us to decrease significantly the error probability. For a detailed derivation on (46), see Gyongyosi. 6 In a multicarrier CVQKD protocol run, there exists an optimal tradeoff between δ and ς; however, it requires to make some preliminary assumptions, as it is concluded in Lemma 1.
Þn min ; ς k >0, of user U k ,k=1,…,K out , the δ k ς k ð Þ extractable manifold is the ratio of p err S ′ k N ð Þ À Á error probability and the n min -normalized private classical capacity 1 n min P ′ N ð Þ; derived at the asymptotic limit of SNR ′ À Á * →∞.
Proof. In particular, at a given ς k and S ′ k N ð Þ (see (31)), the δ k ς k ð Þ manifold parameter of user U k , k=1,…,K out is as follows: where p err S ′ k N ð Þ À Á is the error probability of U k at S ′ k N ð Þ, while SNR ′ À Á * is the SNR of N in an SVDassisted AMQD modulation for private information transmission (see (22)). Specifically, assuming that the condition of holds, where c>0 is a constant and ∂ k is the minimum distance of the 2 S ′ k N ð Þ normalized constellation points precisely as then at a given secret key rate S ′ k N ð Þ, the p err S ′ k N ð Þ À Á error probability of the transmission of U k decays as where Q · ð Þ is the Gaussian tail function. Note that the condition of (48) follows from the fact that the separation (ie, ∂ k ) of the constellation points of C S N ð Þ has to be significantly larger than σ N ; otherwise, the Q · ð Þ Gaussian tail function yields in high error probabilities. 53 Without loss of generality, for an N i dedicated to , and Exploiting the argumentation of (40) on the averaging over the S statistics of the channel transmittance coefficients, and the related result in (38), the p err S ′ which at SNR ′ i À Á * →∞ coincidences with (51).
Thus, using P i ∈ U, i=2,…,l drawn from a U uniform distribution, the private permutation phase space con- ð Þ can be defined for the private multicarrier transmission as The private permutation phase space constellation of (53) can be used as a corresponding C ′ S N i ð Þ, for each N i subchannels.
In particular, assuming the use of can be rewritten precisely as where Pr F T i N i ð Þ ð Þ j j 2 < x À Á ≈ x, by theory at the distribution of (40), 53-55 and δ k;i ς k;i Þ for all N i , then for the S ′ N ð Þ secret key rate in the low SNR regimes the following result yields, precisely: while in the high SNR regimes and from the law of large numbers 53 : , the density of (52) is depicted in Figure 1.
Note that for the transmission of classical (ie, nonprivate) information, C S has a cardinality of An error event E err of (33) for a subchannel N i can be rewritten as thus, introducing Z>1 which brings up by the use of a corresponding C S , a typical error probability is precisely expressed as follows: where (59), the T i transmittance coefficients are arbitrarily distributed, in contrast to (54)). As one immediately can conclude, the phase space constellation C S provides a further decreased p err in comparison to (46). Putting the pieces together, the optimal manifold-degree of freedom ratio tradeoff curve 53 f for a singlecarrier scheme (eg, if l=1 we trivially have a single-carrier scheme) can be expressed as where 0<ς k ≤ 1. Specifically, some calculations then straightforwardly reveal that any phase space constellation C S N ð Þ that satisfies the condition of achieves the optimal f tradeoff curve, for any constant q>0. The recently proposed permutation phase space constellation C P S N ð Þ for SVD-assisted AMQD provably satisfies this condition. Without loss of generality, assuming a constant g>0, p err can be rewritten precisely as Thus, for E g err , the manifold parameter is δ g k ς k ð Þ as In Section 3, we give a proof on the multicarrier CVQKD scenario and show that there exists an optimal tradeoff between ς k and δ k 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.
Proof. The proof assumes a K in ,K out multiuser scenario. First we express p err as follows: where λ 2 i are the squared random singular values of F T N ð Þ ð Þ. 53-55 Assuming ς k >0, the λ i singular values can be decomposed into subsets s 0 and s 1 such that set The remaining n min − ς k singular values formulate the subset s 1 , as In particular, from (65) and (67), for the rank of F T N ð Þ ð Þ, the following relation identifies an error event E err : Thus, the p err at a given ς k is precisely referred to as Specifically, at ς k =0, p err ς k ð Þ is evaluated as At ς k →0, in (64) the corresponding relation is thus for the sum of the n min squared eigenvalues λ i , where N U k refers to the logical channel of U k , which consists of the allocated Gaussian subcarriers of that user.
