Nonlocal games with synchronous correlations are a natural generalization of functions between two finite sets. In this work we examine analogues of Bell's inequalities for such correlations, and derive a synchronous device-independent quantum key distribution protocol. This protocol has the advantage of symmetry between the two users and self-testing while generating shared secret key without requiring a preshared secret. We show that, unlike general correlations and the CHSH inequality, there can be no quantum Bell violation among synchronous correlations with two measurement settings. However we exhibit explicit analogues of Bell's inequalities for synchronous correlations with three measurement settings and two outputs, provide an analogue of Tsirl'son's bound in this setting, and prove existence and rigidity of quantum correlations that saturate this bound. We conclude by posing a security assumption that bypasses the locality, or causality, loophole and examine the protocol's robustness against measurement error and depolarization noise.
Nonlocal games with synchronous correlations are a natural generalization of functions between two finite sets. In this work we examine analogues of Bell's inequalities for such correlations, and derive a synchronous device-independent quantum key distribution protocol. This protocol has the advantage of symmetry between the two users and self-testing while generating shared secret key without requiring a preshared secret. We show that, unlike general correlations and the CHSH inequality, there can be no quantum Bell violation among synchronous correlations with two measurement settings. However we exhibit explicit analogues of Bell's inequalities for synchronous correlations with three measurement settings and two outputs, provide an analogue of Tsirl'son's bound in this setting, and prove existence and rigidity of quantum correlations that saturate this bound. We conclude by posing a security assumption that bypasses the locality, or causality, loophole and examine the protocol's robustness against measurement error and depolarization noise.
We derive a device-independent quantum key distribution protocol based on synchronous correlations and their Bell inequalities. This protocol offers several advantages over other device-independent schemes including symmetry between the two users and no need for pre-shared randomness. We close a "synchronicity" loophole by showing that an almost synchronous correlation inherits the self-testing property of the associated synchronous correlation. We also pose a new security assumption that closes the "locality" (or "causality") loophole: an unbounded adversary with even a small uncertainty about the users' choice of measurement bases cannot produce any almost synchronous correlation that approximately maximally violates a synchronous Bell inequality.
Device-independent quantum key distribution allows two users to set up shared cryptographic key without the need to trust the quantum devices used. Doing so requires nonlocal correlations between the users. However, in Farkas et al. [Phys. Rev. Lett. 127, 050503 (2021)PRLTAO0031-900710.1103/PhysRevLett.127.050503] it was shown that for known protocols nonlocality is not always sufficient, leading to the question of whether there is a fundamental lower bound on the minimum amount of nonlocality needed for any device-independent quantum key distribution implementation. Here, we show that no such bound exists, giving schemes that achieve key with correlations arbitrarily close to the local set. Furthermore, some of our constructions achieve the maximum of 1 bit of key per pair of entangled qubits. We achieve this by studying a family of Bell inequalities that constitute all self-tests of the maximally entangled state with a single linear Bell expression. Within this family there exist nonlocal correlations with the property that one pair of inputs yield outputs arbitrarily close to perfect key. Such correlations exist for a range of Clauser-Horne-Shimony-Holt values, including those arbitrarily close to the classical bound. Finally, we show the existence of quantum correlations that can generate both perfect key and perfect randomness simultaneously, while also displaying arbitrarily small Clauser-Horne-Shimony-Holt violation. This opens up the possibility of a new class of cryptographic protocol.
Bell nonlocality is the experimentally testable incompatibility between quantum correlations and explanations based on local hidden variables. Its significance now extends beyond the foundations of quantum mechanics: a Bell-inequality violation can certify entanglement, randomness, measurement incompatibility, and cryptographic security without requiring a detailed model of the devices producing the observed data. This article develops the mathematical structure of Bell experiments and explains its role in device-independent quantum information. Conditional probability distributions are classified into local, quantum, and no-signalling sets; the Clauser-Horne-Shimony-Holt inequality is derived, and its classical, quantum, and algebraic limits are distinguished. Tsirelson’s bound is obtained from the CHSH operator, while Popescu-Rohrlich correlations demonstrate that no-signalling alone does not identify the quantum set. The relation between entanglement and nonlocality is examined through local entangled states, hidden nonlocality, and activation. Nonlocal games, semidefinite relaxations, self-testing, certified randomness, and device-independent quantum key distribution are treated as operational consequences of Bell violation. Particular attention is given to entropy accumulation, finite-data security, loophole-free experiments, and the limitations of device independence. Bell nonlocality emerges as both a foundational separation between classical and quantum causal models and a quantitative resource for cryptographic and informational tasks.
