Finite rate randomness expansion (RE) which is secure against quantum adversaries is a crucial information processing task which is well studied in the case of two non-communicating parties sharing Bell non-local correlations in a device independent manner. In this paper we show that contextual correlations with only a single localized party are sufficient for RE in a semi-device independent (SDI) manner. We propose a template SDI-RE protocol which is applicable for arbitrary $N$-cycle non-contextuality (NC) inequalities and derive its RE rate under finite statistics. As a special case, we also highlight the advantages of the situation when RE is constrained by Clauser-Horne-Shimony-Holt (CHSH) inequality but with characterized measurements implemented by only a single party. Our template protocol also incorporates non-ideal statistics which may not satisfy perfect commutativity and perfect repeatability, both of which are necessary in a loophole-free test of NC inequalities. As such, several recent experimental implementations of significantly loophole-free tests of NC inequalities can be modified for secure RE following our approach.
Finite rate randomness expansion (RE) which is secure against quantum adversaries is a crucial information processing task which is well studied in the case of two non-communicating parties sharing Bell non-local correlations in a device independent manner. In this paper we show that contextual correlations with only a single localized party are sufficient for RE in a semi-device independent (SDI) manner. We propose a template SDI-RE protocol which is applicable for arbitrary N-cycle non-contextuality (NC) inequalities and derive its RE rate under finite statistics. As a special case, we also highlight the advantages of the situation when RE is constrained by Clauser-Horne-Shimony-Holt (CHSH) inequality but with characterized measurements implemented by only a single party. Our template protocol also incorporates non-ideal statistics which may not satisfy perfect commutativity and perfect repeatability, both of which are necessary in a loophole-free test of NC inequalities. As such, several recent experimental implementations of significantly loophole-free tests of NC inequalities can be modified for secure RE following our approach.
One of the distinguishing features of quantum theory is that its measurement outcomes are usually unpredictable or, equivalently, random. Moreover, this randomness is certifiable with minimal assumptions in the so-called device-independent (DI) paradigm, where a device's behavior does not need to be presupposed but can be verified through the statistics it produces. In this work, we explore various forms of randomness that are certifiable in this setting, where two users can perform two binary-outcome measurements on their shared entangled state. In this case, even though the Clauser-Horne-Shimony-Holt (CHSH) Bell-inequality violation is a pre-requisite for the generation of DI certifiable randomness, the CHSH value alone does not generally give a tight bound on the certifiable randomness. Here, we determine the certifiable randomness when zero-probability constraints are incorporated into the task of DI randomness expansion for the standard local and global randomness and the so-called "blind" randomness. Asymptotically, we observe consistent improvements in the amount of DI certifiable randomness (of all kinds) as we increase the number zero constraints for a wide range of given CHSH Bell violations. However, if we further optimize over the allowed CHSH values, then benefits of these additional constraints over the standard CHSH-based protocol are only found in the case of global and blind randomness. In contrast, in the regimes of finite data, these zero constraints only give a slight improvement in the local randomness rate when compared with all existing protocols.
Semi-device-independent randomness generation protocols based on Kochen-Specker contextuality offer the attractive features of compact devices, high rates, and ease of experimental implementation over fully device-independent (DI) protocols. Here, we investigate this paradigm and derive four results to improve the state of the art. Firstly, we introduce a family of simple, experimentally feasible orthogonality graphs (measurement compatibility structures) for which the maximum violation of the corresponding noncontextuality inequalities allows the certification of the maximum amount of log_{2}d bits of randomness from a qudit system with projective measurements for d≥3. We analytically derive the Lovász theta and fractional packing number for this graph family, and thereby prove their utility for optimal randomness generation in both randomness expansion and amplification tasks. Secondly, a central additional assumption in contextuality-based protocols over fully DI ones is that the measurements are repeatable and satisfy an intended compatibility structure. We frame a relaxation of this condition in terms of ϵ-orthogonality graphs for a parameter ϵ>0, and derive quantum correlations that allow the certification of randomness for arbitrary relaxation ϵ∈[0,1). Thirdly, it is well-known that a single qubit is noncontextual, i.e., the qubit correlations can be explained by a noncontextual hidden variable model. We show however that a single qubit is almost contextual in that there exist qubit correlations that cannot be explained by ϵ-faithful noncontextual hidden variable models for small ϵ>0. Finally, we point out possible attacks by quantum and general consistent (nonsignaling) adversaries for certain classes of contextuality tests over and above those considered in DI scenarios.
In quantum cryptography, secure randomness expansion involves using a short private string of random bits to generate a longer one, even in the presence of an adversary who may have access to quantum resources. In this work, we demonstrate that local contextuality-based self-tests are sufficient to construct a randomness expansion protocol that is secure against computationally unbounded quantum adversaries. Our protocol is based on self-testing from non-contextuality inequalities and we prove that our scheme asymptotically produces secure random numbers which are O(mε)-close to uniformly distributed and private, where ε is the robustness parameter of the self-test and m is the length of the generated random bit string. Our protocol is semi-device-independent in the sense that it inherits any assumptions necessary for the underlying self-test.
