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Security Thresholds of Multicarrier Continuous-Variable Quantum Key Distribution

We prove the secret key rate formulas and derive security threshold parameters of multicarrier continuous-variable quantum key distribution (CVQKD). In a multicarrier CVQKD scenario, the Gaussian input quantum states of the legal parties are granulated into Gaussian subcarrier CVs (continuous-variables). The multicarrier communication formulates Gaussian sub-channels from the physical quantum channel, each dedicated to the transmission of a subcarrier CV. The Gaussian subcarriers are decoded by a unitary CV operation, which results in the recovered single-carrier Gaussian CVs. We derive the formulas through the AMQD (adaptive multicarrier quadrature division) scheme, the SVD-assisted (singular value decomposition) AMQD, and the multiuser AMQD-MQA (multiuser quadrature allocation). We prove that the multicarrier CVQKD leads to improved secret key rates and higher tolerable excess noise in comparison to single-carrier CVQKD. We derive the private classical capacity of a Gaussian sub-channel and the security parameters of an optimal Gaussian collective attack in the multicarrier setting. We reveal the secret key rate formulas for one-way and two-way multicarrier CVQKD protocols, assuming homodyne and heterodyne measurements and direct and reverse reconciliation. The results reveal the physical boundaries of physically allowed Gaussian attacks in a multicarrier CVQKD scenario and confirm that the improved transmission rates lead to enhanced secret key rates and security thresholds.


Introduction
The continuous-variable quantum key distribution (CVQKD) allows for legal parties to transmit information with unconditional security over the currently established telecommunication networks [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. The CVQKD systems, in contrast to discrete variable (DV) QKD protocols, do not require single-photon sources and can be implemented by standard devices and modulation techniques, allowing an efficient signal processing in practical scenarios. The CVQKD schemes in general are based on Gaussian modulation, which is a well-applicable practical finding in the experiment. In CVQKD, the information is carried via a Gaussian-modulated position and momentum quadratures in the phase space. The Gaussian quantum states (referred to as single-carriers throughout) are sent through a noisy quantum channel by the sender, Alice. The quantum channel is attacked by an eavesdropper (Eve), and the receiver (Bob) gets a noisy system. Since the optimal attack against CVQKD is a Gaussian attack [2][3][17][18][19], the noise of the quantum channel can be provably modeled as an additive white Gaussian noise. The security of CVQKD has been already proven against several of the most powerful optimal Gaussian collective attacks, where the eavesdropper is allowed to use quantum memory and to perform a collective (joint) measurement on her quantum register at the end of the protocol run.
Besides the attractive properties of CVQKD, in comparison with traditional telecommunication techniques [21][22][23][24][25], the efficiency of the protocol still requires several improvements, both in secret key rates and the maximum tolerable excess noise. In particular, with this in mind the multicarrier CVQKD has been recently introduced, and the AMQD (adaptive multicarrier quadrature division) scheme has been defined [4]. Specifically, while in the standard CVQKD the information is transmitted by single-carriers, in an AMQD modulation the information is granulated into Gaussian subcarrier CVs. These subcarriers divide the physical quantum link into several Gaussian sub-channels, leading to an improved transmission rate and higher tolerable excess noise overall. An important property of AMQD is that it significantly improves the transmission rates in the low-SNR regimes, which is specifically crucial in experimental CVQKD scenarios. The AMQD modulation has been extended to SVD-assisted (singular value decomposition) AMQD [6], where the SVD of the Gaussian quantum channel derives the eigenchannels of the link. Precisely, in the SVD-AMQD, the transmit information of the users are pre-coded by a corresponding unitary operation in the encoding phase, which are then sent through the eigenchannels of the Gaussian channel. The received quantum states are decoded by the inverse post-unitary operation. The SVD-assistance leads to further improved transmission rates and the maximum tolerable excess noise, which can also be exploited in multiuser CVQKD scenarios. The benefits of AMQD can also be extended to a multiuser environment. The AMQD-MQA (multiuser quadrature allocation) mechanism [5] is based on the AMQD modulation, more precisely on the sophisticated allocation mechanism of the Gaussian subcarrier CVs. In an AMQD-MQA setting, the legal parties use a common, shared physical Gaussian link in parallel. In particular, the multicarrier transmission improves the multiuser transmission rates, leading to an enhanced simultaneous communication through the Gaussian sub-channels. Here we show that the improved classical information trans-mission rates allow for the parties to establish higher secret key rates, and to extend the boundaries on the maximal amount of tolerable excess noise of single-carrier CVQKD.
We derive the security thresholds and prove the secret key rate formulas of multicarrier CVQKD. We demonstrate the results for one-way and two-way CVQKD, homodyne and heterodyne measurements, and direct and reverse reconciliation. The multicarrier CVQKD transmission is analyzed through the AMQD, the SVD-assisted AMQD, and the multiuser AMQD-MQA scheme. We study the impacts of optimal Gaussian attacks on multicarrier CVQKD and determine the private classical capacity [22,[26][27][28] of a Gaussian sub-channel. The results reveal that the improved transmission rates of multicarrier modulation lead to improved secret key rates, higher tolerable excess noise, and better security thresholds. We show that the enhanced secret key rates can be extended to a multiuser setting, allowing the legal parties a reliable simultaneous private communication over a noisy Gaussian link. For the multiuser scenario of multicarrier CVQKD, we define the private classical capacity regions of the users and derive the sum and symmetric private classical capacities.
This paper is organized as follows. In Section 2, the preliminary findings are summarized. Section 3 proposes the proofs of the secret key rate formulas and the security thresholds of multicarrier CVQKD. Section 4 discusses the physical boundaries of optimal Gaussian attacks. Finally, Section 5 concludes the results. Supplementary information is included in the Appendix.

