Information Theoretic Resources in Quantum Theory

Resource identification and quantification is an essential element of both classical and quantum information theory. Entanglement is one of these resources, arising when quantum communication and nonlocal operations are expensive to perform. In the first part of this thesis we quantify the effective entanglement when operations are additionally restricted. For an important class of errors we find a linear relationship between the usual and effective higher dimensional generalization of concurrence, a measure of entanglement. In the second chapter we focus on nonlocality in the presence of superselection rules, where we propose a scheme that may be used to activate nongenuinely multipartite nonlocality with multiple copies of the state. We show that whenever the number of particles is insufficient, the genuinely multipartite nonlocality is degraded to nongenuinely multipartite. While in the first few chapters we focus on understanding the resources present in quantum states, in the final part we turn the picture around and instead treat operations themselves as a resource. We provide our observers with free access to classical operations - ie. those that cannot detect or generate quantum coherence. We show that the operation of interest can then be used to either generate or detect quantum coherence if and only if it violates a particular commutation relation. Using the relative entropy, the commutation relation provides us with a measure of nonclassicality of operations. We show that the measure is a sum of two contributions, the generating power and the distinguishing power, each of which is separately an essential ingredient in quantum communication and information processing. The measure also sheds light on the operational meaning of quantum discord, which we show can be interpreted as the difference in superdense coding capacity between a quantum state and a classical state.Comment: Thesis, 109 page

Cold atoms for deterministic quantum computation with one qubit

In this thesis, I describe a proposal for experiments with dense ensembles of ultracold Rubidium atoms optically excited to highlying electronic states known as Rydberg states. The proposed experiment is designed to implement a model of quantum computation known as deterministic quantum computation with one qubit (DQC1). My proposal is novel among other proposals for Rydberg atoms because it employs mixed states and could provide insights into the source of the enhancements of quantum technology over classical technology. I demonstrate that ultracold atoms and Rydberg interactions offer new perspectives for exploring the DQC1 model. This is mainly due to the large strength, long range and controllability of Rydberg interactions. My specific contributions include: a proposal that solves one of the main issues with implementing quantum logic gates between ensembles of atoms; numerical simulations of the proposed cold atom implementation of a DQC1 protocol; and the design and laboratory implementation of a high-performance optical system for imaging ultracold atoms. These contributions bring the experimental exploration of the DQC1 model closer to realisation

Signatures of quantumness: identification, quantification and dynamical preservation

2014 - 2015The quanti cation of quantumness is necessary to assess how much a physical system departs from a classical behaviour and thus gauge the quantum enhancement in opera- tional tasks such as information processing and computation. For arbitrary multiparti- cle systems, the quanti cation of quantumness typically involves nontrivial optimisation problems, and may require demanding tomographical techniques. We have developed an experimentally feasible approach to the evaluation of geometric measures of quantumness, according to which the distance from the state of the system to a suitable set of classi- cal states is considered. Our approach provides analytical results for particular classes of mixed states of N qubits, and computable lower bounds to global, partial, and gen- uine multiparticle entanglement, as well as to quantum coherence, for any general state. For global and partial entanglement, as well as quantum coherence, useful bounds have been obtained with minimum e ort, requiring local measurements in just three settings for any N. For genuine entanglement, a number of measurements scaling linearly with N is required. We have demonstrated the power of our approach to estimate and quantify di erent types of multiparticle entanglement in a variety of N-qubit states useful for quan- tum information processing and recently engineered in laboratories with quantum optics and trapped ion setups... [edited by author]XIV n.s

Entanglement and other measures of non-classicality

Quantum information theory (QIT) is an emerging field of physics which aims to develop new methods of dealing with information by harnessing the power of quantum mechanics. Besides its potential to revolutionize the techniques of information processing and communication, it also provides novel approaches to better comprehend the foundations of quantum mechanics. Among many important problems in QIT, manipulation and dynamical characterization of correlations present in quantum systems stand out due to their relevance for the practical applications of the theory. This thesis intends to explore such correlations of quantum and classical nature from various perspectives. In particular, our discussions involve the investigation of local transformations among a class of entangled states and the examination of correlation measures in some physical models. We first examine the classification of the flip (0-1) and exchange symmetric (FES) states under local quantum operations. We study the optimal local one-shot conversions of FES states to determine the entanglement transformations that relate multiqubit FES states with the maximum possible probability of success. Next, we investigate the exchange symmetry properties of certain symmetric states when the qubits evolve according to a dephasing model which is also invariant under swap operation. We find that there exist states which do not preserve the exchange symmetry with unit probability during the time evolution, leading to the spontaneous breaking of exchange symmetry. Later, we turn our attention to the dynamics of quantum and classical correlations for qubit-qutrit systems in independent and global dephasing environments. In these cases, we demonstrate several interesting phenomena such as the transition from classical to quantum decoherence. Lastly, we investigate the thermal quantum and total correlations in the one-dimensional anisotropic XY model in transverse field. We discuss the ability of different measures to estimate the critical point of the quantum phase transition at finite temperature. We also consider the relation between correlations and the factorized ground state in this model. Furthermore, we study the effect of temperature on long-range correlations

