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Optimal quantum correlations for (semi)-device-independent applications

Liu, Yuan, 刘圆

Published Jan 1, 2024• 0 citations
OpenAlex

Abstract

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 $\alpha$-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.

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The question of certifying quantum nonlocality under a relaxation of the assumptions in the Bell theorem has gained traction, with potential for device-independent applications under weak seeds and cross-talk. Recently, it was shown that quantum nonlocality can be certified even under a simultaneous arbitrary (but not full) relaxation of the assumptions of Measurement Independence (MI) and Parameter Independence (PI), using states of local dimension $d = poly((1-ε)^{-1})$ for an $ε\in [0,1)$-relaxation. Here, we derive three results strengthening the state-of-art. Firstly, we show that states of constant local dimension $d$ are already sufficient to certify quantum nonlocality under arbitrary MI and PI relaxation, albeit in a non-robust manner. Secondly, and as a theoretical paradigm to derive the above, we introduce the notion of \textit{measurement-dependent parameter-dependent locality} as the set of input-output behaviors under simultaneous relaxations of measurement and parameter independence. We provide a rigorous characterisation of the vertices of the polytope of joint input-output behaviors that obey a $μ$-relaxation of MI and $ε$-relaxation of PI. We highlight a relation between nonlocality certification under PI relaxation and that under detection inefficiencies by pointing out alternative extremal correlations to the Eberhard correlations that also allow to achieve detection efficiency of $η= 2/3$ in the two-input scenario. Finally, we study the implication of the relaxed assumptions for device-independent randomness certification. We analytically derive the quantum guessing probability for one player's outcomes in the CHSH Bell test, as a function of the noise in the test as well as of a leakage of an average amount of $I(X:B) < 1$ bits of input information per measurement round.

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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\alphaα-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.
Investigations of the boundary of the quantum correlation set have gained increased attention in recent years. This is done through the derivation of quantum Bell inequalities, which are related to Tsirelson's problem and have significant applications in device-independent (DI) information processing. However, determining quantum Bell inequalities is a notoriously difficult task and only isolated examples are known. In this paper, we present families of (almost-)quantum Bell inequalities and highlight four foundational and DI applications. Firstly, it is known that quantum correlations on the non-signaling boundary are of crucial importance in the task of DI randomness extraction from weak sources. In the practical Bell scenario of two players with two k -outcome measurements, we derive quantum Bell inequalities that demonstrate a separation between the quantum boundary and certain portions of the boundaries of the no-signaling polytope of dimension up to 4k−8 , extending previous results from nonlocality distillation and the collapse of communication complexity. Secondly as an immediate by-product, we give a general proof of Aumann’s Agreement theorem for quantum systems as well as the almost-quantum correlations, which implies Aumann’s agreement theorem is a reasonable physical principle in the context of epistemics to pick out both quantum theory and almost-quantum correlations from general no-signaling theories. Thirdly, we present a family of quantum Bell inequalities in the two players with m binary measurements scenarios, that we prove serve to self-test the two-qubit singlet and the corresponding 2m measurements. Interestingly, this claim generalizes the result for m=2 discovered by Tsirelson-Landau-Masanes and shows an improvement over the state-of-the-art Device-Independent Randomness-Amplification (DIRA). Lastly, we use our quantum Bell inequalities to derive the general form of the principle of no advantage in nonlocal computation, which is an information-theoretic principle that serves to characterize the quantum correlation set.
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.
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.
Quantum theory admits correlations between distant parties that are stronger than those permitted by classical physics. Such correlations, which are termed nonlocal, have been studied extensively since their discovery by John Bell in 1964. This thesis makes progress on various aspects of quantum nonlocality both from a foundational and practical perspective. We begin by studying a foundational question on the relationship between nonlocality and quantum entanglement. We show that the strongest form of nonlocality, also called pseudo-telepathy or full nonlocality, can be demonstrated by non-maximally entangled states and prove that all pure entangled states can demonstrate full nonlocality in the many-copy scenario. We also provide non-trivial examples of fully nonlocal mixed states. We then turn to the study of another phenomenon in quantum theory which is essential in almost all demonstrations of non-classicality—measurement incompatibility. We generalise the notion of measurement incompatibility and show that our notion has strong operational relevance: it captures the randomness in the outcomes of an untrusted measurement device against a classical adversary. We use our formalism to derive new bounds on the experimental losses and noise that can be tolerated in quantum communication protocols before they become insecure. We then begin a systematic study of routed Bell experiments, a variant of standard Bell experiments that was introduced to demonstrate nonlocality over long distances in the presence of losses and noise. We formalize the notion of genuine long-range quantum correlations in routed Bell experiments and develop numerical and analytical tools to characterize them. We show that routed Bell experiments allow for the demonstration of long-range quantum correlations at arbitrarily large distances using only entangled qubit states. We then prove the security of fully device-independent (DI) quantum key distribution (QKD) protocols based on routed Bell tests and quantify the resulting improvement in key rate and distance compared to standard DI QKD protocols.
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.
The question of certifying quantum nonlocality under a relaxation of the assumptions in the Bell theorem has gained traction, with potential for device-independent applications under weak seeds and cross-talk. Recently, it was shown that quantum nonlocality can be certified even under a simultaneous arbitrary (but not full) relaxation of the assumptions of Measurement Independence (MI) and Parameter Independence (PI), using states of local dimension d=poly((1−ε)−1)d = poly((1-ε)^{-1})d=poly((1−ε)−1) for an ε∈[0,1)ε\in [0,1)ε∈[0,1)-relaxation. Here, we derive three results strengthening the state-of-art. Firstly, we show that states of constant local dimension ddd are already sufficient to certify quantum nonlocality under arbitrary MI and PI relaxation, albeit in a non-robust manner. Secondly, and as a theoretical paradigm to derive the above, we introduce the notion of \textit{measurement-dependent parameter-dependent locality} as the set of input-output behaviors under simultaneous relaxations of measurement and parameter independence. We provide a rigorous characterisation of the vertices of the polytope of joint input-output behaviors that obey a μμμ-relaxation of MI and εεε-relaxation of PI. We highlight a relation between nonlocality certification under PI relaxation and that under detection inefficiencies by pointing out alternative extremal correlations to the Eberhard correlations that also allow to achieve detection efficiency of η=2/3η= 2/3η=2/3 in the two-input scenario. Finally, we study the implication of the relaxed assumptions for device-independent randomness certification. We analytically derive the quantum guessing probability for one player's outcomes in the CHSH Bell test, as a function of the noise in the test as well as of a leakage of an average amount of I(X:B)<1I(X:B) < 1I(X:B)<1 bits of input information per measurement round.