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.
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.
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 allows for proving the security of a shared cryptographic key between two distant parties with potentially untrusted devices. The security proof is based on the measurement outcome statistics (correlation) of a Bell experiment, and security is guaranteed by the laws of quantum theory. While it is known that the observed correlation must be Bell nonlocal in order to prove security, recent results show that Bell nonlocality is in general not sufficient for standard device-independent quantum key distribution. In this work, we show that conversely, there is no lower bound on the amount of nonlocality that is sufficient for device-independent quantum key distribution. Even more so, we show that from certain correlations that exhibit arbitrarily small nonlocality, one can still extract unbounded device-independent key rates. Therefore, a quantitative relation between device-independent key rates and Bell nonlocality cannot be drawn in general. Our main technique comprises a rigorous connection between self-testing and device-independent quantum key distribution, applied to a recently discovered family of Bell inequalities with arbitrarily many measurement outcomes.
Two parties sharing entangled quantum systems can generate correlations that cannot be produced using only shared classical resources. These nonlocal correlations are a fundamental feature of quantum theory but also have practical applications. For instance, they can be used for device-independent random number generation, whose security is certified independently of the operations performed inside the devices. The amount of certifiable randomness that can be generated from some given nonlocal correlations is a key quantity of interest. Here, we derive tight analytic bounds on the maximum certifiable randomness as a function of the nonlocality as expressed using the Clauser-Horne-Shimony-Holt (CHSH) value. We show that for every CHSH value greater than the local value (2) and up to 3sqrt[3]/2≈2.598 there exist quantum correlations with that CHSH value that certify a maximal two bits of global randomness. Beyond this CHSH value the maximum certifiable randomness drops. We give a second family of Bell inequalities for CHSH values above 3sqrt[3]/2, and show that they certify the maximum possible randomness for the given CHSH value. Our work hence provides an achievable upper bound on the amount of randomness that can be certified for any CHSH value. We illustrate the robustness of our results, and how they could be used to improve randomness generation rates in practice, using a Werner state noise model.
Given a pair of isolated devices that accept random binary inputs and return binary outputs, a user can deduce from the observed data alone if the underlying mechanism can be explained classically. Bell's theorem further states that a classical explanation can be ruled out if the devices perform certain measurements on an entangled quantum state, underpinning the security of cryptographic protocols that are device-independent (DI). For certain protocols, such as those performed in a tight space, it might be difficult to perfectly enforce the non-signaling assumption required in Bell's theorem. This prompts the question: is quantum nonlocality robust to such imperfections? We show that if a binary channel sends a noisy copy of one party's input to the other before any measurements take place, the answer is yes. We completely characterize the vertices and facets of the local polytope in this scenario, and identify Bell inequalities that certify non-signaling quantum correlations. This is possible even when a near perfect copy of the input is sent. We go on to show that the identified inequalities are more robust to depolarizing noise than the Clauser-Horne-Shimony-Holt inequality when certifying DI randomness in this scenario. In addition, we characterize the local polytope when both parties receive a noisy copy of each other's input and make similar conclusions, leaving many new potential Bell inequalities to be explored.
Abstract One of the most striking features of quantum theory is that it allows spatially separated observers to share correlations that resist local-causal (classical) explanations, a phenomenon referred to as Bell nonlocality[1, 2]. Besides their foundational relevance, the nonlocal correlations enable distant observers to accomplish classically inconceivable information processing and cryptographic feats such as unconditionally secure device-independent key distribution schemes[3-5]. However, the distances over which nonlocal correlations can be realized in state-of-the-art Bell experiments remain severely limited owing to the high threshold efficiencies of the detectors and the fragility of the nonlocal correlations to experimental noise[6, 7]. Instead of looking for quantum strategies with marginally lower threshold requirements, we exploit the properties of loophole-free nonlocal correlations, which are experimentally attainable today, albeit at short distances, to extend them over arbitrarily large distances. Specifically, we consider Bell experiments wherein the spatially separated parties randomly choose the location of their measurement devices in addition to their measurement settings. We demonstrate that when devices close to the entanglement source are perfect and witness extremal loophole-free nonlocal correlations, such correlations can be extended to devices placed arbitrarily far from the source, with almost zero detection efficiency and visibility. To accommodate imperfections close to the source, we demonstrate a specific analytical tradeoff: the higher the loophole-free nonlocality close to the source, the lower the threshold requirements away from the source. We utilize this analytical tradeoff paired with optimal quantum strategies to estimate the critical requirements of a measurement device placed away from the source in the simplest non-trivial scenario and formulate a versatile numerical method applicable to generic network scenarios. Our results demonstrate that the properties of already feasible short-distance loophole-free nonlocal correlations can be exploited to extend them to longer distances, enabling device-independent information processing and cryptography fuelled by long-distance loophole-free nonlocal correlations to become near-term commonplace technologies.
The question of “non-locality of a single photon'', which started with a paper by Tan, Walls and Collett (TWC, 1991) stirred a thirty years long debate. This hampered attempts to use the TWC interferometric scheme in quantum cryptography. The scheme involves a single photon 50-50 beam-split into two modes propagating to two spatially separated observation stations at which weak homodyne measurements are made. The physics and non-classicality of such an arrangement has been understood only recently, and points out that an unquestionable Bell non-classicality, as was suggested by Hardy (1994), can be observed when the local measurement settings differ by the weak local oscillator being on or off, and additionally the homodyning for the on case is not balanced. Based on that, we present a single-photon based device-independent quantum key distribution scheme secure even against no-signaling eavesdropping. In our protocol the random bits of the cryptographic key are obtained by measurements on the single photon, that is for off settings at both Alice and Bob sides, while the security is positively tested if for eavesdropping testing runs one observes a violation of a specific Bell inequality involving the on and off weak homodyne measurements as alternative local settings. The security analysis presented here is based on a decomposition of the correlations into extreme points of a no-signaling polytope, which allows for identification of the optimal strategy for any eavesdropping constrained only by the no-signaling principle. For this strategy, the key rate is calculated, which is then connected with the violation of a specific Clauser-Horne inequality. We also adapt this analysis to propose a self-testing quantum random number generator based on the old idea that employs the randomness of reflection and transmission events of a quantum light impinged on a 50-50 beamsplitter.