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Theoretical Computer Science

Multicarrier continuous-variable quantum key distribution

László Gyöngyösi

Published Nov 27, 2019• 45 citations
Publisher

Abstract

The multicarrier continuous-variable quantum key distribution (CVQKD) protocol is defined. In a CVQKD protocol, the information is conveyed by coherent quantum states. The quantum continuous variables are sent through a noisy quantum channel. For a quantum channel with additive-multiplicative noise both additive and multiplicative disturbances are present in the transmission. The multiplicative disturbance is an inherent attribute of diverse physical environments. Physical links with additive and multiplicative disturbances can represent a more general approach than purely additive noise links in several practical scenarios. In a standard CVQKD setting, the noise is modeled as an additive white Gaussian noise caused by an eavesdropper (Gaussian quantum link). As a corollary, standard CVQKD protocols are not optimal for arbitrary Gaussian quantum channels if multiplicative disturbances are also present in the physical link. Here, we define the adaptive multicarrier quadrature division (AMQD) modulation technique for CVQKD. The AMQD method is optimal for arbitrary Gaussian quantum channels with arbitrary multiplicative disturbances. The protocol granulates the Gaussian random input into Gaussian subcarrier continuous variables in the encoding phase, which are then decoded by a continuous unitary transformation. The subcarrier coherent variables formulate sub-channels from the physical link which leads to improved transmission efficiency, higher tolerable loss, and excess noise in comparison to standard CVQKD protocols. We also derive the security proof of multicarrier CVQKD at optimal Gaussian attacks in the finite-size and asymptotic regimes.

DOI
10.1016/j.tcs.2019.11.026

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The multicarrier continuous-variable quantum key distribution (CVQKD) protocol is defined. In a CVQKD protocol, the information is conveyed by coherent quantum states. The quantum continuous variables are sent through a noisy quantum channel. For a quantum channel with additive-multiplicative noise both additive and multiplicative disturbances are present in the transmission. The multiplicative disturbance is an inherent attribute of diverse physical environments. Physical links with additive and multiplicative disturbances can represent a more general approach than purely additive noise links in several practical scenarios. In a standard CVQKD setting, the noise is modeled as an additive white Gaussian noise caused by an eavesdropper (Gaussian quantum link). As a corollary, standard CVQKD protocols are not optimal for arbitrary Gaussian quantum channels if multiplicative disturbances are also present in the physical link. Here, we define the adaptive multicarrier quadrature division (AMQD) modulation technique for CVQKD. The AMQD method is optimal for arbitrary Gaussian quantum channels with arbitrary multiplicative disturbances. The protocol granulates the Gaussian random input into Gaussian subcarrier continuous variables in the encoding phase, which are then decoded by a continuous unitary transformation. The subcarrier coherent variables formulate sub-channels from the physical link which leads to improved transmission efficiency, higher tolerable loss, and excess noise in comparison to standard CVQKD protocols. We also derive the security proof of multicarrier CVQKD at optimal Gaussian attacks in the finite-size and asymptotic regimes.
In a continuous-variable quantum key distribution (CVQKD) system, the information is conveyed by coherent state carriers. The quantum continuous variables are sent through a quantum channel, where the presence of the eavesdropper adds a white Gaussian noise to the transmission. The amount of tolerable noise and loss is a crucial point in CVQKD, since it determines the overall performance of the protocol, including the secure key rates and transmission distances. In this work, we propose the adaptive multicarrier quadrature division (AMQD) modulation technique for CVQKD. The method granulates the Gaussian random input into Gaussian subcarrier continuous variables in the encoding phase, which are then decoded by a continuous unitary transformation. The subcarrier coherent variables formulate Gaussian sub-channels from the physical link with strongly diverse transmission capabilities, which leads to significantly improved transmission efficiency, higher tolerable loss, and excess noise. We also investigate a modulation-variance adaption technique within the AMQD scheme, which provides optimal capacity-achieving communication over the sub-channels in the presence of a Gaussian noise.
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.
Abstract The subcarrier domain of multicarrier continuous-variable quantum key distribution (CVQKD) is defined. In a multicarrier CVQKD scheme, the information is granulated into Gaussian subcarrier CVs and the physical Gaussian link is divided into Gaussian sub-channels. The subcarrier domain injects physical attributes to the description of the subcarrier transmission. We prove that the subcarrier domain is a natural representation of the subcarrier-level transmission in a multicarrier CVQKD scheme. We also extend the subcarrier domain to a multiple-access multicarrier CVQKD setting. We demonstrate the results through the adaptive multicarrier quadrature-division (AMQD) CVQKD scheme and the AMQD-MQA (multiuser quadrature allocation) multiple-access multicarrier scheme. The subcarrier domain representation provides a general apparatus that can be utilized for an arbitrary multicarrier CVQKD scenario.
Summary 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.
We propose the Gaussian quadrature inference (GQI) method for multicarrier continuous-variable quantum key distribution (CVQKD). A multicarrier CVQKD protocol utilizes Gaussian subcarrier quantum continuous variables (CV) for information transmission. The GQI framework provides a minimal error estimate of the quadratures of the CV quantum states from the discrete, measured noisy subcarrier variables. GQI utilizes the fundamentals of regularization theory and statistical information processing. We characterize GQI for multicarrier CVQKD, and define a method for the statistical modeling and processing of noisy Gaussian subcarrier quadratures. We demonstrate the results through the adaptive multicarrier quadrature division (AMQD) scheme. We introduce the terms statistical secret key rate and statistical private classical information, which quantities are derived purely by the statistical functions of GQI. We prove the secret key rate formulas for a multiple access multicarrier CVQKD via the AMQD-MQA (multiuser quadrature allocation) scheme. The framework can be established in an arbitrary CVQKD protocol and measurement setting, and are implementable by standard low-complexity statistical functions, which is particularly convenient for an experimental CVQKD scenario.
We introduce the multidimensional manifold extraction for multicarrier continuous-variable (CV) quantum key distribution (QKD). The manifold extraction utilizes the resources that are injected into the transmission by the additional degrees of freedom of the multicarrier modulation. We demonstrate the results through the AMQD (adaptive multicarrier quadrature division) scheme, which granulates the information into Gaussian subcarrier CVs and divides the physical link into several Gaussian sub-channels for the transmission. We prove that the exploitable extra degree of freedom in a multicarrier CVQKD scenario significantly extends the possibilities of single-carrier CVQKD. The manifold extraction allows for the parties to reach decreased error probabilities by utilizing those extra resources of a multicarrier transmission that are not available in a single-carrier CVQKD setting. We define the multidimensional manifold space of multicarrier CVQKD and the optimal tradeoff between the available degrees of freedom of the multicarrier transmission. We also extend the manifold extraction for the multiple-access 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.