Timing the State of Light with Anomalous Dispersion and a Gradient Echo Memory

We study the effects of anomalous dispersion on the continuous-variable entanglement of EPR states (generated using four-wave mixing in 85Rb) by sending one part of the state through a fast-light medium and measuring the state's quantum mutual information. We observe an advance in the maximum of the quantum mutual information between modes. In contrast, due to uncorrelated noise added by a small phase-insensitive gain, we do not observe any statistically significant advance in the leading edge of the mutual information. We also study the storage and retrieval of multiplexed optical signals in a Gradient Echo Memory (GEM) at relevant four-wave mixing frequencies in 85Rb. Temporal multiplexing capabilities are demonstrated by storing multiple classical images in the memory simultaneously and observing the expected first-in last-out order of recall without obvious cross-talk. We also develop a technique wherein selected portions of an image written into the memory can be spatially targeted for readout and erasure on demand. The effect of diffusion on the quality of the recalled images is characterized. Our results indicate that Raman-based atomic memories may serve as a flexible platform for the storage and retrieval of multiplexed optical signals

Quantum Apices: Identifying Limits of Entanglement, Nonlocality, & Contextuality

This work develops analytic methods to quantitatively demarcate quantum reality from its subset of classical phenomenon, as well as from the superset of general probabilistic theories. Regarding quantum nonlocality, we discuss how to determine the quantum limit of Bell-type linear inequalities. In contrast to semidefinite programming approaches, our method allows for the consideration of inequalities with abstract weights, by means of leveraging the Hermiticity of quantum states. Recognizing that classical correlations correspond to measurements made on separable states, we also introduce a practical method for obtaining sufficient separability criteria. We specifically vet the candidacy of driven and undriven superradiance as schema for entanglement generation. We conclude by reviewing current approaches to quantum contextuality, emphasizing the operational distinction between nonlocal and contextual quantum statistics. We utilize our abstractly-weighted linear quantum bounds to explicitly demonstrate a set of conditional probability distributions which are simultaneously compatible with quantum contextuality while being incompatible with quantum nonlocality. It is noted that this novel statistical regime implies an experimentally-testable target for the Consistent Histories theory of quantum gravity.Comment: Doctoral Thesis for the University of Connecticu

Classical & quantum dynamics of information and entanglement properties of fermion systems

Due to their great importance, both from the fundamental and from the practical points of view, it is imperative that the various facets of the concepts of information and entanglement are explored systematically in connection with diverse physical systems and processes. These concepts are at the core of the emerging field of the Physics of Information. In this Thesis I investigate some aspects of the dynamics of information in both classical and quantum mechanical systems and then move on to explore entanglement in fermion systems by searching for novel ways to classify and quantify entanglement in fermionic systems. In Chapter 1 a brief review of the different information and entropic measures as well as of the main evolution equations of classical dynamical and quantum mechanical systems is given. The conservation of information as a fundamental principle both at the classical and quantum levels, and the implications of Landauer's theorem are discussed in brief. An alternative and more intuitive proof of the no-broadcasting theorem is also provided. Chapter 2 is a background chapter on quantum entanglement, where the differences between the concept of entanglement in systems consisting of distinguishable subsystems and the corresponding concept in systems of identical fermions are emphasized. Different measures of entanglement and relevant techniques such as majorization, are introduced. To illustrate some of the concepts reviewed here I discuss the entanglement properties of an exactly soluble many-body model which was studied in paper (E) of the publication list corresponding to the present Thesis. An alternative approach to the characterization of quantum correlations, based on perturbations under local measurements, is also briefly reviewed. The use of uncertainty relations as entanglement indicators in composite systems having distinguishable subsystems is then examined in some detail. Chapter 3 is based on papers (A) and (B) of the list of publications. Extended Landauer-like principles are developed, based amongst others on the conservation of information of divergenceless dynamical systems. Conservation of information within the framework of general probabilistic theories, which include the classical and quantum mechanical probabilities as particular instances, is explored. Furthermore, Zurek's information transfer theorem and the no-deleting theorem are generalized. Chapter 4 is based on articles (C) and (D) mentioned in the publication list, and investigates several separability criteria for fermions. Criteria for the detection of entanglement are developed based either on the violation of appropriate uncertainty relations or on inequalities involving entropic measures. Chapter 5 introduces an approach for the characterization of quantum correlations (going beyond entanglement) in fermion systems based upon the state disturbances generated by the measurement of local observables. Chapter 6 summarizes the conclusions drawn in the previous chapters. The work leading up to this Thesis has resulted in five publications in peer reviewed science research journals.Thesis (PhD)--University of Pretoria, 2012.Physicsunrestricte

New concepts in quantum-metrology: From coherent averaging to Hamiltonian extensions

