This thesis deals with a detailed study of the unconditional security of Continuous-Variable (CV) Quantum Key Distribution (QKD). We consider general communication architectures based on both point-to-point and end-to-end principle. For the first case we develop an extensive analysis of the unconditional security of both one-way and two-way protocol under several eavesdropping conditions. We describe an effective post-processing strategy, to apply to one-way communications, that allows to neutralise general coherent attacks. This result motivates us to formulate the conjecture that the "de Finettization" of the classical data, typically adopted in the asymptotic limit of many signals exchanged in order to reduce the attacks from a general coherent to a collective one, may not be necessary for Gaussian one-way communication. For what it concerns two-way protocols we show that after the reduction of the general attack using the de Finetti symmetrization, two-mode coherent attacks are the optimal eavesdropping. Our cryptanalysis shows that the parties can exploit a strategy, in combination with the results obtained on the security of one-way protocols against two-mode coherent attacks. In fact we prove explicitly that two-way communication with Gaussian continuous variables is immune to general coherent attacks. This is possible exploiting the random opening and closing of the circuit by Alice (ON/OFF switching), in the limit of many uses of the channel. We then conclude that collective attacks represent, strictly, the best strategy available to Eve. We have then studied the general security of one-way protocol considering not just lossy channel, but extending the cryptanalysis to all physically allowed canonical forms attacks. Finally we extended the study of thermal CV-QKD to two-way communication at different frequencies, in the framework of optimal collective attacks. In the last part of this thesis we focused on CV-QKD considering a modern end-to-end configuration. We then extend CV quantum cryptography to a network configuration. We develop a detailed cryptoanalysis of a communication scheme based on an untrusted relay, assisting the parties during the preparation of the secret key. We find the optimal eavesdropping and prove, both theoretically and experimentally, the feasibility of high-rate CV-QKD over metropolitan distances with of-the-shelves devices.
This thesis deals with a detailed study of the unconditional security of Continuous-Variable (CV) Quantum Key Distribution (QKD). We consider general communication architectures based on both point-to-point and end-to-end principle. For the first case we develop an extensive analysis of the unconditional security of both one-way and two-way protocol under several eavesdropping conditions. We describe an effective post-processing strategy, to apply to one-way communications, that allows to neutralise general coherent attacks. This result motivates us to formulate the conjecture that the "de Finettization" of the classical data, typically adopted in the asymptotic limit of many signals exchanged in order to reduce the attacks from a general coherent to a collective one, may not be necessary for Gaussian one-way communication. For what it concerns two-way protocols we show that after the reduction of the general attack using the de Finetti symmetrization, two-mode coherent attacks are the optimal eavesdropping. Our cryptanalysis shows that the parties can exploit a strategy, in combination with the results obtained on the security of one-way protocols against two-mode coherent attacks. In fact we prove explicitly that two-way communication with Gaussian continuous variables is immune to general coherent attacks. This is possible exploiting the random opening and closing of the circuit by Alice (ON/OFF switching), in the limit of many uses of the channel. We then conclude that collective attacks represent, strictly, the best strategy available to Eve. We have then studied the general security of one-way protocol considering not just lossy channel, but extending the cryptanalysis to all physically allowed canonical forms attacks. Finally we extended the study of thermal CV-QKD to two-way communication at different frequencies, in the framework of optimal collective attacks. In the last part of this thesis we focused on CV-QKD considering a modern end-to-end configuration. We then extend CV quantum cryptography to a network configuration. We develop a detailed cryptoanalysis of a communication scheme based on an untrusted relay, assisting the parties during the preparation of the secret key. We find the optimal eavesdropping and prove, both theoretically and experimentally, the feasibility of high-rate CV-QKD over metropolitan distances with of-the-shelves devices.
This thesis is concerned with the study and analysis of two quantum cryptographic protocols: quantum key distribution (QKD) and unforgeable quantum money in the continuous-variable (CV) framework. The main advantage of CV protocols is that their implementation only requires standard telecom components. QKD allows two distant parties, Alice and Bob, to establish a secure key, even in the presence of an eavesdropper, Eve. The remarkable property of QKD is that its security can be established in the information-theoretic setting, without appealing to any computational assumptions. Proving the security of CV-QKD protocols is challenging since the protocols are described in an infinite-dimensional Fock space. One of the open questions in CV-QKD was establishing security for two-way QKD protocols against general attacks. We exploit the invariance of Unitary group U(n) of the protocol to establish composable security against general attacks. We answer another pressing question in the field of CV-QKD with a discrete modulation by establishing the asymptotic security of such protocols against collective attacks. We provide a general technique to derive a lower bound on the secret key rate by formulating the problem as a semidefinite program. Quantum money exploits the no-cloning property of quantum mechanics to generate unforgeable tokens, banknotes, and credit cards. We propose a CV private-key quantum money scheme with classical verification. The motivation behind this protocol is to facilitate the process of practical implementation. Previous classical verification money schemes use single-photon detectors for verification, while our protocols use coherent detection.
We consider two-way continuous-variable quantum key distribution, studying its security against general eavesdropping strategies. Assuming the asymptotic limit of many signals exchanged, we prove that two-way Gaussian protocols are immune to coherent attacks. More precisely we show the general superadditivity of the two-way security thresholds, which are proven to be higher than the corresponding one-way counterparts in all cases. We perform the security analysis first reducing the general eavesdropping to a two-mode coherent Gaussian attack, and then showing that the superadditivity is achieved by exploiting the random on/off switching of the two-way quantum communication. This allows the parties to choose the appropriate communication instances to prepare the key, accordingly to the tomography of the quantum channel. The random opening and closing of the circuit represents, in fact, an additional degree of freedom allowing the parties to convert, a posteriori, the two-mode correlations of the eavesdropping into noise. The eavesdropper is assumed to have no access to the on/off switching and, indeed, cannot adapt her attack. We explicitly prove that this mechanism enhances the security performance, no matter if the eavesdropper performs collective or coherent attacks.