Quantum entanglement is a unique property of quantum particles. It can correlate them in such a way, that further correlations to an additional unwanted party is prevented. This renders quantum theory as the prime candidate to implement a secure distribution of an encryption key. Quantum key distribution (QKD) is dedicated to this task. A striking feature of correlations in a statistical experiment is, that they can, in principle, always be approximated. Even more so, correlations can be obtained without a detailed knowledge about the underlying physical process that generated the experimental data. This no-characterization approach is at the heart of the device-independent (DI) paradigm. Here, Bell inequalities are an imperative tool to detect nonlocal correlations, which are necessary for a DI secret key. One central subject of this thesis is multipartite DIQKD and the implications of Bell nonlocality in this context. Any DIQKD protocol involves test rounds in which the violation of a Bell inequality is checked. For multipartite DIQKD, however, not all Bell inequalities are suitable. We identify the crucial properties a Bell test requires to be a viable option. In this light, a published protocol for multipartite DIQKD is examined. We establish an incompatibility which is inherent to the proposed Bell test in combination with the required quantum states. This leads us to the conclusion, that the proposed protocol necessarily aborts. In a subsequent work, we develop a family of multipartite Bell inequalities, specifically tailored to the task of DI conference key agreement (CKA). Several features of this Bell inequality are analytically characterized. In addition, we prove its usefulness for the purposes of DICKA. To this end, semidefinite programming techniques are employed and extended to the multipartite scenario which allows us to quantify asymptotic DI conference key rates. In a second part, we explore the challenges of implementing (bipartite) long-distance DIQKD with quantum repeaters. For two repeater protocols, we describe how experimental parameters manifest themselves in DI secret-key rates. In doing so, we shed light on the fundamental differences between the usual and the DI scenario and we benchmark the threshold requirements of quantum devices to make profitable DI secret-key rates feasible. Finally, we develop a general method to describe in a non-DI setting the propagation of an important class of errors through quantum circuits in arbitrary dimensions. With this, we discuss the potential of error-corrected quantum repeaters to overcome fundamental point-to-point limitations.
Recently, Wooltorton et al. [Phys. Rev. Lett. 132, 210802 (2024)] and Farkas [Phys. Rev. Lett. 132, 210803 (2024)] have exhibited the mismatch between Bell inequality violations and their cryptographic application in device-independent quantum key distribution, by showing that arbitrarily close to the set of local behaviours there exist quantum correlations guaranteeing a constant rate of secret key. While these results require correlations attaining the maximum quantum value of a suitable Bell observable (aka the Tsirelson bound) and rely on a kind of ideal self-testing of a maximally entangled state and associated Bell measurement, here we show that the effect is robust: for every one of the Bell inequalities considered by Wooltorton \emph{et al.}, a constant rate of secret key ensues if the observed Bell violation is sufficiently close to the respective Tsirelson bound. For these and also the Bell inequalities of Farkas, we furthermore present numerical results based on semidefinite relaxations of the minimum min-entropy consistent with a certain Bell violation, which demonstrate that small but nonzero key rates can be guaranteed (in principle) by practical and efficient means.
Device-independent quantum key distribution is a secure quantum cryptographic paradigm that allows two honest users to establish a secret key, while putting minimal trust in their devices. Most of the existing protocols have the following structure: first, a bipartite nonlocal quantum state is distributed between the honest users, who perform local projective measurements to establish nonlocal correlations. Then, they announce the implemented measurements and extract a secure key by postprocessing their measurement outcomes. We show that no protocol of this form allows for establishing a secret key when implemented on any correlation obtained by measuring local projective measurements on certain entangled nonlocal states, namely, on a range of entangled two-qubit Werner states. To prove this result, we introduce a technique for upper bounding the asymptotic key rate of device-independent quantum key distribution protocols, based on a simple eavesdropping attack. Our results imply that either different reconciliation techniques are needed for device-independent quantum key distribution in the large-noise regime, or Bell nonlocality is not sufficient for this task.