As follows, (72) holds if only thus, Þfor a K in , K out multiple-access scenario is as follows 53 : Particularly, the M manifold space can be represented as a rank−ς k matrix M (see later (78)); thus, it precisely has a dimension of Specifically, for any ς k >0, where Þ, and M is a matrix with rank ς k . Thus, for ς k >0, p err ς k ð Þ is related to as the difference between matrix F T N ð Þ ð Þand a rank-ς k matrix M, since Þis a K in ×K out matrix with rank=ς k ; that is, it has ς k >0 linearly independent row vectors from the K out rows. 53-55 Without loss of generality, M can be characterized by K in ς k þ K out − ς k ð Þ ς k parameters by theory, from which (77) straightforwardly follows. Putting the pieces together, the Particularly, from (78), the multidimensional optimal tradeoff function is yielded as The N dim ⊥ in function of dim M ð Þ, at K in =K out −1, ς k =0,3;0.6;0.9 is depicted in Figure 2. . For user U k , the manifold extraction at l Gaussian subchannels leads to an optimal h tradeoff curve h: Þ¼lf , for any multicarrier scheme, where f is the optimal tradeoff curve of a single-carrier CVQKD protocol, f : 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 Without loss of generality, for the total constraint of SVD-assisted AMQD, one has precisely In particular, by further exploiting the results of SVD-assisted AMQD and following the derivations, 2 here, we determine the private random codeword difference for two l-dimensional input codewords Þis evaluated precisely as follows: where ∂ i is the normalized difference of p A,i and p B,i , calculated as follows: Assuming the case that in (83), the condition holds, one obtains for any constant c>0 and for an arbitrary pair of p A and p B .
In particular, at a secret key rate S ′ k per N i , the cardinality of C ′ S N i ð Þ is as follows: Thus, in the private transmission each Specifically, evaluating the Q · ð Þ Gaussian tail function at where υ Eve is Eve's corresponding security parameter in an optimal Gaussian attack. The result in (89) leads to a worst-case scenario precisely as such that for the l N i Gaussian subchannels, the following constraint is satisfied: In particular, the optimal manifold extraction δ k ς k ð Þ requires the maximization of the product distance ∂ 1 …l j j 2=l at (90); thus, without loss of generality, the optimizing condition at l Gaussian subchannels is a maximization as follows: Since for C P S N i ð Þ this condition is satisfied, by using the C P S N i ð Þ random permutation operators as C S N i ð Þ for the Gaussian subchannels, the optimality of δ k ς k ð Þ can be satisfied. Using (84), the constraint of (91) can be rewritten as follows: and Without loss of generality, from (95) and (96), the Gaussian tail function in (90) can be precisely rewritten as follows: and Particularly, from these derivations, the manifold extraction for the multicarrier scenario is yielded as follows. The E err error event can be rewritten as thus, for p errS ′ k N ð ÞÞ À , Specifically, it can be further evaluated as and at ς k,i , the optimal manifold extraction for each N i is Thus, without loss of generality, As follows, from (103), the manifold extraction for the l Gaussian subchannels, and the optimal manifolddegree of freedom ratio tradeoff curve h 53-55 for the multicarrier transmission, is precisely expressed as where 0 ≤ ς k .
The single-carrier and multicarrier tradeoff curves f and h are compared in Figure 3.
To conclude the results, the multicarrier CVQKD with l Gaussian subchannels provides an l-fold manifold gain over the single-carrier CVQKD protocols, for all ς k .

| Manifold extraction for AMQD-MQA
Theorem 3. (Manifold extraction in a multiple-access multicarrier CVQKD). For any K in , K out multipleaccess multicarrier CVQKD scenario, the manifold extraction is maximized via δ k ς k ð Þ: max where η is expressed as where M j stands for the private codeword difference matrix, In particular, using (107) and (105) can be written precisely as Precisely, the result of (109) follows from the fact that for a K in ×K out matrix M j , the following relation holds for M j and its smallest eigenvalueλ, by theory: Some calculations then straightforwardly reveal that for Q 1 Introducing a covariance matrix K o as where I K in is the K in ×K in identity matrix, E err can be rewritten as where without loss of generality, Let the SNIR (signal-to noise plus interference ratio) of N i be SNIR ′ i À Á * in an SVD-assisted AMQD setting, then E err can be rewritten precisely as Then, let us assume that r subchannels are interfering with each other in the SVD-assisted multicarrier transmission. Specifically, at r interfering subchannels, after some calculations, it can be verified that the S ′ k N ð Þ secret key rate reduces to precisely Thus, the resulting p err S ′ k N ð Þ À Á error probability is 53 where I K out is the K out × K out identity matrix. Then, by exploiting a union bound averaged over the S statistics (see (40)) for each N i , 53,54 the h K in >K out optimal tradeoff curve is yielded as follows: Assuming the situation K in ≤ K out , some further results can also be derived. By using (115) and the properties of the multidimensional manifold space M (see Theorem 1), and by averaging over the S statistics, the h K in ≤K out tradeoff function 53 without loss of generality is Putting the pieces together, for each function h K in >K out and h K in ≤K out , the manifold extraction is optimized via the maximization of the n min smallest singular values as λ 1 …n min , where for each 0 ≤ λ i ≤ 2 ffiffiffiffiffiffiffi K in p and are determined from M j (see (108)). Exploiting the argumentation of (86), the corresponding condition on λ 1 …n min j j for the optimal tradeoff curve h is precisely as for any constant c>0.
