Entanglement and nonlocality are two of the features demonstrated by quantum systems which give rise to new possibilities in a variety of fields such as communication technology, cryptography and the study of fundamental physics. In this thesis, we explore three such applications of entanglement and nonlocality. \n \n In the first problem, we show that currently available noisy intermediate-scale quantum (NISQ) computers can be used for versatile quantum simulations of chaotic systems. We introduce a novel classical-quantum hybrid approach for exploring the dynamics of the chaotic quantum kicked top (QKT) on a universal quantum computer. The programmability of this approach allows us to experimentally explore the complete range of QKT chaoticity parameter regimes inaccessible to previous studies. Furthermore, the number of gates in our simulation does not increase with the number of kicks, thus making it possible to study the QKT evolution for an arbitrary number of kicks without fidelity loss. Using a publicly accessible NISQ computer (IBMQ), we observe periodicities in the evolution of the 2-qubit QKT, as well as signatures of chaos in the time-averaged 2-qubit entanglement. We also demonstrate a connection between entanglement and delocalization in the 2-qubit QKT, confirming theoretical predictions. \n \n In the second problem, we perform a device independent study of controlled teleportation of a qubit with an untrusted receiver. We construct a device independently testable scenario in a way that allows us to certify in the context of controlled teleportation, whether quantum resources are being used by the device despite the receiver being untrusted. We find in this case that the well known Svetlichny inequality must be violated to certify quantum correlations. While a maximal violation of Mermin's inequality guarantees maximum control power in controlled teleportation with trusted parties, we find that the Svetlichny inequality must be maximally violated to guarantee maximal control power with an untrusted receiver. This indicates that a stronger form of nonlocality is required to device independently test the controlled quantum teleportation with an untrusted receiver. By taking the example of the total depolarized and the qubit depolarized GHZ states, we show that controller's authority is a monotonically increasing function of the Svetlichny inequality violation. We find a window of non-maximal Svetlichny inequality violation where the controller's authority is non-zero. This shows that the controlled teleportation scheme with an untrusted receiver is robust to depolarizing noise present in the device. \n \n In the third problem, we study the quantum circuit implementation of a cryptographic object called quantum lightning. Quantum lightning is a strengthening of the public key quantum money scheme. We review an existing construction of quantum lightning, build the quantum circuit implementing an important step of the scheme and identify the difficulties in realizing others.
Entanglement and nonlocality are two of the features demonstrated by quantum systems which give rise to new possibilities in a variety of fields such as communication technology, cryptography and the study of fundamental physics. In this thesis, we explore three such applications of entanglement and nonlocality. \n \n In the first problem, we show that currently available noisy intermediate-scale quantum (NISQ) computers can be used for versatile quantum simulations of chaotic systems. We introduce a novel classical-quantum hybrid approach for exploring the dynamics of the chaotic quantum kicked top (QKT) on a universal quantum computer. The programmability of this approach allows us to experimentally explore the complete range of QKT chaoticity parameter regimes inaccessible to previous studies. Furthermore, the number of gates in our simulation does not increase with the number of kicks, thus making it possible to study the QKT evolution for an arbitrary number of kicks without fidelity loss. Using a publicly accessible NISQ computer (IBMQ), we observe periodicities in the evolution of the 2-qubit QKT, as well as signatures of chaos in the time-averaged 2-qubit entanglement. We also demonstrate a connection between entanglement and delocalization in the 2-qubit QKT, confirming theoretical predictions. \n \n In the second problem, we perform a device independent study of controlled teleportation of a qubit with an untrusted receiver. We construct a device independently testable scenario in a way that allows us to certify in the context of controlled teleportation, whether quantum resources are being used by the device despite the receiver being untrusted. We find in this case that the well known Svetlichny inequality must be violated to certify quantum correlations. While a maximal violation of Mermin's inequality guarantees maximum control power in controlled teleportation with trusted parties, we find that the Svetlichny inequality must be maximally violated to guarantee maximal control power with an untrusted receiver. This indicates that a stronger form of nonlocality is required to device independently test the controlled quantum teleportation with an untrusted receiver. By taking the example of the total depolarized and the qubit depolarized GHZ states, we show that controller's authority is a monotonically increasing function of the Svetlichny inequality violation. We find a window of non-maximal Svetlichny inequality violation where the controller's authority is non-zero. This shows that the controlled teleportation scheme with an untrusted receiver is robust to depolarizing noise present in the device. \n \n In the third problem, we study the quantum circuit implementation of a cryptographic object called quantum lightning. Quantum lightning is a strengthening of the public key quantum money scheme. We review an existing construction of quantum lightning, build the quantum circuit implementing an important step of the scheme and identify the difficulties in realizing others.
