This thesis describes experimental generation, manipulation and measurement of quantum information using photon pairs emitted in bulk crystals. Multi-photon sources engineered during the course of this thesis have proven to be ideal for original contributions in the field of optical quantum information. In the first part of this dissertation, we study nonlocality, bound entanglement and measurement-based quantum computing using entangled resources produced by our source. First, we produced and characterised three-photon GHZ polarisation states. We then experimentally violate the long-standing Svetlichny's inequality with a value of 4.51, which is greater than the classical bound by 3.6 standard deviations. Our results agree with the predictions of quantum mechanics, rule out nonlocal hidden-variable theories and certify the genuine tripartite entanglement achievable by our source. Second, with four-photon polarisation states, we demonstrate bound entanglement in Smolin states and realize all of their conceptually important characteristics. Our results highlight the difficulties to achieve the critical condition of undistillability without completely losing entanglement. We conclude the first part by simulating, for the first time, valence-bond solid states and use them as a resource for measurement-based quantum computing. Affleck-Kennedy-Lieb-Tasaki states are produced with 87% fidelity and single-qubit quantum logic gates reach an average fidelity of 92% over all input states and rotations. In the second part of this dissertation, we explore controlled waveform manipulation at the single-photon level. Specifically, we shrink the spectral bandwidth of a single photon from 1740 GHz to 43 GHz and demonstrate tunability over a range 70 times that bandwidth. The results are a considerable addition to the field of quantum frequency conversion and have genuine potential for technological applications.

In this thesis, we experimentally test fundamental properties of quantum mechanics, namely non-locality (in the form of three new families of Bell's inequalities) and the symmetry of envariance. To accomplish these we use a Sagnac source of polarization entangled photon pairs. In chapters one and two we discuss the relevant background information in quantum information theory, nonlinear optics, experimental realization of polarization entangled photons and a trouble-shooting and maintenance guide for a Saganc source. In chapter three we experiment with a set of three newly derived families of Bell's inequalities. These three families are predicted to yield the largest volume of violation of the local hidden variable models (LHVM). Our experimental results are in good agreement with those predictions and therefore, represent the largest volume of experimental violation of LHVM to date. We showed a violation of up to 30 sigma from what is predicted by LHVM, and our results followed closely to the predictions of quantum mechanics. In chapter four we experimentally test envariance, an assisted-symmetry exhibited by specifi c quantum systems. Envariance is a fundamental property in the quantum world that has lacked, until now, extensive experimental study. The symmetry has ramifi cations in the foundations of quantum mechanics, and plays an integral role in a proof of Born's rule [1]. Our results serve as a benchmark the property of envariance. We show that experimental quantum states can be (99.66+/- 0.04)% envariant over a wide range of transformations, as measured using the average quantum fi delity [2], and (99.963 +/- 0.005)% as measured using a modifi ed average Bhattacharya Coeffi cient [3], a measure of the overlap of two probability distributions.

Most applications in quantum information processing make either explicit or implicit use of entanglement. It is thus important to have a good understanding of entanglement and the role it plays in these protocols. However, especially when it comes to multipartite entanglement, there still remain a lot of mysteries. This thesis is devoted to getting a better understanding of multipartite entanglement, and its role in various quantum information protocols. First, we investigate transformations between multipartite entangled states that only use local operations and classical communication (LOCC). We mostly focus on three qubit states in the GHZ class, and derive upper and lower bounds for the successful transformation probability between two states. We then focus on absolutely maximally entangled (AME) states, which are highly entangled multipartite states that have the property that they are maximally entangled for any bipartition. With them as a resource, we develop new parallel teleportation protocols, which can then be used to implement quantum secret sharing (QSS) schemes. We further prove the existence of AME states for any number of parties, if the dimension of the involved quantum systems is chosen appropriately. An equivalence between threshold QSS schemes and AME states shared between an even number of parties is established, and further protocols are designed, such as constructing ramp QSS schemes and open-destination teleportation protocols with AME states as a resource. As a framework to work with AME states, graph states are explored. They allow for efficient bipartite entanglement verification, which makes them a promising candidate for the description of AME states. We show that for all currently known AME states, absolutely maximally entangled graph states can be found, and we were even able to use graph states to find a new AME state for seven three-dimensional systems (qutrits). In addition, the implementation of QSS schemes from AME states can be conveniently described within the graph state formalism. Finally, we use the insight gained from entanglement in QSS schemes to derive necessary and sufficient conditions for quantum erasure channel and quantum error correction codes that satisfy the quantum Singleton bound, as these codes are closely related to ramp QSS schemes. This provides us with a very intuitive approach to codes for the quantum erasure channel, purely based on the entanglement required to protect information against losses by use of the parallel teleportation protocol.

