Secret key rate proof of multicarrier continuous‐variable quantum key distribution

Abstract

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 continuous variables (CVs). The multicarrier communication formulates Gaussian subchannels 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 adaptive multicarrier quadrature division (AMQD) scheme, the singular value decomposition (SVD)-assisted AMQD, and the multiuser AMQD multiuser quadrature allocation (MQA). We prove that the multicarrier CVQKD leads to improved secret key rates and higher tolerable excess noise in comparison with single-carrier CVQKD. We derive the private classical capacity of a Gaussian subchannel 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. INTRODUCTIONContinuous-variable quantum key distribution (CVQKD) allows for legal parties to transmit information with unconditional security over the currently established telecommunication networks. [33][34][35][36][37][38] 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. 29 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 39-50 by the sender, Alice. The quantum channel is attacked by an eavesdropper (Eve), and the receiver (Bob) gets a noisy system. The optimal attack against CVQKD is a Gaussian attack; thus, the noise of the quantum channel can be provably modeled as an additive white Gaussian noise. 1-3,7-13,16-19 The security of CVQKD has been already proven against several of the most Int J Commun Syst. 2019;32:e3865.wileyonlinelibrary.com/journal/dac powerful optimal Gaussian collective attacks, [8][9][10][11][12][13][16][17][18][19] 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 compari...

Diversity space of multicarrier continuous‐variable quantum key distribution

“…The detailed description of AMQD and AMQD‐MQA can be found in Gyongyosi and Imre . For the secret key rate formulas, see Gyongyosi and Imre …”

“…The complete derivation of the secret key rate formulas can be found in Gyongyosi and Imre; here, we give a brief overview on the transmission rates of multicarrier CVQKD.…”

“…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 subchannels. The secret key rates and security thresholds of multicarrier transmission have been proven in Gyongyosi and Imre, leading to enhanced secret key rates in both one‐ and two‐way CVQKD . For further information on the bounds of private quantum communications, we suggest Pirandola et al 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.…”

“…Particularly, the AMQD divides the physical Gaussian channel into several Gaussian subchannels; each subchannel 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 . Specifically, the benefits of AMQD can be extended by SVD‐assistance (singular value decomposition) through the SVD‐assisted AMQD .…”

“…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 …”