The present thesis can be divided in three parts: 1) Quantum gate teleportation; 2) Numerical search of universal entanglers; 3) Connections between quantum information and number theory. Regarding the quantum gate teleportation, a separability criterion of normal matrices is used to find the analytical conditions of the preservation of separability under conjugation. That analytical condition allowed to find the general formula of an element of $mathbb{C}^{4}$ Clifford group, as well to understand the role of the basis of measurement in the quantum gate teleportation protocol. Considering the searching for universal entanglers, the same separability criterion of normal matrices was used as fitness function in a computational heuristics, in prder to find good candidates for universal entanglers in $mathbb{C}^{3} otimes mathbb{C}^{4}$ and $mathbb{C}^{4} otimes mathbb{C}^{4}$ Hilbert spaces. At last, in the connection of quantum information with the number theory, it is presented the study of the preparation and entanglement of several multi-qubit quantum states based in integer sequences, and the Riemannian quantum circuit, a quantum circuit whose eigenvalues are related to the zeros of the Riemann zeta function. The existence of such circuit proves that is always possible to construct a physical system related to a finite amount of zeros.
The present thesis can be divided in three parts: 1) Quantum gate teleportation; 2) Numerical search of universal entanglers; 3) Connections between quantum information and number theory. Regarding the quantum gate teleportation, a separability criterion of normal matrices is used to find the analytical conditions of the preservation of separability under conjugation. That analytical condition allowed to find the general formula of an element of mathbbC4 Clifford group, as well to understand the role of the basis of measurement in the quantum gate teleportation protocol. Considering the searching for universal entanglers, the same separability criterion of normal matrices was used as fitness function in a computational heuristics, in prder to find good candidates for universal entanglers in mathbbC3otimesmathbbC4 and mathbbC4otimesmathbbC4 Hilbert spaces. At last, in the connection of quantum information with the number theory, it is presented the study of the preparation and entanglement of several multi-qubit quantum states based in integer sequences, and the Riemannian quantum circuit, a quantum circuit whose eigenvalues are related to the zeros of the Riemann zeta function. The existence of such circuit proves that is always possible to construct a physical system related to a finite amount of zeros.
The present thesis can be divided in three parts: 1) Quantum gate teleportation; 2) Numerical search of universal entanglers; 3) Connections between quantum information and number theory. Regarding the quantum gate teleportation, a separability criterion of normal matrices is used to find the analytical conditions of the preservation of separability under conjugation. That analytical condition allowed to find the general formula of an element of mathbbC4 Clifford group, as well to understand the role of the basis of measurement in the quantum gate teleportation protocol. Considering the searching for universal entanglers, the same separability criterion of normal matrices was used as fitness function in a computational heuristics, in prder to find good candidates for universal entanglers in mathbbC3otimesmathbbC4 and mathbbC4otimesmathbbC4 Hilbert spaces. At last, in the connection of quantum information with the number theory, it is presented the study of the preparation and entanglement of several multi-qubit quantum states based in integer sequences, and the Riemannian quantum circuit, a quantum circuit whose eigenvalues are related to the zeros of the Riemann zeta function. The existence of such circuit proves that is always possible to construct a physical system related to a finite amount of zeros.
The main topic of this compilation thesis is the investigation of multipartite entanglement of finite dimensional systems. We developed a numerical algorithm that detects if a multipartite state is entangled or separable in a finite number of steps of a semi-definite optimization task. This method is an extension of previously known semi-definite methods, which are inconclusive when the state is separable. In our case, if the state is separable, an explicit decomposition into a mixture of separable states can be extracted. This was achieved by mapping the entanglement problem onto the mathematically well studied truncated moment problem. Additionally, a new way of writing the partially transposed state for symmetric multi-qubit states was developed which simplifies many results previously known in entanglement theory. This new way of writing the partial transpose criterion unifies different interpretations and alternative formulations of the partial transpose criterion and it is also a part in the aforementioned semi-definite algorithm. The geometric properties of entangled symmetric states of two qubits were studied in detail: We could answer the question of how far a given pure state is from the convex hull of symmetric separable states, as measured by the Hilbert-Schmidt distance, by giving an explicit formula for the closest separable symmetric state. For mixed states we could provide a numerical upper and analytical lower bound for this distance. For a larger number of qubits we investigated the ball of absolutely classical states, i.e. symmetric multi-qubit states that stay separable under any unitary transformation. We found an analytical lower bound for the radius of this ball around the maximally mixed symmetric state and gave a numerical upper bound on this radius, by searching for an entangled state as close as possible to the maximally mixed symmetric state. The tensor representation of a symmetric multi-qubit state, or spin-j state, allowed us to study entanglement properties based on the spectrum of the tensor via tensor eigenvalues. The definiteness of this tensor relates to the entanglement of the state it represents and, hence, the smallest tensor eigenvalue can be used to detect entanglement. However, the tensor eigenvalues are more difficult to determine than the familiar matrix eigenvalues which made the investigation computationally more challenging. The relationship between the value of the smallest tensor eigenvalue and the amount of entanglement in the state was also investigated. It turned out that they are strongly correlated for small system sizes, i.e. for up to six qubits. However, to investigate this correlation we needed an independent way to gauge the amount of entanglement of a state and in order to do so we improved existing numerical methods to determine the distance of a state to the set of separable symmetric states, using a combination of linear and quadratic programming. The tensor representation of symmetric multi-qubit states was also used to formally define a new tensor class of regularly decomposable tensors that corresponds to the set of separable symmetric multi-qubit states.