Preliminaries
First we summarize briefly the notations and basic terms. For further information, see the detailed descriptions of [4][5][6].

Multicarrier CVQKD
First we summarize the basic notations of AMQD from [4]. 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 Particularly, this Gaussian single-carrier is transmitted through a Gaussian quantum channel  .
In the multicarrier scenario, the information is carried by Gaussian subcarrier CVs, 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 Specifically, the real and imaginary variables (i.e., the position and momentum quadratures) formulate n-dimensional real Gaussian random vectors, where z K is the n ń Hermitian covariance matrix of z : while † z is the adjoint of z .
for any 0, 2 g p é ù Î ë û . The density of z is as follows (if z K is 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 The Fourier transformation ( ) 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 , The ( ) 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 Re Im where ( ) ( )

Re Im
The Fourier-transformed transmittance of the i-th sub-channel i  (resulted from CVQFT operation at Bob) is denoted by The n-dimensional zero-mean, circular symmetric complex Gaussian noise vector with independent, zero-mean Gaussian random components with variance 2 i s  , for each i D of a Gaussian sub-channel i  , which identifies the Gaussian 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 with independent components ( ) Precisely, it also defines an n-dimensional zero-mean, circular symmetric complex Gaussian random vector ( ) where ( ) At a constant subcarrier modulation variance 2 i w s for the n Gaussian subcarrier CVs, the corresponding relation is For the method of the determination of these l Gaussian sub-channels, see [4]. Alice's i-th Gaussian subcarrier is precisely expressed as follows: ( ) Specifically, the result of (10) defines n, independent i  Gaussian sub-channels, each with noise  [4]. For the extension of the analysis to a multiuser setting, see the derivations of [5]. The description of the SVD-assisted multiuser AMQD CVQKD scheme can be found in [6].