The results for any K in >K out and K in ≤ K out at K in =2,K out =4 are summarized in Figure 4.

| 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 FIGURE 4 The optimal tradeoff curves h Kin>Kout for any K in >K out , and h Kin≤Kout at K in =2,K out =4. The h Kin>Kout curve is maximized in δ k ς k ð Þ ¼ 4 at ς k =0, and picks up the minimum δ k ς k ð Þ ¼ 0 at ς k =2, for any K in >K out . For any K in ≤ K out , the h Kin≤Kout curve has the max. in δ k ς k ð Þ ¼ K in K out , ς k =0, and the min. δ k ς k ð Þ ¼ 0 at ς k ¼ min K in ; K out ð Þ 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 single-carrier 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 long-distance scenarios. First, we summarize the basic notations of AMQD. 1 The following description assumes a single user, and the use of n Gaussian subchannels N i for the transmission of the subcarriers, from which only l subchannels will carry valuable information.
In the single-carrier modulation scheme, the j-th input single-carrier state φ j is a Gaussian state in the phase space S, with i.i.d. Gaussian random position and momentum quadratures x j ∈ N 0; σ 2 where σ 2 ω 0 is the modulation variance of the quadratures. For simplicity, σ 2 ω 0 is referred to as the single-carrier modulation variance, throughout. Particularly, this Gaussian single-carrier is transmitted through a Gaussian quantum channel N. In the multicarrier scenario, the information is carried by Gaussian subcarrier CVs, where σ 2 ω is the modulation variance of the subcarrier quadratures, which are transmitted through a noisy Gaussian subchannel N i . Each N i Gaussian subchannel is dedicated for the transmission of one Gaussian subcarrier CV from the n subcarrier CVs. (Note: Index i refers to the subcarriers, while index j to the single-carriers throughout the manuscript.) The single-carrier state φ j in the phase space S can be modeled as a zero-mean, circular symmetric complex Gaussian random variable z j ∈ CN 0; σ 2 ω z j , with variance σ 2 and imaginary zero-mean Gaussian random components Re z j À Á ∈ N 0; σ 2 ω 0 , Im z j À Á ∈ N 0; σ 2 ω 0 .
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 j 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 ¼ x 1 ; …; x n ð Þ T and p ¼ p 1 ; …; p n ð Þ T , with zero-mean Gaussian random variables with densities f (x j ) and f (p j ) as where K z is the n×n Hermitian covariance matrix of z: while z † is the adjoint of z. For vector z, holds, and for any γ ∈ 0; 2π ½ . 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 N 0; 1 ð Þ n . This vector is characterized by its covari- with independent, zero-mean Gaussian random components and with variance σ 2 N i , for each Δ i of a Gaussian subchannel N i , which identifies the Gaussian noise of the i-th subchannel N i on the quadrature components in the phase space S.
The CVQFT-transformed noise vector can be rewritten as It also defines an n-dimensional zero-mean, circular symmetric complex Gaussian random vector where K F Δ ð Þ ¼ K Δ , by theory. At a constant subcarrier modulation variance σ 2 ω i for the n Gaussian subcarrier CVs, the corresponding relation is where σ 2 ω i is the modulation variance of the quadratures of the subcarrier ϕ i j i transmitted by subchannel N i . Assuming l good Gaussian subchannels from the n with constant quadrature modulation variance σ 2 ω i , where σ 2 ω i ¼ 0 for the i-th unused subchannel, In particular, from the relation of (A27), for the transmittance parameters the following relation follows at a given modulation variance σ 2 ω 0 , precisely, where T N ð Þ j j 2 is the transmittance of N in a single-carrier scenario, and For the method of the determination of these l Gaussian subchannels, see Gyongyosi and Imre. 1 Alice's i-th Gaussian subcarrier is expressed as The notations of the manuscript are summarized in Table A1. i Index for the i-th subcarrier Gaussian CV, l Number of Gaussian sub-channels N i for the transmission of the Gaussian subcarriers. The overall number of the sub-channels is n. The remaining n−l subchannels do not transmit valuable information.
x i ; p i ð Þ Position and momentum quadratures of the i-th Gaussian subcarrier, Noisy position and momentum quadratures of Bob's i-th noisy subcarrier Gaussian CV, x j ; p j Position and momentum quadratures of the j-th Gaussian single-carrier φ j x ′ j ; p ′ j Noisy position and momentum quadratures of Bob's j-th recovered single-carrier Gaussian CV jφ ′ x A,i ,p A,i Alice's quadratures in the transmission of the i-th subcarrier.