This thesis studies the manipulation of entanglement in three-qubit quantum systems. I consider the operational setting in which the qubits are distributed to three spatially separated parties. The parties act locally on their quantum systems and share classical communication with one another, a scenario commonly called local operations and classical communication (LOCC). In the LOCC setting, there are two different classes of entanglement in multipartite systems, called the GHZ and W classes, which are inequivalent in the sense that states from one class cannot be transformed into the other without the consumption of additional entanglement. In this thesis, I first show that the LOCC conversion of certain GHZ and W-class states becomes possible by using only one additional ebit (“entangled bit”) of shared entanglement. In many cases, this can be proven as the minimal amount of needed entanglement. I then consider the problem of one-way communication transformations from general three-qubit states into two-qubit maximally entangled states, known as EPR states. An optimal protocol in terms of success probability is provided for W-class states. The success probability of this protocol coincides with the optimal success probability if two of the parties are allowed to act jointly within the same laboratory. In other words, forcing the locality constraint on all three parties does not weaken their capabilities for obtaining bipartite entanglement when starting from a W-class state. I also present that this property holds for certain types of GHZ-class states as well.

Due to its great importance, both from the fundamental and from the practical points of view, it is imperative that the concept of entanglement is explored. In this thesis I investigate the connection between information measures, entanglement and the “speed” of quantum evolution. In Chapter 1 a brief review of the different information and entanglement measures as well as of the concept of “speed” of quantum evolution is given. An illustration of the quantum no-cloning theorem in connection with closed timelike curves is also provided. The work leading up to this thesis has resulted in the following three publications and in one conference proceeding: (A) C. Zander and A.R. Plastino, “Composite systems with extensive Sq (power-law) entropies”, Physica A 364, (2006) pp. 145-156 (B) S. Curilef, C. Zander and A.R. Plastino, “Two particles in a double well: illustrating the connection between entanglement and the speed of quantum evolution”, Eur. J. Phys. 27, (2006) pp. 1193-1203 (C) C. Zander, A.R. Plastino, A. Plastino and M. Casas, “Entanglement and the speed of evolution of multi-partite quantum systems”, J. Phys. A: Math. Theor. 40 (11), (2007) pp. 2861-2872 (D) A.R. Plastino and C. Zander, “Would Closed Timelike Curves Help to Do Quantum Cloning?”, AIP Conference Proceedings: A century of relativity physics, ERE 841, (2005) pp. 570-573. Chapter 2 is based on (A) and is an application of the Sq (powerlaw) entropy. A family of models for the probability occupancy of phase space exhibiting an extensive behaviour of Sq and allowing for an explicit analysis of the thermodynamic limit is proposed. Chapter 3 is based on (B). The connection between entanglement and the speed of quantum evolution as measured by the time needed to reach an orthogonal state is discussed in the case of two quantum particles moving in a one-dimensional double well. This illustration is meant to be incorporated into the teaching of quantum entanglement. Chapter 4 is based on (C). The role of entanglement in time evolution is investigated in the cases of two-, three- and N-qubit systems. A clear correlation is seen between entanglement and the speed of evolution. States saturating the speed bound are explored in detail. Chapter 5 summarizes the conclusions drawn in the previous chapters. / Dissertation (MSc)--University of Pretoria, 2007. / Physics / MSc / Unrestricted

Quantum entanglement is a phenomenon which has shown great potential use in modern technical implementations, but there is still much development needed in the field. One major problem is how to measure the amount of entanglement present in a given entangled state. There are numerous different entanglement measures suggested, all satisfying some conditions being of either operational, or more abstract, mathematical nature. However, in contradiction to what one might expect, the measures show discrepancies in the ordering of entangled states. Concretely this means that with respect to one measure, a state can be more entangled than another state, but the ordering may be opposite for the same states using another measure. In this thesis we take a closer look at some of the most commonly occurring entanglement measures, and find examples of states showing inequivalent entanglement ordering for the different measures. / Kvantmekanisk sammanflätning är ett fenomen som visat stor potential för framtida tekniska tillämpningar, men för att kunna använda oss av detta krävs att vi hittar lämpliga modeller att mäta omfattningen av sammanflätningen hos ett givet tillstånd. Detta har visat sig vara en svår uppgift, då de modeller som finns idag är otillräckliga när det gäller att konsekvent avgöra till vilken grad olika tillstånd är sammanflätade. Exempelvis kan en modell visa att ett tillstånd är mer sammanflätat än ett annat, medan en annan modell kan visa på motsatsen - att det första tillståndet är mindre sammanflätat än det andra. En möljig orsak kan ligga i de olika modellernas deifnition, då vissa utgår från operativa definitioner, medan andra grundas på matematiska, abstrakta villkor. I denna uppsats tittar vi lite närmre på några av de mätmodeller som finns, och hittar exempel på tillstånd som uppvisar olika ordning av sammanflätningsgrad beroende på vilken modell som används.

