Quantum key distribution (QKD) protocols are intended to allow cryptographic keys to be generated and distributed in way that is provably secure based on inherent limitations, such as the no-cloning principle, imposed by quantum mechanics. This unique advantage compared with classical cryptography comes with an added difficulty: key bits in QKD protocols are encoded in analogue quantum states and their preparation is consequently subject to the usual imprecisions inevitable in any real world experiment. The negative impact of such imprecisions is illustrated for the BB84 QKD protocol. Following this, the main part of this thesis is concerned with the incorporation of such imprecisions in security proofs of the BB84 and two semi-device-independent protocols against the class of collective attacks. On a technical level, by contrast with the vast majority of security proofs developed since the turn of the century, in which recasting the protocol into an equivalent entanglement-based form features heavily in the analysis, the main results obtained here are approached directly from the prepare-and-measure perspective and in particular the connection with the no-cloning theorem and an early security proof by Fuchs et al. against the class of individual attacks is emphasised. This thesis also summarises, as an appendix, a separate project which introduces and defines a hierarchy of polytopes intermediate between the local and no-signalling polytopes from the field of Bell nonlocality.
Quantum key distribution (QKD) protocols are intended to allow cryptographic keys to be generated and distributed in way that is provably secure based on inherent limitations, such as the no-cloning principle, imposed by quantum mechanics. This unique advantage compared with classical cryptography comes with an added difficulty: key bits in QKD protocols are encoded in analogue quantum states and their preparation is consequently subject to the usual imprecisions inevitable in any real world experiment. The negative impact of such imprecisions is illustrated for the BB84 QKD protocol. Following this, the main part of this thesis is concerned with the incorporation of such imprecisions in security proofs of the BB84 and two semi-device-independent protocols against the class of collective attacks. On a technical level, by contrast with the vast majority of security proofs developed since the turn of the century, in which recasting the protocol into an equivalent entanglement-based form features heavily in the analysis, the main results obtained here are approached directly from the prepare-and-measure perspective and in particular the connection with the no-cloning theorem and an early security proof by Fuchs et al. against the class of individual attacks is emphasised. This thesis also summarises, as an appendix, a separate project which introduces and defines a hierarchy of polytopes intermediate between the local and no-signalling polytopes from the field of Bell nonlocality.
In this thesis we develop practical tools for quantum key distribution (QKD) security proofs. We apply the tools to provide security proofs for several protocols, ranging from discrete variable protocols in high dimensions, protocols with realistic implementations, measurement device independent QKD and continuous-variable QKD. The security proofs are based on the Devetak-Winter security framework [I. Devetak and A. Winter, Proc. of the Roy. Soc. of London Series A, 461, 207 (2005); B. Kraus, N. Gisin, and R. Renner, Phys. Rev. Lett., 95 080501 (2005)]. \n \nIn the key rate calculation, it is often convenient to assume that the optimal attack is symmetric. \nUnder the assumption that the parameter estimation is based on coarse-grained observations, we show that the optimal attack is symmetric, if the protocol and the postselection have sufficient symmetries. \nAs an example we calculate the key rates of protocols using 2, d and d+1 mutually unbiased bases in d-dimensional Hilbert spaces. \n \nWe investigate the connection between the optimal collective eavesdropping attack and the optimal cloning attack, in which the eavesdropper employs an optimal cloner to attack the protocol. \nWe find that, in general, it does not hold that the optimal attack is an optimal cloner. However, there are classes of protocols, for which we can identify the optimal attack by an optimal cloner. \nWe analyze protocols with mutually unbiased bases in d dimensions, and show that for the protocols with 2 and d+1 mutually unbiased bases the optimal attack is an optimal cloner, but for the protocols with d mutually unbiased bases, it is not. \n \nIn optical implementations of the phase-encoded BB84 protocol, the bit information is usually encoded in the phase of two consecutive photon pulses generated in a Mach-Zehnder interferometer. In the actual experimental realization, the loss in the arms of the Mach-Zehnder interferometer is not balanced, for example because only one arm contains a lossy phase modulator. \nSince the imbalance changes the structure of the signals states and measurements, the BB84 security analysis no longer applies in this scenario. \nWe provide a security proof for the unbalanced phase-encoded BB84. The loss does lower the key rate compared to a protocol without loss. However, for a realistic parameter regime, the same key rate is found by applying the original BB84 security analysis. \n \nRecently, the security of a measurement device-independent QKD setup with BB84 signal states was proven in Refs. [H.-K. Lo, M. Curty, and B. Qi, Phys. Rev. Lett., 108 (2012); S. L. Braunstein and S. Pirandola, Phys. Rev. Lett., 108 (2012)]. In this setup Alice and Bob send quantum states to an intermediate node, which performs the measurement, and is assumed to be controlled by Eve. \nWe analyze the security of a measurement device-independent QKD protocol with B92 signal states, and calculate the key rates numerically for a realistic implementation. \nBased on our security proof we were able to prove the security of the strong reference pulse B92 protocol. \n \nWe analyze the security of continuous-variable protocols using the entropic uncertainty relations established in Ref. [M. Berta, M. Christandl, R. Colbeck, J. M. Renes, and R. Renner, Nature Physics 6, 659 (2010)] to provide an estimate of the key rate based on the observed first and second moments. We analyze a protocol with squeezed coherent states and the 2-state protocol with two coherent states with opposite phases.