Quantum Key Distribution (QKD) provides a method for securely sharing a secret key between two distant parties, Alice and Bob, by leveraging the principles of quantum mechanics. This thesis focuses on Continuous-Variable (CV) QKD, a variant of QKD where information is encoded in the quadrature components of coherent light states, allowing for efficient and cost-effective detection using homodyne or heterodyne receivers. The thesis is divided into three main parts: an introduction to quantum cryptography concepts, the presentation of a CV-QKD protocol, and a detailed security proof of the protocol against different types of attacks. The security proof considers both finite and infinite symbol transmission and tackles various eavesdropping attacks, including individual, collective, and coherent attacks. The use of Gaussian states and operations provides a promising approach to enhance QKD security, particularly in the context of practical quantum cryptography. The final sections provide a mathematical framework for the optimization of secure key rates and strategies for addressing the photon-number cutoff assumption in practical implementations.
Quantum information processing uses the laws of quantum physics to perform tasks that are infeasible or impossible classically. Within this broader programme, quantum communication has emerged as one of the most practically mature sub-fields of quantum information science, with several proposed protocols having already been demonstrated over deployed optical fibre and free-space links. Of the various platforms available for encoding quantum information for communication, the continuous-variable (CV) framework encodes information in the quadratures of the electromagnetic field and is therefore directly compatible with the existing machinery of classical coherent optical communications. The most prominent CV protocol is continuous-variable quantum key distribution (CV-QKD), which establishes an information-theoretically secure cryptographic key between two spatially separated parties. Because CV-QKD reuses much of the hardware of classical coherent telecommunications, including off-the-shelf lasers, modulators, and balanced detectors, it is a leading candidate for the deployment of quantum-secured communication over standard optical fibre networks. The development of CV-QKD has typically proceeded with the underlying physics dictating the engineering. Security proofs, often relying on the Gaussian optimality of collective attacks, set the structure of the protocol, and the practical system is then built to satisfy the assumptions on which those proofs depend. Contributions of this thesis adopt the opposite view, taking the practical engineering of high-rate CV-QKD systems as the starting point and developing the theoretical machinery required to support it. The first contribution is the formalisation of one-round sliced reconciliation with guard-band post-selection as a key distillation scheme designed for real-time, high-throughput operation. Originally introduced experimentally as a means to stabilise bit error rates and enable the use of fixed-rate error correction codes at high symbol rates, guard-band post-selection is shown here to identify the guard-band span as a free parameter that can be tuned to hold the quantum bit error rate approximately constant across a range of channel conditions. However, conditioning on accepted data produces non-Gaussian post-selected statistics that lie outside the reach of standard Gaussian-optimality security arguments, which motivates the remaining two contributions. The second contribution is an asymptotic security analysis of guard-band post-selected CV-QKD against arbitrary collective attacks, no longer restricted to Gaussian adversaries. By using a quantum channel estimate as side information and by lifting reduced-state distances to purification distances via entropic continuity bounds, security is certified against the class of channels lying close to a Gaussian thermal-loss reference, with a key-rate penalty that scales with the energy-constrained diamond distance to that reference. The third contribution is the experimental basis required to apply this framework in practice. A lightweight three-stage protocol for quantum channel estimation is proposed and demonstrated, which repurposes the Gaussian probe constellation already present in the CV-QKD signal and requires no additional hardware or modifications to the underlying protocol. Over single-mode optical fibre at distances from 10 km to 50 km, the protocol produces physically valid process tensor estimates whose energy-constrained diamond distances to a Gaussian reference channel remain on the order of 10^(-3) throughout. Together, these three contributions provide a treatment of post-selected Gaussian-modulated CV-QKD in which the engineering, the security theory, and the channel characterisation are designed to operate together within a deployable system.
We consider the continuous-variable protocol of Pirandola et al. [arXiv:1312.4104] where the secret key is established by the measurement of an untrusted relay. In this network protocol, two authorized parties are connected to an untrusted relay by insecure quantum links. Secret correlations are generated by a continuous-variable Bell detection performed on incoming coherent states. In the present work we provide a detailed study of the symmetric configuration, where the relay is midway between the parties. We analyze symmetric eavesdropping strategies against the quantum links explicitly showing that, at fixed transmissivity and thermal noise, two-mode coherent attacks are optimal, manifestly outperforming one-mode collective attacks based on independent entangling cloners. Such an advantage is shown both in terms of security threshold and secret-key rate.
Establishing the security of continuous-variable quantum key distribution against general attacks in a realistic finite-size regime is an outstanding open problem in the field of theoretical quantum cryptography if we restrict our attention to protocols that rely on the exchange of coherent states. Indeed, techniques based on the uncertainty principle are not known to work for such protocols, and the usual tools based on de Finetti reductions only provide security for unrealistically large block lengths. We address this problem here by considering a new type of Gaussian de Finetti reduction, that exploits the invariance of some continuous-variable protocols under the action of the unitary group U(n) (instead of the symmetric group S_{n} as in usual de Finetti theorems), and by introducing generalized SU(2,2) coherent states. Crucially, combined with an energy test, this allows us to truncate the Hilbert space globally instead as at the single-mode level as in previous approaches that failed to provide security in realistic conditions. Our reduction shows that it is sufficient to prove the security of these protocols against Gaussian collective attacks in order to obtain security against general attacks, thereby confirming rigorously the widely held belief that Gaussian attacks are indeed optimal against such protocols.