No description available.

Quantum theories have had an unprecedented success in providing a framework for studying physical systems. A fundamental implication of these theories is the existence of so-called entangled states, that is states whose description cannot be reduced to their constituents. These states are purely quantum and there is no such analogue in classical physics, where knowing the state of every particle is sufficient to infer the state of the system they compose. Entanglement is a core element of many quantum algorithms, quantum teleportation, quantum communications and quantum cryptographic scenarios. Furthermore, entanglement is present in nearly all solid-state systems, when they are at, or close to, their state of lowest energy. Therefore, it is both a technological resource and also a property which needs to be investigated in order to achieve understanding of real world materials at a fundamental level. The most concise demonstration of entanglement is perhaps in the case of maximal entanglement between two spin-l/2 particles. These maximally entangled two- particle states are called Bell states and they have been used to demonstrate experimentally that quantum mechanics is inequivalent to classical mechanics. A gen- eralization of this setting comes from studying entanglement between two physical systems, these can be either pure or mixed (e.g. in contact with a thermal bath). Entanglement between two systems, also knows as bipartite entanglement, has been studied in depth and quantified through various measures. However bipartite entanglement, by definition, is not the only quantity of in- terest. In some cases, entanglement is global and its properties cannot be reduced to studying bi-partitions. This type of entanglement, so-called multipartite entanglement, is harder to quantify and to study in general. Its presence is profound in physical systems that are at the point of undergoing a quantum phase transition and it is also a core ingredient for quantum error correcting codes, performing classical computation with quantum resources and some cryptographic scenarios. In this thesis we study properties of systems with multi-partite entanglement in the context of renormalization and quantum phase transitions, we show that multi- partite entanglement can be used to perform cryptographic tasks and we investigate what classes of Hamiltonians generate multiartite entanglement, while at the same time, their action can be simulated efficiently by a classical computer.

In this thesis we examine methods for measuring entanglement entropy in spin-1/2 Heisenberg systems using quantum Monte Carlo in the valence bond basis. We begin by presenting the quantum Monte Carlo techniques used in this research. We then use these techniques to directly compare the recently proposed valence bond entanglement entropy to the standard definition of entanglement entropy: the von Neumann entanglement entropy. We find that the valence bond entanglement entropy does not give a bound on the von Neumann entanglement entropy, and that it exhibits a multiplicative logarithmic correction to the area law that is not present in the scaling of the von Neumann entanglement entropy. We then present a method to measure higher orders of the generalized Renyi entanglement entropies using valence bond quantum Monte Carlo, and show results for the second Renyi entropy. We find the results converge to the exact results for one dimensional Heisenberg spin-1/2 chains, and see that the scaling of the second Renyi entropy follows an area law in the two dimensional Heisenberg ground state.

A review is given on the thermodynamical structure of bipartite entanglement. By comparing it to the axiomatic formulation of thermodynamics presented by Giles it is shown that for finite dimensional systems the two theories are formally inequivalent. The same approach is used to demonstrate the full equivalence in the asymptotic limit for pure quantum states. For mixed states a different method for obtaining the second law is described applied to two different classes of operations, PPT-preserving and asymptotically non-entangling operations. / Δίνεται μια επισκόπηση της θερμοδυναμικής δομής του διμερούς εναγκαλισμού. Συγκρίνοντας τον με την αξιωματική θεμελίωση της θερμοδυναμικής όπως παρουσιάστηκε από τον Giles δείχνεται ὸτι για συστήματα πεπερασμένων διαστάσεων οι δύο θεωρίες δεν είναι ισοδύναμες. Η ὶδια προσέγγιση χρησιμοποιείται για να δειχτεί η πλήρης αντιστοιχία στο ασυμπτωτικό ὸριο για καθαρές καταστάσεις. Για τις μικτές καταστάσεις περιγράφεται μια διαφορετική μέθοδος για την κατασκευή του δεύτερου νόμου της θερμοδυναμικής η οποία εφαρμόζεται σε δύο διαφορετικές κατηγορίες μετασχηματισμών, μετασχηματισμοί που διατηρούν την θετικότητα του μερικού ανάστροφου και ασυμπτωτικώς μη εναγκαλιστικούς μετασχηματισμούς.