Gaussian States
where I is the identity matrix, , while Z refers to the Pauli Z matrix, where ( ) s 1

Private Classical Capacity
The ( ) P  private classical capacity of  in direct reconciliation (DR, i.e., from Alice to Bob) is defined as where ( ) : I B E is the mutual information between Bob and Eve.
Assuming collective (joint) measurement in the protocol runs on Bob's and Eve's sides (also referred to as collective CVQKD protocols), the related DR and RR formulas are where c is the Holevo quantity. We use collective protocols in the evaluation of the secret key rates in Section 3.
Note that for collective protocols, the ( ) : I B E function causes negative divergence in (37) (except in one-way collective CVQKD at heterodyne measurement) [2]; thus these RR rate formulas are derived via

Optimal Gaussian Attack on Multicarrier CVQKD
First we analyze the correlation of Alice and Bob's quadratures in an AMQD modulation, for a homodyne and heterodyne measurement, hom M and het M . Then we characterize the correlation measures at an optimal Gaussian attack on the multicarrier transmission. : The : To express these rates, we first define the ( ) i P  private classical capacity of an i  Gaussian quantum channel in the AMQD modulation.
In particular, the * SNR i (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 is expressed as and where 2 s  is the vacuum noise and i N is the excess noise of the Gaussian sub-channel i  defined as where i W is the variance of Eve's EPR state used for the attacking of i  , while secret key rate formulas for the AMQD transmission are given in an extended form in Section 3.

Secret Key Rates of AMQD and AMQD-MQA
In this section, we prove the secret key rates of AMQD [4], SVD-assisted AMQD [6], and AMQD-MQA [5] for collective CVQKD protocols, following the formalisms of [1] and [2]. The secret key rates of AMQD are evaluated for the case of n Gaussian subcarriers and l Gaussian sub-channels with i E v e n n < . The variance of Eve's EPR ancilla is i W for the attack of a sub-channel i  [2][3]. For the multiuser scenario of AMQD-MQA, K users are assumed for the parallel use of the Gaussian quantum channel. The subcarrier Gaussian CVs of Alice, Bob, and Eve will be referred to as i A , i B , and i E , respectively. The single-carrier Gaussian CVs are referred to as j A , j B , and j E .
Theorem 1 (AMQD secret key rates). The AMQD leads to improved secret key rates in a CVQKD scenario in comparison to a single-carrier CVQKD. Proof.
In the first part of the proof, we reveal the secret key rates of AMQD for one-way CVQKD, assuming homodyne and heterodyne measurements with RR and DR. In the second part, we reveal the secret key rates of AMQD for two-way CVQKD. Finally, we conclude the results with the analysis of the secret key rates of the SVD-assisted AMQD.

AMQD Secret Key Rates in One-Way CVQKD
Following the formalism of [2] throughout the analysis, the quadratures particularly can be precisely rewritten as and thus (78) can be rewritten as Let the variance of Eve's EPR ancilla used for the attack of the i-th Gaussian sub-channel i  to be referred as i W . The covariance matrix of Eve's i-th density matrix Assuming that Bob receives a noisy Gaussian subcarrier CV i Specifically, the conditional variances of Alice's variables and Eve's noisy subcarrier quadratures are , , Specifically, for the quadratures , i i x p of the i-th Gaussian subcarrier CV, the conditional covariance matrices [2] are precisely and ( ) Without loss of generality, assuming l n < Gaussian subcarriers for the information transmission, from the conditional variances 2 Particularly, using the covariance matrices of (91)-(94), the symplectic spectra [2] can be constructed for the Gaussian subcarriers. We omit this step and show only the symplectic spectra of the Gaussian single-carriers recovered from the Gaussian subcarriers via the CVQFT operation; because the secret key rate formulas will also use this representation.