This thesis addresses a number of problems within the emerging field of quantum information science. Quantum information science can be said to encompass the more-established disciplines of quantum computation and quantum information, as well as rather more recent attempts to apply concepts, tools and techniques from these disciplines to gain greater understanding of quantum systems in general. The role of entanglement — non-classical correlation — has been of particular interest to date. Part I contributes to this later goal. In particular, we establish a connection between the energy of a many-body quantum system and the idea of an entanglement witness from the theory of mixed-state entanglement. This connection allows mathematical results about entanglement witnesses to be translated into physical results about many-body quantum systems, specifically energy and temperature thresholds for entanglement. For the case of two qubits we are able to establish fairly detailed results about the behaviour of entanglement with temperature. We also study entanglement in systems of indistinguishable particles, where even the question of which quantum states should be regarded as entangled has been the subject of much controversy. We aim to clarify this issue by applying Wiseman and Vaccaro’s notion of entanglement of particles to a number of wellunderstood model systems. We discuss the advantages of the entanglement of particles approach compared with other methods in common use. Finally, we study the operational meaning of superselection rules in quantum physics, in particular the connection to the existence or not of an appropriate reference frame. We propose an experiment that aims to exhibit a coherent superposition of an atom and a molecule, apparently in violation of the commonly-accepted particle-number superselection rule. This result sheds light on the entanglement of particles approach to entanglement of indistinguishable particles. Part II returns to a fundamental question at the heart of quantum computation and quantum information, namely: how many quantum gates are required to perform a particular quantum computation? In other words, how efficiently can a quantum computer solve a particular computational problem? We establish a connection between this question and the field of Riemannian geometry. Intuitively, optimal quantum circuits correspond to “free-falling” along the shortest path between two points in a curved space. This opens up the possibility of using Riemannian geometry to study quantum computation, a possibility that was previously unknown. We provide explicit calculations of all the basic geometric quantities associated with the space, and give some preliminary results of applying geometric ideas to quantum computing. Finally, we explore more generally the connection between optimal control and quantum circuit complexity, of which the Riemannian metric described above can be viewed as a special case.
A quantum computer is similar to an ordinary computer, except that instead of bits it uses two-level quantum systems (qubits), and the usual logical operations on bits are replaced with noiseless quantum dynamics in the form of unitary gates or Hamiltonian interactions. Many sets of resources that are universal for quantum computation (that is, capable of implementing any quantum computation) have been found, but general conditions for universality are still only partially understood. Improving our understanding of these conditions contributes to two broad goals. On the one hand, understanding what class of physical systems is suitable for quantum computation allows us choose the systems that are most easily controlled or built in the laboratory. On the other hand, identifying the unifying characteristics of such a class provides insight into the fundamental physics responsible for the much greater computational power of quantum computers. The original results presented in this thesis are a contribution to our understanding of the physical requirements for universal quantum computation. These results lead to the identification of a new, large class of universal dynamics. More specifically, the main problem that is addressed is the following: consider a physical system that is made up of a finite number of qubits that are subject to a natural, fixed interaction described by a two-body Hamiltonian. Suppose we have the ability to control the system only by applying fast one-qubit operations. Under what conditions is this system capable of universal quantum computation? The first original result of the thesis is to prove that universal quantum computation is possible under these conditions given the single extra constraint that the Hamiltonian must entangle, or connect, all of the qubits. In other words, since a two-body Hamiltonian is expressible as a sum of coupling terms each of which acts nontrivially on a pair of qubits, consider the graph whose vertices correspond to qubits and whose edges connect qubits that are coupled by a term in the Hamiltonian. Then this Hamiltonian, together with fast one-qubit unitaries, is universal for quantum computation precisely when the corresponding graph is connected. This result suggests many potential generalizations, three of which are explored in this thesis. The first generalization is to show that the same result holds for a two-body Hamiltonian that acts on D-dimensional quantum systems (qudits), given fast one-qudit unitaries. A second extension is to the case of a many-body Hamiltonian, that is, a Hamiltonian that has coupling terms that act on more than two qubits. Again, we find that under mild restrictions, such Hamiltonians are universal provided they are connected and that arbitrary one-qubit unitaries are available. We also discover and explore some surprising structure in many-body Hamiltonians that is associated with the parity of the couplings. A third related problem that is addressed is the question of what two-qubit unitaries are universal for quantum computation. This problem was originally solved by J.