Noisy intermediate-scale quantum (NISQ) computers provide a new experimental platform for investigating the behaviour of complex quantum systems. We show that currently available NISQ devices can be used for versatile quantum simulations of chaotic systems. We introduce a classical-quantum hybrid approach for exploring the dynamics of the chaotic quantum kicked top (QKT) on a quantum computer. The programmability of this approach allows us to experimentally explore a broad range of QKT chaoticity parameter regimes inaccessible to previous studies. Furthermore, the number of gates in our simulation does not increase with the number of kicks, thus making it possible to study the QKT evolution for arbitrary number of kicks without fidelity loss. Using a publicly accessible NISQ computer (IBMQ), we observe periodicities in the evolution of the 2-qubit QKT, as well as signatures of chaos in the time-averaged 2-qubit entanglement. We also demonstrate a connection between entanglement and delocalization in the 2-qubit QKT, confirming theoretical predictions.
This dissertation sets out to examine some fundamental open questions in quantum physics regarding quantum-classical correspondence in regular versus chaotic systems. Specifically, we study these questions using approaches in quantum information science in an experimentally realized textbook model of quantum chaos - the quantum kicked top (QKT). \n \nThe effect of classical chaos on the generation of entanglement in spin systems has been a field of active research for a couple of decades. Whether high entanglement in these systems is a hallmark of chaos or not remains a widely debated topic. We explain the connection between entanglement and chaos in spin systems and resolve previous conflicting results. The previous studies have mostly drawn conclusions from numerical work on a few initial states in regular versus chaotic regions. We instead focus on stable and unstable periodic orbits because chaos emerges around unstable periodic orbits. We first propose a new set of criteria for determining whether quantum evolution will correspond to the classical trajectory in a localized manner at stable periodic orbits in periodically driven systems. These criteria can be used to calculate the quantum numbers that will lead to quantum-classical correspondence even in a deep quantum regime, and thus to quantify the well-known Bohr correspondence principle. Next, we analytically show a direct connection between entanglement generation and a measure of delocalization of a quantum state in spin systems. More concretely, we describe a method to calculate an upper bound on entanglement generation in any bipartition of spin systems, where the upper bound is a function of trace distance between the evolved state and the most localized classical-like separable states. This method along with our criteria for localized evolution enables us to explain the behaviour of entanglement in both deep quantum and semiclassical regimes for regular as well as chaotic regions. Hence, our analysis resolves the long-standing debates regarding the connection between classical chaos and quantum entanglement in deep quantum and semiclassical regimes. \n \nIn addition to the study of entanglement, we perform the first study of nonlocality, and the effect of chaos on its generation in the QKT. Since nonlocality and entanglement are inequivalent quantum resources, the effect of chaos on nonlocality merits an explicit study. Violations of Bell inequalities in the presence of spacelike separation among the subsystems imply nonlocality - meaning nonlocal correlations between subsystems of the total spin system. We show that the QKT evolution can lead to states that violate multiqubit Bell inequalities and hence provides a deterministic method to prepare nonlocal quantum states. Our numerical results suggest a correlation between delocalized evolution of a pure quantum state and generation of nonlocality in the quantum state. We further demonstrate that dynamical tunnelling - a classically forbidden phenomenon - in the QKT leads to the generation of Greenberger-Horne-Zeilinger (GHZ)-like states for even numbers of qubits. We analytically prove that these states are maximally nonlocal. On the other hand, we numerically show that any reduced state of the QKT obtained by tracing out a subsystem of the total spin system does not violate Bell inequalities. We provide an analytical explanation of the numerical results for 2−qubit reduced states by formulating and proving two general theorems regarding 2−qubit Bell inequalities. These theorems imply that any 2−qubit mixed state having a symmetric extension or symmetric purification cannot violate the Clauser-Horne-Shimony-Holt inequality. This highlights fundamental connections between two important and distinct concepts in quantum information science - Bell inequalities and symmetric extension of quantum states. \n \nApart from providing deeper insights into the fundamental questions of quantum-classical correspondence and new approaches to analyze quantum chaos, the methods developed in this thesis can be used to design quantum systems that can efficiently generate entanglement and nonlocality. Thus, our results could have interesting applications in quantum computing and quantum information science.