Homodyne Measurement, Reverse Reconciliation
In the case of homodyne measurement, Bob first applies for all collected i f subcarriers the CVQFT operation to recover the Gaussian single-carrier CV i where 0 N is the shot-noise variance and s is the squeezing factor, which is and for Eve's estimators Precisely, from (98) and (99) l o g , and from (57)  l o g , The reverse reconciliation RR x S , RR p S secret key rates from (61) and (62) are as follows: : Particularly, the private classical capacity

Secret key rate
In the next step, the hom , RR M one way Ssecret key rate is evaluated for the recovered single-carrier Gaussian CVs. Introducing the notation ⋅ for the variance of the quadratures, for Bob's noisy quadratures (which noise arise from Eve's Gaussian attack, which will noise be corrected in the post- and since Specifically, a commutation relation argument [1] then yields Without loss of generality, assuming that for the l Gaussian subchannels i  and 0 2 1 w s  holds, the following symplectic spectra can be constructed for Bob's system, j B : and for Eve's system, j E , precisely: In particular, in case of a reverse reconciliation, the related symplectic spectra  are evaluated as follows [2]: Finally, for the joint system : I B E causes negative divergence in the rate formulas of (37) [2].

Private classical capacity of a Gaussian sub-channel
Without loss of generality, using a slightly modified version of the corresponding terms of Section , 0 Then the commutation relation [1] leads to Specifically, assuming the use of coherent Gaussian CVs for the transmission, a corresponding relation yields as 2 , , , thus , , , For Eve's system j E , the corresponding covariance matrices are evaluated as follows: In particular, for the direct reconciliation, the symplectic spectra Without loss of generality, by utilizing the covariance matrices of (121) and (139), the symplectic spectra of (123)-(125), and (139)        , , 0

Heterodyne Measurement, Reverse Reconciliation
, , , , Specifically, the het M leads to a covariance matrix , , Precisely, for Eve's system j E , the corresponding covariance matrix is In particular, assuming that ( ) ( ) 1 w s  holds, the following symplectic spectra [2] can be constructed for Bob's system: and for Eve's system the symplectic spectra is Precisely, in case of reverse reconciliation and het M , Particularly, for the joint system Without loss of generality, using the related covariance matrices of (147)-(148), symplectic spectra of (125) and (149)

Private classical capacity of a Gaussian sub-channel
Without loss of generality, using a slightly modified version of the related formulas of Section , and ( ) and , , , , Eve's corresponding covariance matrix is The symplectic spectra In particular, using the covariance matrices of (147)-(148), symplectic spectra of (125), (149)- The derivation of the AMQD secret key rates for two-way CVQKD is included in the Appendix.

SVD-Assisted AMQD
The SVD-assisted AMQD modulation [6] leads to a virtually improved modulation variance ( ) where 0 c > for the subcarrier transmission, for further details and analysis see [6]. The SVD defines a logical layer above the physical layer of the multicarrier transmission, and the virtual increment of the multicarrier modulation variance from 2 w s ¢¢ to 2 w s in (164) where ( ) ( ) is precisely the largest eigenvalue of ( ) ( ) † F F T T [6], and where T G is the transpose of G (e.g., elements of the SVD of ( ) F T ). For further details, see [6]. In particular, the SVD-assisted rate formulas can be computed by the previously obtained AMQD secret key rate formulas, using the following for the transmittance of the i  Gaussian sub-channels, without loss of generality, as

AMQD-MQA Secret Key Rates
In this section, we derive the private classical capacity regions of users 1 U and 2 U and the corresponding secret key rates in an AMQD-MQA multiuser scenario. First, we have to summarize the derivation of the classical capacity region of the users, from [5].
Theorem 2 (AMQD-MQA secret key rates). The AMQD-MQA leads to improved secret key rates in a K K  multiuser CVQKD scenario in comparison to a single-carrier K K  multiuser CVQKD scenario.

