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arXiv:quant-ph/0608191

Simulation of an entangled state in a chain of three nuclear spins system

Gustavo V. López, Lorena Lara

Published Aug 24, 2006• 0 citations
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Abstract

We study the formation of an entangled state in a one-dimensional chain of three nuclear spins system which interact weakly through the Ising type of interaction and taking into account first and second neighbor interactions. We can get this entangled state using two pulses ($π/2$ and $π$ pulses), and we study the efficiency of getting this entangled state as a function of the ratio of the second neighbor interaction coupling constant to the first neighbor interaction coupling constant ($J'/J$). We found that for $J'/J\ge 0.04$, the entangled state is well defined.

DOI
10.48550/arxiv.quant-ph/0608191

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We study the formation of an entangled state in a one-dimensional chain of three nuclear spins system which interact weakly through the Ising type of interaction and taking into account first and second neighbor interactions. We can get this entangled state using two pulses (π/2π/2π/2 and πππ pulses), and we study the efficiency of getting this entangled state as a function of the ratio of the second neighbor interaction coupling constant to the first neighbor interaction coupling constant (J′/JJ'/JJ′/J). We found that for J′/J≥0.04J'/J\ge 0.04J′/J≥0.04, the entangled state is well defined.
For a one-dimensional chain of three nuclear spins (one half), we make the numerical simulation of quantum teleportation of a given state from one end of the chain to the other end, taking into account first and second neighbor interactions among the spins. It is shown that a well defined teleportation protocol is achieved for a ratio of the first to second neighbor interaction coupling constant of J′/J≥0.04J'/J\ge 0.04J′/J≥0.04. We also show that the optimum Rabi's frequency to control the non-resonant effects is dominated by the second neighbor interaction coupling parameter (J′J'J′).
We present the study of a quantum Controlled-Controlled-Not gate, implemented in a chain of three nuclear spins weakly Ising interacting between all of them, that is, taking into account first and second neighbor spin interactions. This implementation is done using a single resonant πππ-pulse on the initial state of the system (digital and superposition). The fidelity parameter is used to determine the behavior of the CCN quantum gate as a function of the ratio of the second neighbor interaction coupling constant to the first neighbor interaction coupling constant (J′/JJ'/JJ′/J). We found that for J′/J≥0.02J'/J\ge 0.02J′/J≥0.02 we can have a well defined CCN quantum gate.
We present the study of a quantum Controlled-Controlled-Not gate, implemented in a chain of three nuclear spins weakly Ising interacting between all of them, that is, taking into account first and second neighbor spin interactions. This implementation is done using a single resonant pi\\pipi-pulse on the initial state of the system (digital and superposition). The fidelity parameter is used to determine the behavior of the CCN quantum gate as a function of the ratio of the second neighbor interaction coupling constant to the first neighbor interaction coupling constant (J′/JJ'/JJ′/J). We found that for J′/Jge0.02J'/J\\ge 0.02J′/Jge0.02 we can have a well defined CCN quantum gate.
Quantum simulations, where controllable quantum systems are used to simulate other quantum systems, originally proposed by Richard Feynman, are one of the most remarkable applications of quantum information science. Compared to computation, quantum simulations require much less number of qubits for the m to be practical. In the work described in this thesis, we have performed a few quantum simulations of small quantum systems using Nuclear Magnetic Resonance(NMR) techniques. These simulations have been used to experimentally demonstrate the underlying interesting quantum protocols. All the experiments presented have been carried out using liquid-state or liquid crystal NMR. Numerical pulse optimization techniques have been utilized in some of the experiments, to achieve better control over the spin systems. The first chapter contains “Introduction” to quantum information processing, NMR, and numerical pulse optimization techniques. In chapter 2, we describe quantum simulation of a 3-spin Heisenberg-XY spin chain having only nearest neighbour interactions. Recently, spin chains having pre-engineered short-range interactions have been proposed to efficiently transfer quantum information between different parts of a quantum information processor. Other important proposals involving these spin chains include generating entangled states and universal quantum computation. However, such engineered interactions do not occur naturally in any system. In such a scenario, the experimental viability of these proposals can be tested by simulating the spin chains in other controllable quantum systems. In this work, we first theoretically study the time evolution of bipartite and tripartite entanglement measures for a 3-spin open ended XY spin chain. Then, by simulating the XY interactions in a 3-spin nuclear spin system, we experimentally generate, (i)a bipartite maximally(pseudo-)entangled state(Bell state) between end qubits, and(ii) multipartite(pseudo-)entangled states(Wand GHZ states),starting from separable pseudo-pure states. Bell state has been generated by using only the natural unitary evolution of the XY spin chain. W-state and GHZ-state have been generated by applying a single-qubit rotation to the second qubit, and a global rotation of all the three qubits respectively after the unitary evolution of the spin chain. In chapter 3, we simulate a 3-spin quantum transverse Ising spin system in a triangular configuration, and show that multipartite quantum correlations can be used to distinguish between the frustrated and non-frustrated regimes in the ground state of this spin system. The ground state of the spin system has been prepared by using adiabatic state preparation method. Gradient ascent pulse engineering technique has been utilized to efficiently realize the adiabatic evolution of the spin system. To analyse the experimental ground state of the system, we employ two different multipartite quantum correlation measures, generated from monogamy studies of bipartite quantum correlations. Chapter 4 contains a digital…
The evolution of entanglement in a 3-spin chain with nearest-neighbor Heisenberg-XY interactions for different initial states is investigated here. In an NMR experimental implementation, we generate multipartite entangled states starting from initial separable pseudo-pure states by simulating nearest-neighbor XY interactions in a 3-spin linear chain of nuclear spin qubits. For simulating XY interactions, we follow algebraic method of Zhang et al. [Phys. Rev. A72 (2005) 012331]. Bell state between end qubits has been generated by using only the unitary evolution of the XY Hamiltonian. For generating W-state and GHZ-state a single qubit rotation is applied on second and all the three qubits, respectively after the unitary evolution of the XY Hamiltonian.
The evolution of entanglement in a one-dimensional Ising chain is numerically studied under various initial conditions. We analyze two problems concerning the dynamics of entanglement: (i) generation of the entanglement from the pseudopure separable state and (ii) transportation of the entanglement from one end of the chain to the other. The model investigated is a one-dimensional Ising spin-1/2 chain with nearest-neighbor interactions placed in an external magnetic field and irradiated by a weak resonant transverse field. The possibility of selective initialization of partially entangled states is considered. It was shown that, in spite of the use of a model with direct interactions between the nearest neighbors, entanglement between remote spins is generated.