The squashing model proof technique is a powerful tool in proving security of quantum key distribution (QKD) protocols since it reduces the dimension of the associated computational problem from infinite to finite. In the first half of this thesis, we formally prove that the unconditional asymptotic security of an arbitrary QKD protocol after applying a squashing model implies the same security for the original protocol. We also prove rigorously that the squashing map of the flag-state squashing model is indeed a quantum channel for an infinite-dimensional input Hilbert space. \n \nIn the second half of the thesis, we apply the flag-state squashing model to the phase-encoded BB84 protocol. Since all phase-encoded BB84 implementations have signal states with unbalanced amplitudes in practice, the original security analyses a priori do not apply to them. Previous security proofs use signal tagging of multi-photon pulses to recover the behaviour of regular BB84. This is overly conservative, as for unbalanced signals, the photon-number splitting attack does not leak full information to Eve. In this work, we exploit the flag-state squashing model to preserve some parts of the multi-photon generated private information in our analysis. Using a numerical proof technique, we obtain significantly higher key rates compared with previously published results in the low-loss regime. It turns out that the usual scenario of untrusted dark counts runs into conceptual difficulties in some parameter regime. Thus, we discuss the trusted dark count scenario in this paper as well. We also report a gain in key rates when part of the total loss is known to be induced by a trusted device. We highlight that all these key rate improvements can be achieved without modification of the experimental setup.
The squashing model proof technique is a powerful tool in proving security of quantum key distribution (QKD) protocols since it reduces the dimension of the associated computational problem from infinite to finite. In the first half of this thesis, we formally prove that the unconditional asymptotic security of an arbitrary QKD protocol after applying a squashing model implies the same security for the original protocol. We also prove rigorously that the squashing map of the flag-state squashing model is indeed a quantum channel for an infinite-dimensional input Hilbert space. \n \nIn the second half of the thesis, we apply the flag-state squashing model to the phase-encoded BB84 protocol. Since all phase-encoded BB84 implementations have signal states with unbalanced amplitudes in practice, the original security analyses a priori do not apply to them. Previous security proofs use signal tagging of multi-photon pulses to recover the behaviour of regular BB84. This is overly conservative, as for unbalanced signals, the photon-number splitting attack does not leak full information to Eve. In this work, we exploit the flag-state squashing model to preserve some parts of the multi-photon generated private information in our analysis. Using a numerical proof technique, we obtain significantly higher key rates compared with previously published results in the low-loss regime. It turns out that the usual scenario of untrusted dark counts runs into conceptual difficulties in some parameter regime. Thus, we discuss the trusted dark count scenario in this paper as well. We also report a gain in key rates when part of the total loss is known to be induced by a trusted device. We highlight that all these key rate improvements can be achieved without modification of the experimental setup.
All phase-encoded BB84 implementations have signal states with unbalanced amplitudes in practice.Thus the original security analyses a priori do not apply to them.Previous security proofs use signal tagging of multiphoton pulses to recover the behavior of regular BB84.This is overly conservative as for unbalanced signals the photon number splitting attack does not leak full information to Eve.In this work we exploit the flag-state squashing model to preserve some parts of the multiphoton-generated private information in our analysis.Using a numerical proof technique we obtain significantly higher key rates compared with previously published results in the low-loss regime.It turns out that the usual scenario of untrusted dark counts runs into conceptual difficulties in some parameter regimes.Thus we discuss the trusted dark-count scenario in this paper as well.We also report a gain in key rates when part of the total loss is known to be induced by a trusted device.We highlight that all these key rate improvements can be achieved without modification of the experimental setup.
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.
Satellite based quantum key distribution (QKD) aims to establish secure key exchange over long distances despite significant technological challenges. To alleviate some of these challenges, Ghalaii et al. [PRX Quantum 4, 040320 (2023)] proposed that any airborne eavesdropper up to a certain size can be detected by classical monitoring techniques, limiting the transmission efficiencies of any undetected Eve. This creates a new QKD scenario in which some of the transmitted signal from Alice to Bob bypasses Eve entirely. In this manuscript, we develop a general framework for computing key rates in this "bypass" scenario for discrete variable protocols. We first numerically support a conjecture that the performance of BB84 with single photons does not improve under bypass constraints, and go on to find new regimes that do. Specifically, we find improvements when the receiver's detectors have an efficiency mismatch and when BB84 is implemented using weak coherent pulses under certain squashing assumptions. Technically, our framework is realized by including marginal constraints on the source to account for bypass effects, combined with existing numerical approaches for minimizing the key rate and squashing and dimension reduction techniques to handle photonic states of unbounded dimension.
Quantum key distribution (QKD) can be proved to be secure by laws of quantum mechanics. In this thesis, we review security proof methods in Renner's framework and discuss numerical methods to calculate asymptotic and finite key rates. These methods are highly versatile and applicable to general device-dependent QKD protocols. We also discuss analytical tools that extend the applicability of these numerical methods. We then present the asymptotic security proof against collective attacks for a variant of the twin-field QKD protocol, which can overcome the repeaterless secret-key capacity bound. Our variant reduces the sifting cost and uses non-phase-randomized coherent states as both signals and test states. We confirm the loss scaling of this protocol. Another important family of protocols that we investigate here are discrete-modulated continuous-variable QKD protocols. They are interesting due to their experimental simplicity and their great potential for massive deployment in the quantum-secured networks. Our security proof method can provide tight asymptotic key rates. We demonstrate that the postselection of data in combination with reverse reconciliation can improve the key rates. We analyze both untrusted and trusted detector noise scenarios. Our results in the trusted detector noise scenario show that we can thus cut out most of the effect of detector noise and obtain asymptotic key rates similar to those had we access to ideal detectors. Finally, we present several simple examples to illustrate our newly developed method for the numerical finite-key analysis against the most general attacks via the entropy accumulation theorem.
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.
The Bennett-Brassard 1984 (BB84) protocol is the most widely implemented quantum key distribution (QKD) scheme. However, despite enormous theoretical and experimental efforts in the past decades, the security of this protocol with imperfect sources has not yet been rigorously established. In this paper, we address this shortcoming and prove the security of the BB84 protocol in the presence of multiple source imperfections, including state preparation flaws and side channels, such as Trojan-horse attacks, mode dependencies and classical correlations between the emitted pulses. To do so, we consider a modified BB84 protocol that exploits the basis mismatched events, which are often discarded in standard security analyses of this scheme; and employ the reference technique, a powerful mathematical tool to accommodate source imperfections in the security analysis of QKD. Moreover, we compare the achievable secret-key rate of the modified BB84 protocol with that of the three-state loss-tolerant protocol, and show that the addition of a fourth state, while redundant in ideal conditions, significantly improves the estimation of the leaked information in the presence of source imperfections, resulting in a better performance. This paper demonstrates the relevance of the BB84 protocol in guaranteeing implementation security, taking us a step further towards closing the existing gap between theory and practice of QKD.