Proof.
The proof assumes two users, k U , 1, 2. k = Let l be the number of good i  Gaussian subchannels. (i.e., the noise of the sub-channels is below the critical security parameter Eve n , which identifies the optimal Gaussian collective attack, see the properties of AMQD and AMQD-MQA in [4] and [5]). Let the transmittance of the i-th sub-channel be ( ) The outputs of 1 U and 2 U are expressed as follows: Specifically, the output d-dimensional output of the k-th user is In particular, the sum capacity [5] is the total throughput over the l sub-channels of  at a constant modulation variance 2 w s is as follows: The symmetric capacity [5] is the maximum common rate at which both 1 U and 2 U can reliably transmit information over the l sub-channels of  , as follows: The C capacity region [5] is the region of the rates of ( ) Particularly, if the equality holds, then only user k U is allowed to transmit over the l subchannels. These rates define the corner points At the corner points, the rate of the given user is maximal whereas rate of the other user is zero. On the line between the corner points, both users are allowed to simultaneously communicate at rates 1 R and 2 R .
Hence, in the corner points 1 C , 2 C , only 1 U , 2 U is allowed to transmit over the l sub-channels, whereas the rate of the other user is zero. Taking the  convex hull of all possible independent input distributions leads to the capacity region C as ( ) The inputs 1 z and 2 z are zero-mean Gaussian random variables, for which follows that all information quantities that characterize capacity region C are simultaneously maximized because the capacity region ( ) with zero-mean, circular symmetric complex Gaussian random distribution variables formulates the superset S of all other capacity regions (for the proof of optimality, see [5]) with arbitrary x p ¢ distributions, that is, H ⋅ is the differential entropy. For a CV variable x with probability density function x F , the differential entropy is evaluated as follows: The conditional differential entropy, at given output y is , log d d .
x y From these quantities, the continuous mutual information function is and (Note: If the l Gaussian subcarriers are not completely mutually independent, then £ stands in the second line of (180), whereas if the l subcarriers are derived from not the optimal zero-mean circular symmetric complex Gaussian random   distribution, then £ stands in the third line of (180).) The input maximization leads to the following sum capacity in an AMQD-MQA setting: where ( ) ( ) ( ) , with a total transmit variance constraint In particular, for two users, 1 U and 2 U , the C capacity region ( ) 1 2 , R R of AMQD-MQA is as follows:  The results on the C capacity region of ( ) 1 2 , R R of 1 U and 2 U in AMQD-MQA are summarized as follows. The corner points 1 C and 2 C identify the maximal rates at with a single user can communicate. The line between the two corner points represents that trade-off between the rates of users 1 U and 2 U , at which simultaneously reliable transmission is possible. The transmission is realized through l subcarriers, each having a constant modulation variance 2 w s per subcarrier quadrature components. The two users communicate over the Gaussian quantum channel with rates 1 R and 2 R . At the corner points 1 C and 2 C , only one user is allowed to transmit and all degrees of freedom is allocated to that user.
Specifically, from the derivation of the C classical capacity region, the private classical capacity region P is derived as follows. In particular, the private classical capacity region P is precisely defined as ( ) For two users, 1 U and 2 U , the P region of secret key rates ( ) 1 2 ,

S S
of AMQD-MQA is as follows: For K users 1 ,

SVD-Assisted AMQD-MQA Secret Key Rates
The SVD-assisted AMQD-MQA secret key rates 1 S ¢ and 2 S ¢ of users 1 U and 2 U are the related secret key rates obtained in Section 3.3 and in the Appendix. The corresponding transmission rates are evaluated as follows: where * 2 i s  was shown in (69). Thus for any 1 i u > , without loss of generality, which follows from (168) for i  , since In particular, the ( ) sym P ¢  SVD-assisted symmetric private classical capacity is precisely as where k R is the rate of classical communication between Alice and Bob k , measured by either the ( ) : : is the SVD-assisted rate of Eve, quantified by ( ) The AMQD-MQA secret key rate ratios for users 1 U and 2 U are depicted in Figure 2.  The SVD-assistance leads to an improved ( ) between Alice and Bob. See (165). In particular, this enhancement, overall, results in an increased secret key rates 1 S ¢ and 2 S ¢ for users 1 U and 2 U . ■ These results conclude the proof of Theorem 2. ■