Quantum mechanics is nonlocal, meaning it cannot be described by any classical local hidden variable model. In this thesis we study two aspects of quantum nonlocality. Part I addresses the question of what classical resources are required to simulate nonlocal quantum correlations. We start by constructing new local models for noisy entangled quantum states. These constructions exploit the connection between nonlocality and Grothendieck's inequality, first noticed by Tsirelson. Next, we consider local models augmented by a limited amount of classical communication. After generalizing Bell inequalities to this setting, we show that (i) one bit of communication is sufficient to simulate the correlations of projective measurements on a maximally entangled state of two qubits, and (ii) five bits of communication are sufficient to simulate the joint correlation of two-outcome measurements on any bipartite quantum state. The latter result can be interpreted as a stronger (constrained) version of Grothendieck's inequality. In part II, we investigate the monogamy of nonlocal correlations. In a setting where three parties, A, B, and C, share an entangled quantum state of arbitrary dimension, we: (i) bound the trade-off between AB's and AC's violation of the CHSH inequality, obtaining an intriguing generalization of Tsirelson's bound, and (ii) demonstrate that forcing B and C to be classically correlated prevents A and B from violating certain Bell inequalities. We study not only correlations that arise within quantum theory, but also arbitrary correlations that do not allow signaling between separate groups of parties. These results are based on new techniques for obtaining Tsirelson bounds, or bounds on the quantum value of a Bell inequality, and have applications to interactive proof systems and cryptography.
Nonlocality and quantum entanglement constitute two special aspects of the quantum correlations existing in quantum systems, which are of paramount importance in quantum-information theory. Traditionally, they have been regarded as identical (equivalent, in fact, for pure two qubit states, that is, Gisin's Theorem ), yet they constitute different resources. Describing nonlocality by means of the violation of several Bell inequalities, we obtain by direct optimization those states of two qubits that maximally violate a Bell inequality, in terms of their degree of mixture as measured by either their participation ratio R = 1/Tr(ρ 2 ) or their maximum eigenvalue λ max . This optimum value is obtained as well, which coincides with previous results. Comparison with entanglement is performed too. An example of an application is given in the XY model. In this novel approximation, we also concentrate on the nonlocality for linear combinations of pure states of two qubits, providing a closed form for their maximal nonlocality measure. The case of Bell diagonal mixed states of two qubits is also extensively studied. Special attention concerning the connection between nonlocality and entanglement for mixed states of two qubits is paid to the so-called maximally entangled mixed states. Additional aspects for the case of two qubits are also described in detail. Since we deal with qubit systems, we will perform an analogous study for three qubits, employing similar tools. Relation between distillability and nonlocality is explored quantitatively for the whole space of states of three qubits. We finally extend our analysis to four-qubit systems, where nonlocality for generalized Greenberger–Horne–Zeilinger states of arbitrary number of parties is computed.
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.
Quantum entanglement is a non-classical correlation between quantum systems that cannot be accounted for by any local hidden-variable theory. Since its articulation in the Einstein-Podolsky-Rosen paper of 1935 and its formal testability through Bell's theorem in 1964, entanglement has evolved from a philosophical puzzle into the central resource of quantum information science. This review surveys the conceptual foundations of entanglement, the experimental record culminating in loophole-free Bell tests, and applications including quantum key distribution, quantum teleportation, and measurement-based quantum computation. We summarise quantitative results of representative Bell experiments, discuss entanglement measures for mixed states, and outline recent developments in long-distance entanglement distribution via satellites and quantum repeaters. Persistent challenges decoherence, certification of high-dimensional entanglement, and scalable distribution are identified as active frontiers of research.