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.
The output randomness from a random number generator can be certified by observing the violation of quantum contextuality inequalities based on the Kochen-Specker theorem. Contextuality can be tested in a single quantum system, which significantly simplifies the experimental requirements to observe the violation comparing to the ones based on nonlocality tests. However, it is not yet resolved as to how to ensure compatibilities for sequential measurements that is required in contextuality tests. Here, we employ a modified Klyachko-Can-Binicio\ifmmode \breve{g}\else \u{g}\fi{}lu-Shumovsky contextuality inequality, which can ease the strict compatibility requirement on measurements. On a trapped single
138Ba+ ion system, we experimentally demonstrate violation of the contextuality inequality and realize quantum random number expansion by closing detection loopholes. We perform $1.29\ifmmode\times\else\texttimes\fi{}{10}^{8}$ trials of experiments and extract a randomness of $5.28\ifmmode\times\else\texttimes\fi{}{10}^{5}$ bits with a speed of $270\phantom{\rule{0.2em}{0ex}}\mathrm{bits}\phantom{\rule{0.2em}{0ex}}{\mathrm{s}}^{\ensuremath{-}1}$. Our demonstration paves the way for practical high-speed spot-checking quantum random number expansion and other secure information processing applications.
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.
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.
The generation of certifiable randomness is one of the most promising applications of quantum technologies. Furthermore, the intrinsic non-locality of quantum correlations allow us to certify randomness in a device-independent way, i.e., we do not need to make assumptions about the devices used. Due to the work of Curchod et al. a single entangled two-qubit pure state can be used to produce arbitrary amounts of certified randomness. However, the obtaining of this randomness is experimentally challenging as it requires a large number of measurements, both projective and general. Motivated by these difficulties in the device-independent setting, we instead consider the scenario of one-sided device independence where certain devices are trusted, and others are not; a scenario motivated by asymmetric experimental set-ups such as ion-photon networks. We show how certain aspects of previous works can be adapted to this scenario and provide theoretical bounds on the amount of randomness that can be certified. Furthermore, we give a protocol for unbounded randomness certification in this scenario, and provide numerical results demonstrating the protocol in the ideal case. Finally, we numerically test the possibility of implementing this scheme on near-term quantum technologies, by considering the performance of the protocol on several physical platforms.
The generation of certifiable randomness is one of the most promising applications of quantum technologies. Furthermore, the intrinsic non-locality of quantum correlations allow us to certify randomness in a device-independent way, ie, we do not need to make assumptions about the devices used. Due to the work of Curchod et al. a single entangled two-qubit pure state can be used to produce arbitrary amounts of certified randomness. However, the obtaining of this randomness is experimentally challenging as it requires a large number of measurements, both projective and general. Motivated by these difficulties in the device-independent setting, we instead consider the scenario of one-sided device independence where certain devices are trusted, and others are not; a scenario motivated by asymmetric experimental set-ups such as ion-photon networks. We show how certain aspects of previous works can be adapted to this scenario and provide theoretical bounds on the amount of randomness that can be certified. Furthermore, we give a protocol for unbounded randomness certification in this scenario, and provide numerical results demonstrating the protocol in the ideal case. Finally, we numerically test the possibility of implementing this scheme on near-term quantum technologies, by considering the performance of the protocol on several physical platforms.
One of the striking properties of quantum mechanics is the occurrence of the Bell-type non-locality. They are a fundamental feature of the theory that allows two parties that share an entangled quantum system to observe correlations stronger than possible in classical physics. In addition to their theoretical significance, non-local correlations have practical applications, such as device-independent randomness generation, providing private unpredictable numbers even when obtained using devices delivered by an untrusted vendor. Thus, determining the quantity of certifiable randomness that can be produced using a specific set of non-local correlations is of significant interest. First, we present an experimental realization of recent Bell-type operators designed to provide private random numbers that are secure against adversaries with quantum resources. We use semi-definite programming to give lower bounds on the generated randomness in terms of both min-entropy and von Neumann entropy in a device-independent scenario. Our results demonstrate the first experiment that certifies nearly two bits of randomness from binary measurements of two parties [1], [2]. Apart from single-round certification, we provide an analysis of finite-key protocol for quantum randomness expansion using the Entropy Accumulation Theorem. Second, we present an efficient and practical method for certifying quantum randomness using generalized measurements. Indeed, we have derived a method for self-testing the presence of POVMs. We present the certification of more than one bit of min-entropy from a single POVM on one of the qubits of an entangled state using a variant of the elegant Bell inequality. We also obtain more than one bit of randomness in a prepare and measure scenario of a similar structure with a POVM on a single qubit [3]. We provide numerical simulations to demonstrate the effectiveness of our randomness certification and expansion.