This thesis is dedicated to the understanding of the metrology of quantum systems by using the tools of quantum parameter estimation, in particular the quantum Fisher information (QFI). Our first project deals with a specific protocol of quantum enhanced measurement known as coherent averaging [Braun and Martin, 2011]. This protocol is based on a star topology, with one central object, the so-called quantum bus, connected to N extra subsystems, called probes. For the estimation of a parameter characteristic of the interaction between the quantum bus and the probes, coherent averaging leads to a Heisenberg limited (HL) scaling for the QFI (QFI proportional to N 2 ). Importantly this HL scaling can be obtained while starting with a separable state. This provides an advantage as generally one needs to use entangled states to achieve this scaling. Another important aspect in coherent averaging is the possibility to obtain the HL scaling by performing a measurement on the quantum bus only. These results were obtained using perturbation theory in the regime of weak interactions. In this thesis we go one step further in the study of the coherent averaging protocol. We extend the formalism of perturbation theory to encompass the possibility of estimating any parameter, in the regimes of strong and weak interactions. To illustrate the validity of our results, we introduce two models as examples for a coherent averaging scheme. In these models both the quantum bus and all the probes are qubits. In the ZZXX model, the free Hamiltonians do not commute with the interaction Hamiltonians and we have to rely on numerics to find non-perturbative solutions .In the ZZZZ model the free evolution Hamiltonians commute with the interaction Hamiltonians and we can find the exact solution analytically. Perturbation theory shows that in the strong interaction regime and starting with a separable state, we can estimate the parameter of the free evolution of the probes with a HL scaling if the free Hamiltonians do not commute with the interaction Hamiltonians. This is confirmed by the non-perturbative numerical results for the ZZXX model. In the weak interaction regime we only obtain a standard quantum limit (SQL) scaling for the parameter of the free evolution of the probes (QFI proportional to N ). When one has only access to the quantum bus, we show that the HL scaling found using the perturbation theory does not necessarily survive outside the regime of validity of the perturbation. This is especially the case as N becomes large. It is shown by comparing the exact analytical result to the perturbative result with the ZZZZ model. The same behaviour is observed with the ZZXX model using the non-perturbative numerical results. In our second project we investigate the estimation of the depolarizing channel and the phase-flip channel under non-ideal conditions. It is known that using an ancilla can lead to an improvement of the channel QFI (QFI maximized over input states feeding the channel) even if we act with the identity on the ancilla. This method is known as channel extension. In all generality the maximal channel QFI can be obtained using an ancilla whose Hilbert space has the same dimension as the dimension of the Hilbert space of the original system. In this ideal scenario using multiple ancillas — or one ancilla with a larger Hilbert space dimension — is useless. To go beyond this ideal result we take into account the possibility of loosing either the probe or a finite number of ancillas. The input states considered are GHZ and W states with n + 1 qubits (the probe plus n ancillas). We show that for any channel, when the probe is lost then all the information is lost, and the use of ancillas cannot help. For the phase-flip channel the introduction of ancillas never improves the channel QFI and ancillas are useless. For the depolarizing channel the maximal channel QFI can be reached using one ancilla and feeding the extended channel with a Bell state, but if the ancilla is lost then all the advantage is lost. We show that the GHZ states do not help to fight the loss of ancillas: If one ancilla or more are lost all the advantage provided by the use of ancillas is lost. More interestingly, we show that the W states with more than one ancilla are robust against loss. For a given number of lost ancillas, there always exists an initial number of ancillas for which a W state provides a higher QFI than the one obtained without ancillas. Our last project is about Hamiltonian parameter estimation for arbitrary Hamiltonians. It is known that channel extension does not help for unitary channels. Instead we apply the idea of extension to the Hamiltonian itself and not to the channel. This is done by adding to the Hamiltonian an extra term, which is independent of the parameter and which possibly encompasses interactions with an ancilla. We call this technique Hamiltonian extension. We show that for arbitrary Hamiltonians there exists an upper bound to the channel QFI that is in general not saturated. This result is known in the context of non-linear metrology. Here we show explicitly the conditions to saturate the bound. We provide two methods for Hamiltonian extensions, called signal flooding and Hamiltonian subtraction, that allow one to saturate the upper bound for any Hamiltonian. We also introduce a third method which does not saturate the upper bound but provides the possibility to restore the quadratic time scaling in the channel QFI when the original Hamiltonian leads only to a periodic time scaling of the channel QFI. We finally show how these methods work using two different examples. We study the estimation of the strength of a magnetic field using a NV center, and show how using signal flooding we saturate the channel QFI. We also consider the estimation of a direction of a magnetic field using a spin-1. We show how using signal flooding or Hamiltonian subtraction we saturate the channel QFI. We also show how by adding an arbitrary magnetic field we restore the quadratic time scaling in the channel QFI. Eventually we explain how coherent averaging can be scrutinized in the formalism of Hamiltonian extensions

Quantum-Classical Correspondence and Entanglement in Periodically Driven Spin Systems

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). The 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. In 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. Apart 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