Semi-device-independent random number expansion (SDI-RNE) protocols require some truly random numbers to generate fresh ones, with making no assumptions on the internal working of quantum devices except for the dimension of the Hilbert space. The generated randomness is certified by non-classical correlation in the prepare-and-measure test. Until now, the analytical relations between the amount of the generated randomness and the degree of non-classical correlation, which are crucial for evaluating the security of SDI-RNE protocols, are not clear under both the ideal condition and the practical one. In the paper, first, we give the analytical relation between the above two factors under the ideal condition. As well, we derive the analytical relation under the practical conditions, where devices' behavior is not independent and identical in each round and there exists deviation in estimating the non-classical behavior of devices. Furthermore, we choose a different randomness extractor (i.e., two-universal random function) and give the security proof.
Quantum mechanics dramatically revolutionized our understanding of nature, challenged traditional information theory, and led to significant technological advancements. Two of the most counterintuitive features of quantum mechanics are quantum nonlocality and quantum contextuality. These features rule out hidden-variable models that attempt to explain quantum phenomena based on classical intuition. Quantum non-locality, first revealed by Bell's theorem in 1964, rules out local hidden-variable (LHV) models, originating from the famous Einstein-Podolsky-Rosen paradox. Meanwhile, quantum contextuality, originally pinpointed by the Kochen-Specker (KS) theorem in 1967, focuses on single-system correlations rather than between space-like separated systems, demonstrating quantum mechanics is incompatible with non-contextual hidden-variable (NCHV) models. Quantum correlations are crucial in several applications, both in computing and communication tasks. They also enable the strongest form of cryptographic security through the device-independent (DI) paradigm, where devices are treated as black-boxes, and could even be supplied by potentially malicious adversaries. The semi-device-independent (SDI) paradigm offers a more practical solution by balancing security and feasibility, allowing for some assumptions about devices. Since different DI and SDI applications require specific properties, this thesis focuses on designing optimal quantum correlations for these applications. DI randomness amplification (DIRA) is an important task in DI information processing, that converts a weak, biased seed into a fully random one. Quantum correlations on the boundary of the no-signaling set are necessary for this task. We introduce a class of almost-quantum Bell inequalities and develop a tilted version of the celebrated Hardy’s paradox showing corresponding quantum correlations enhance the robustness and efficiency in state-of-the-art DIRA protocols. Another direction pursued in the thesis is the study of self-testing, a DI approach to certify the state and measurements of quantum systems in a unique manner. We investigated three families of quantum correlations, the tilted Hardy tests, which we prove self-test any two-qubit pure entangled state, except the maximally-entangled one, the α-CHSH inequalities, where we prove robust self-test of the singlet and corresponding measurements, and weighted-Braunstein-Caves chained inequalities, which we prove delineate a boundary for the quantum correlation set, extending beyond the well-known Tsirelson-Landau-Masanes boundary. A third main direction explored in the thesis is the efficient construction of novel state-dependent and state-independent contextuality proof. We provide an optimal and experiment-friendly toolbox for various contextuality-based applications, including demonstrating entanglement-assisted advantages in zero-error communication, separating binary-outcome no-signaling models from quantum theory, and achieving optimal randomness rate in the SDI paradigm. We also discuss and address issues arising from the noise and errors in real-world experimental tests of the KS theorem, and show that quantum mechanics also violates NCHV models under arbitrary relaxation of the so-called non-disturbance conditions. Finally, a counter-intuitive phenomenon, termed ``bound randomness'', is introduced, revealing a deep resource-theoretical relationship between non-locality and DI-certified randomness. Analogous to Werner's famous result that entanglement is not sufficient for non-locality, ``bound randomness'' indicates non-locality is not always sufficient for DI randomness certification under standard spot-checking protocols. In conclusion, this thesis introduces key techniques in contextuality and non-locality for designing targeted correlations for various DI and SDI applications.
Bell nonlocality provides a device-independent (DI) way to certify quantum randomness, based on which true random numbers can be extracted from the observed correlations without detail characterizations on devices for quantum state preparation and measurement. However, the efficiency of current strategies for DI randomness certification is still heavily constrained when it comes to non-maximal Bell values, especially for multiple parties. Here, we present a family of multipartite Bell inequalities that allows to certify optimal quantum randomness and self-test GHZ (Greenberger-Horne-Zeilinger) states, which are inspired from the stabilizer group of the GHZ state. Due to the simple representation of stabilizer group for GHZ states, this family of Bell inequalities is of simple structure and can be easily expanded to more parties. Compared with the Mermin-type inequalities, this family of Bell inequality is more efficient in certifying quantum randomness when non-maximal Bell values achieved. Meanwhile, the general analytical upper bound for the Holevo quantity is presented, and achieves better performance compared with the MABK (Mermin-Ardehali-Belinskii-Klyshko) inequality, Parity-CHSH (Clauser-Horne-Shimony-Holt) inequality and Holz inequality at N=3, which is of particular interests for experimental researches on DI quantum cryptography in quantum networks.