Physical Boundaries of an Optimal Gaussian Attack
Theorem 3 (Tolerable excess noise of AMQD). The AMQD leads to an improved amount of tolerable excess noise in comparison to single-carrier CVQKD. Proof.
As we have seen in Section 3, from the variance 2 s C of the Gaussian link, the excess noise is evaluated as where 2 s  is the vacuum noise. From [1][2][3] for the RR of AMQD, the following inequalities bring up for the , . , RR coh tol AMQD N maximum tolerable excess noise, in a coherent state transmission: . , 1.
Thus, for squeezed states [1][2][3], the tolerable excess noise is precisely as follows: Specifically, the inequality of (219) can be sharpened for the RR AMQD settings as follows. Using coherent states for the transmission, the upper bound on the , tol AMQD N tolerable excess noise in an AMQD modulation in one-way CVQKD, for the case of RR, is as follows.
In particular, the upper bound on the tolerable excess noise, (223) From a security argument linked to the entangling cloning attack [1], the following upper bound can be derived for the maximal tolerable excess noise for the DR of AMQD: Without loss of generality, the inequality of can be sharpened for the DR AMQD settings as follows [3]: The result in follows from the following equation [3]: thus, , , for some 0 Ã > .
In particular, for two-way CVQKD, these upper bounds change as follows: Without loss of generality, the improvement in the tolerable excess noise can be approached by the ratio 1 k ³ between the excess noise ( ) ( ) In a precise form, the maximal amount of tolerable excess noise For the corresponding rate formulas, see Section 3.
The results are summarized in Figure 3. The tolerable excess noise is where ( ) ( ) where 1 M > . In particular, the improvement in the amount of tolerable excess noise is present more significantly in two-way CVQKD, since the multicarrier-based transmission enhances the rates of private classical communication for each channel uses,

Conclusions
The multicarrier CVQKD extends the possibilities of current, standard single-carrier CVQKD. We studied the security thresholds of multicarrier CVQKD, through AMQD modulation, and its extended SVD-assisted version. The results have also been extended to a multiuser environment, through the analysis of AMQD-MQA. We proved the secret key rate formulas of multicarrier CVQKD and determined the security parameters of the transmission. We investigated the private classical capacity of a Gaussian sub-channel and characterized the private classical capacity domain. The physical boundaries of optimal Gaussian attacks against multicarrier transmission have also been revealed, along with the maximal tolerable variance parameters. The results revealed that multicarrier CVQKD provides a considerably improved framework for the experimental implementation of an unconditionally secure communication over standard telecommunication networks.

S.1 Homodyne and Heterodyne Measurements
In a homodyne measurement hom M , only one quadrature, the position or the momentum quadrature is measured. The variances of Alice's quadratures are For these quantities, the following inequalities hold

S.2 AMQD Secret Key Rates in Two-Way CVQKD
In case of the two-way CVQKD, Bob starts the communication by sending via AMQD modulation a quantum system to Alice, who adds her internal secret to the received system, which is then sent back via AMQD modulation to Bob. We use the formalisms of [2] throughout to derive the security parameters.
The covariance matrices (S.25)-(S.27), the symplectic spectra for the single-carrier Gaussians will be constructed next.

S.2.1 Homodyne Measurement, Reverse Reconciliation
Without loss of generality, for Eve's system, j E , the symplectic spectra j E  is precisely as follows: In particular, in case of reverse reconciliation, the symplectic spectra

S.3 Notations
The notations of the manuscript are summarized in Table S.1.      ( ) sum S ¢  The sum secret key rate of K users in an SVD-assisted multicarrier CVQKD setting.

MQA sym
S ¢ The symmetric secret key rate of K users in an SVDassisted multicarrier CVQKD setting.    The  convex hull of independent input distributions.