The SNR of the i-th Gaussian subchannel N i in the SVD-assisted multicarrier transmission,

SNR
The SNR of the Gaussian channel N, SNR ′ The SNR of the Gaussian channel N in an SVD-assisted protocol, SNR′ ¼ σ 2 The SNR of the i-th Gaussian sub-channel N i in a private transmission, The SNR of the i-th Gaussian sub-channel N i in an SVD-assisted private transmission, The SNR of the Gaussian channel N in an SVD-assisted private transmission, SNR ′ À The private classical capacity of a Gaussian sub-channel N i .
The private classical capacity of a Gaussian sub-channel N i at SVD-assistance.
S N ð Þ, S k N ð Þ The secret key rate in a multicarrier setting, and the secret key rate of user U k . (Continues) p err Error probability.
T Set of transmittance coefficients, such that for ∀j of T: It refers to the worst-case scenario at which a S ′ N ð Þ > 0 nonzero secret key rate could exist.
S N i ð Þ A statistical averaging over the distribution of the T i N i ð Þ transmittance coefficients.
Chi-square distribution with 2l degrees of freedom has a density f x Error probability in a single-carrier transmission, p single Error probability in a multicarrier transmission, p AMQD err ¼ 1 ∂ k ,∂ k,i Distance function for the phase space constellation C ′ S N ð Þ and C ′ S N i ð Þ of user U k , ∂ i ¼ 1 , and ∂ k;i ¼ 1 The optimal manifold-degree of freedom ratio tradeoff curve for a single-carrier transmission, f : The optimal manifold-degree of freedom ratio tradeoff curve for multicarrier transmission, f : δ k ς k ð Þ ¼ lZ 1 − ς k ð Þ, where 0<ς k ≤ 1, at l subchannels. r Number of interfering subchannels in an SVD-assisted multicarrier scenario.
h M ð Þ The multidimensional optimal manifold-degree of freedom ratio tradeoff curve over the multidimensional manifold space M.
The squared random singular values of F T N ð Þ ð Þ.
K o An optimizing covariance matrix, defined as where F −1 1 ; F 1 ∈ C Kin×Kin and U 2 ; U −1 2 ∈ C Kout ×Kout are unitary matrices, K in and K out refer to the number of sender and receiver users such that K in ≤ K out , F −1 The nonnegative real diagonal elements of the diagonal matrix Γ∈R, called the eigenchannels of F T ð Þ ¼ U 2 ΓF −1 1 .
n min n min ¼ min K in ; K out ð Þ , equals to the rank of F T ð Þ, where K in ≤ K out .
s i A stream variable s i that identifies the CV state s i j i in the phase space S. Expressed as s i ′ j i ¼ λ i U 2;i F −1 1;i s i j i, and The variance σ 2 γ →0, in the low-SNR regimes.
Δ ∈ CN 0; σ 2 Δ À Á The noise variable of the Gaussian channel N, with i.i.d. zero-mean, Gaussian random noise components on the position and momentum quadratures Δ x ; Δ p ∈ N 0; σ 2 The variable of a Gaussian subcarrier CV state, ϕ i j i ∈ S. Zero-mean, circular symmetric Gaussian random variable, Single-carrier modulation variance.
Multicarrier modulation variance. Average modulation variance of the l Gaussian sub-channels N i .
p di ∈ N 0; σ 2 ωF are i.i.d. zero-mean Gaussian random quadrature components, and σ 2 ωF is the variance of the Fourier transformed Gaussian state.
The decoded single-carrier CV of user U k from the subcarrier CV, φ k;i ¼ CVQFT ϕ i j i ð Þ, also expressed as The modulation variance of the AMQD multicarrier transmission in the SVD environment. Expressed as with a total constraint 1 l ∑ l σ 2 The statistical model of F T ð Þ at a partial channel side information, S F T ð Þ ð Þ¼ξ −1 Kout Γξ Kin , where ξ −1 Kout and ξ Kin are unitaries that formulate the input covariance matrix K s ¼ ξ Kin ℘ξ −1 Kin , while ℘ is a diagonal matrix,

C S
Phase space constellation C S .
C P S N ð Þ Random phase space permutation constellation for the transmission of the Gaussian subcarriers, expressed as C P S N ð Þ ¼ C P S N 1 ð Þ; …; C P S N l ð Þ ¼ ðjϕ 1 …d C P