-L. and R. Brylinski [BB02], who gave a technical proof of the universality of any two-qudit gate that is not the swap gate or a product of one-qudit unitaries, given one-qudit unitaries as a free resource. We develop an elementary and constructive proof for the case of qubits, and prove that our construction is very close to optimal in the number of uses of the two-qubit gate. Given a qualitative understanding of what dynamical resources are universal for quantum computation, it is natural to consider developing a quantitative measure of the usefulness of quantum dynamics for quantum information processing. This is done in the final chapter of the thesis, in which the beginnings of a quantitative theory of quantum dynamics are developed.
In this thesis, we have proposed some novel thought experiments involving foundations of quantum mechanics and quantum information theory, using quantum entanglement property. Concerning foundations of quantum mechanics, we have suggested some typical systems including two correlated particles which can distinguish between the two famous theories of quantum mechanics, i.e. the standard and Bohmian quantum mechanics, at the individual level of pair of particles. Meantime, the two theories present the same predictions at the ensemble level of particles. Regarding quantum information theory, two theoretical quantum communication schemes including quantum dense coding and quantum teleportation schemes have been proposed by using entangled spatial states of two EPR particles shared between two parties. It is shown that the rate of classical information gain in our dense coding scheme is greater than some previously proposed multi-qubit protocols by a logarithmic factor dependent on the dimension of Hilbert space. The proposed teleportation scheme can provide a complete wave function teleportation of an object having other degrees of freedom in our three-dimensional space, for the first time. All required unitary operators which are necessary in our state preparation and Bell state measurement processes are designed using symmetric normalized Hadamard matrix, some basic gates and one typical conditional gate, which are introduced here for the first time.
This thesis contains four projects: entanglement and quantum phase transition in one-dimensional quantum dots system; entanglement and electron correlation in molecular systems; quantum teleportation in one-dimensional quantum dots system and simulation of molecular systems on a quantum computer. Chapter 1 presents a general introduction to quantum computing. A brief history and motivation for quantum computing is introduced. In Chapter 2, we studied the entanglement of an array of quantum dots system modeled by one-dimensional Hubbard Hamiltonian and the relation between quantum entanglement and quantum phase transition. In Chapter 3, entanglement in molecular systems is studied. It is proved that the full configuration interaction (FCI) wave function violates Bell’s inequality, thus it is entangled; while the Hartree-Fock (HF) wave function does not violate Bell’s inequality, so it is unentangled. Based on this, we studied the relation between quantum entanglement and electron correlation in molecular systems. We suggested that quantum entanglement might be used as an alternative way to measure electron correlation in molecular systems. The application of entanglement in an array of quantum dots system in quantum teleportation is studied in Chapter 4. We showed that the space-spin entanglement in the quantum dots system can be filtered to obtain pure spin (space) entanglement, and it can be used in quantum teleportation. We also showed that by selecting appropriate Hamiltonian for the C-NOT gate, the spin-based information can be transformed into charge-based information. The pulse sequence for realizing the C-NOT operation is given. In Chapter 5, we studied the simulation of molecular systems on a quantum computer. We proposed a quantum algorithm to solve the Schrödinger equation using the basis set method based on the multi-configuration self-consistent (MCSCF) wave function. By using this algorithm, the potential energy surfaces of molecules, even some complicated regions, can be studied more efficiently than if the simpler HF wave function was employed. This algorithm can give high success probability, almost deterministic results for ground state and excited states. A small increase in the number of configuration state functions in the MCSCF wave function can dramatically increase the success probability of the quantum algorithm. A multi-reference configuration interaction approach is suggested for the treatment of larger systems. We generalized this method to the finite element method and simulated a quantum system of one particle in Yukawa potential. We showed that numerical methods for solving differential equations can be implemented efficiently on a quantum computer.