This thesis is concerned with rigorous security analyses of practical Quantum Key Distribution (QKD) protocols, using a variety of modern proof techniques. The main results are as follows. First, we establish a security proof for variable-length QKD protocols against IID collective attacks, and extend this result to coherent attacks using the postselection technique. In doing so, we resolve a long-standing flaw in the application of the postselection technique to QKD, thereby placing it on a rigorous mathematical footing. Second, we develop a method to bound phase error rates in entropic uncertainty relation-based and phase error rate-based proofs, using only the observed statistics of the protocol, even when detectors are imperfect and only approximately characterized. This removes a key assumption of identical detector behaviour and enables these techniques to be applied in realistic settings. Third, we present a very general security analysis based on the marginal-constrained entropy accumulation theorem. The resulting framework can be readily adapted to practical imperfections and side channels, and is suitable for certification efforts. Finally, we show that the security of QKD protocols under realistic authentication assumptions can be reduced to the standard idealized setting, where authentication is assumed to behave honestly, with only minor protocol modifications. A distinctive feature of this thesis is its unified presentation of several major QKD security proof frameworks using consistent protocol descriptions and notation. Consequently, this thesis is intended not only as a collection of new technical results, but also as a useful reference for understanding rigorous security analysis in quantum key distribution.
We introduce a quantum cloning bound which we apply to a straightforward and relatively direct security proof of the prepare-and-measure Bennett-Brassard 1984 (BB84) quantum key distribution (QKD) protocol against collective attacks. The approach we propose is able to handle the practical problem of source and detector alignment imprecisions in a simple way. Specifically, we derive a key-rate bound for a BB84 implementation in which Alice's source emits four given but arbitrary pure states, where the usual equivalence between prepare-and-measure and entanglement-based QKD no longer applies. Our result is similar to a key rate derived by Mar\o{}y et al. [Phys. Rev. A 82, 032337 (2010)] and generally an improvement over the key rate derivable from the entropic uncertainty relation in situations where it applies. We also provide a stronger result for a source emitting arbitrary qubit states, and a further improved result if the detector is additionally assumed two dimensional.
Every quantum key distribution (QKD) protocol must limit the information available outside the users' laboratories. In device-independent QKD this is essentially the only assumption made on the devices. In a prepare-and-measure setting it cannot be imposed strictly, since the sender's device must emit a signal that depends on her private input. The natural replacement, and a minimal one, is a leakage bound: a bound on how much information about the input leaves the sender's laboratory, through the intended signal or through any side channel. We ask how far this unavoidable requirement can replace a detailed model of the source. We show that this single bound, quantified by the probability of guessing the input from everything the source emits, suffices to prove security against collective attacks. In our analysis, nothing is assumed about the precise form of the emitted states, their dimension, energy or purity, nor about the receiver's measurement device, beyond the standard requirement that it does not leak Bob's data. The standard B92 and BB84 protocols do not certify any key under this minimal leakage bound assumption. But closely related protocols can and, in the ideal limit, achieve one secret bit per key round. For noisy implementations, we compute key rates using semidefinite programming relaxations. Our best protocols tolerate visibilities down to 0.84, corresponding to an 8% quantum bit error rate, compared with the 11% threshold of standard device-dependent BB84.
By testing nonlocality, the security of entanglement-based quantum key distribution (QKD) can be enhanced to being ``device-independent.'' Here we ask whether such a strong form of security could also be established for one-way (prepare and measure) QKD. While fully device-independent security is impossible, we show that security can be guaranteed against individual attacks in a semi-device-independent scenario. In the latter, the devices used by the trusted parties are noncharacterized, but the dimensionality of the quantum systems used in the protocol is assumed to be bounded. Our security proof relies on the analogies between one-way QKD, dimension witnesses, and random-access codes.
This chapter introduces Quantum Key Distribution (QKD), a key cryptographic technology that employs quantum mechanics to enhance communication security. Central to QKD is its use of quantum principles such as superposition, entanglement, and the no-cloning theorem, which collectively facilitate the generation of secure cryptographic keys. These keys are secure against both traditional cryptographic attacks and those posed by emerging quantum computing technologies. The chapter begins with a detailed introduction to the core concepts of QKD, explaining how it fundamentally differs from classical cryptographic methods by ensuring security based on the laws of physics rather than computational complexity. It traces the historical evolution of QKD, from early theoretical formulations to its current implementations and the development of key protocols like the BB84, which marked a significant milestone in practical QKD applications. As the discussion progresses, the chapter explores various sophisticated QKD protocols, including E91, decoy-state, and Measurement Device Independent (MDI) protocols. Each protocol is analysed for its operational mechanics, security features, and the specific challenges it addresses in protecting data against quantum threats. The technical complexities and requirements for implementing these protocols in real-world scenarios are also examined, showing their potential and limitations for extending secure communication networks. Moreover, the chapter evaluates the practical challenges associated with QKD, such as managing transmission distances, environmental susceptibilities, and the infrastructural demands of deploying quantum networks. It concludes by highlighting the future directions of QKD, underscoring its growing integration into critical global communication infrastructures, and its significant potential to revolutionize data security in the quantum age. This chapter presents the significance of QKD in modern cryptography, demonstrating its evolution from a conceptual framework to a fundamental technology set to counteract the challenges of quantum decryption. Through this approach, the chapter demonstrates the critical role of QKD in fortifying the foundations of secure communications worldwide.