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Cálculo do entrelaçamento de estados puros com quatro a seis qubits

David Sena Oliveira

Published Jan 1, 2012• 0 citations
OpenAlex

Abstract

This thesis is divided in three parts. The first one brings a brief review of the basic concepts about entanglement of quantum states with two and multiple qubits, including the entanglement monotones and the Groverian entanglement measure, which is calculated by a genetic algorithm. The second part is focused on the generation of a entanglement measures that identify only four qubit states with genuine entanglement. Some examples of the calculus of the entanglement of four qubit states, using the proposed measure, when these states are processed by quantum circuits, are presented. In the third part, the proposed measure is extended, to calculate the genuine entanglement of pure six qubits quantum states. Numerical simulations were realized aiming to show the variation, creation and annihilation of the entanglement of pure six qubits states.

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This thesis is divided in three parts. The first one brings a brief review of the basic concepts about entanglement of quantum states with two and multiple qubits, including the entanglement monotones and the Groverian entanglement measure, which is calculated by a genetic algorithm. The second part is focused on the generation of a entanglement measures that identify only four qubit states with genuine entanglement. Some examples of the calculus of the entanglement of four qubit states, using the proposed measure, when these states are processed by quantum circuits, are presented. In the third part, the proposed measure is extended, to calculate the genuine entanglement of pure six qubits quantum states. Numerical simulations were realized aiming to show the variation, creation and annihilation of the entanglement of pure six qubits states.
The Groverian entanglement measure $G(\ensuremath{\psi})$ is applied to characterize pure quantum states $\ensuremath{\mid}\ensuremath{\psi}⟩$ of multiple qubits. This is an operational measure of entanglement in the sense that it quantifies the utility of the state $\ensuremath{\mid}\ensuremath{\psi}⟩$ as an initial state for the search algorithm. A convenient parametrization is presented, which allows us to calculate the Groverian measure analytically for certain states of high symmetry. A numerical procedure is used in order to calculate it for arbitrary pure states of multiple qubits. Using the Groverian measure to evaluate the entanglement produced by quantum algorithms may provide useful insight into the role of entanglement in making quantum algorithms powerful. Here we calculate $G(\ensuremath{\psi})$ for the intermediate states generated during the evolution of Grover's algorithm for various initial states and for different sets of marked states. It is shown that Grover's iterations generate highly entangled states in intermediate stages of the quantum search process, even if the initial state and the target state are product states.
This thesis presents some results about two important subjects of the quantum information theory: quantum entanglement and quantum algorithms. Regarding the entanglement, a relationship between negativity and minors principals of Hermitian matrices was stablished, allowing the analysis of the entanglement of two qubits states, three and four qubits pure states, using the minors principals. It was also proposed a new measure, using negativity, for calculating the entanglement of pure states of six qubits. Finally, with respect to quantum algorithms it was proposed a quantum search algorithm able to finding the minimum of a function by performing only one measurement. The algorithm is described and an example of its usage in the calculation of the period of a periodic function is presented.
The Groverian entanglement measure, introduced earlier for pure quantum states of multiple qubits [O. Biham, M.A. Nielsen, and T. Osborne, Phys. Rev. A 65, 062312 (2002)], is generalized to the case of mixed states. The Groverian measure of a mixed state of nnn qubits is obtained by a purification procedure into a pure state of 2n2n2n qubits, followed by an optimization process, before the resulting state is fed into Grover's search algorithm. It is expressed in terms of the maximal success probability of the algorithm and in this sense provides an operational measure of entanglement.
The Groverian entanglement measure of pure quantum states of nnn qubits is generalized to the case in which the qubits are divided into any $p\ensuremath{\leqslant}n$ parties. The entanglement between these parties is evaluated numerically using an efficient parametrization. To demonstrate this measure we apply it to symmetric states such as the Greenberg-Horne-Zeiliner state and the W state. Interestingly, this measure is equivalent to an entanglement measure introduced earlier [H. Barnum and N. Linden, J. Phys. A 34, 6787 (2001)], using different considerations.
Entanglement is a key resource in many quantum information schemes andin the last years the research on multi-qubit entanglement has drawn lots ofattention. In this thesis the experimental generation and characterization ofmulti-qubit entanglement is presented. The qubits are implemented in the polarizationdegree of freedom of photons and we have prepared genuine entangledstates of two, four and six photons. We emphasize that one type of states thatwe produce are invariant entangled states, remaining unchanged under simultaneousidentical unitary transformations of all their individual constituents.Such states can be applied to e.g. decoherence-free encoding, quantum communicationwithout sharing a common reference frame, quantum telecloning,secret sharing and remote state preparation schemes.In the experimental implementation we use a single source of entangledphoton pairs and extract the first, second and third order parametric downconversion.The multi-order processes are not entirely spontaneous, as we getthe right states utilizing bosonic emission enhancement due to indistinguishability.Despite the achievement of six-photon entangled states, is the setupcompletely free from interferometric overlaps making it robust and contributingto high fidelities of the generated states. The analysis results of our experimentalstates are in very good agreement with theory and also show very highvisibilities in their correlations.