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International Journal of Modern Physics B

Entanglement and quantum spin glass

C.Y. Koh

Published Jun 11, 2014• 4 citations
Publisher

Abstract

The paper reviews the entanglement behavior of a 2-qubit system in a quantum spin glass using the Heisenberg XX model and the interaction of the system with a spin glass bath environment. In the first part, we study the entanglement (concurrence) for a 3- and 4-qubit with nearest neighbor interaction. With a fixed mean and varying standard deviation for the J coupling, the concurrence is numerically plotted with temperature for the different configurations. A general formula for the concurrence is given for n qubits at low temperature. In the second part, we study the concurrence of a 2-qubit system coupled to a spin glass bath environment with n = 2 to ≥ 4 qubits. The bath sites are coupled with random J coupling and varying applied magnetic field. A general formula for concurrence is given for mean J = 0 and B = 0 for n bath sites. For small random J and magnetic field B, a steady state is obtained with an approximate concurrence of 0.5, showing that the entanglement is preserved.

DOI
10.1142/s0217979214300126

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The paper reviews the entanglement behavior of a 2-qubit system in a quantum spin glass using the Heisenberg XX model and the interaction of the system with a spin glass bath environment. In the first part, we study the entanglement (concurrence) for a 3- and 4-qubit with nearest neighbor interaction. With a fixed mean and varying standard deviation for the J coupling, the concurrence is numerically plotted with temperature for the different configurations. A general formula for the concurrence is given for n qubits at low temperature. In the second part, we study the concurrence of a 2-qubit system coupled to a spin glass bath environment with n = 2 to ≥ 4 qubits. The bath sites are coupled with random J coupling and varying applied magnetic field. A general formula for concurrence is given for mean J = 0 and B = 0 for n bath sites. For small random J and magnetic field B, a steady state is obtained with an approximate concurrence of 0.5, showing that the entanglement is preserved.
Well-known theories like Edwards-Anderson and Sherrington-Kirkpatrick models together with some important experimental results for spin glass are introduced in chapter 1. However, there is no single model, capable of explaining all the experimental results obtained from the spin glass materials. We use the theory of quantum entanglement (concurrence) to explore the quantum correlations of a spin glass. Quantum entanglement is a useful physical resource for quantum information and communication. In chapter 2, we discuss how information is encoded and measured. The applications of quantum information in algorithms are also mentioned. As quantum information science is built upon the foundation of quantum mechanics, two important aspects of this area: Einstein, Podolsky and Rosen (EPR) gedanken experiment and the Bell's inequality are introduced in chapter 3. In chapter 4, we discuss the idea of density operator, together with the use of trace and partial trace, to describe the state of the composite system. In chapter 5, a short introduction is made on the various entanglement measures. In chapter 6, the analytical calculations of the concurrence for a 3- and 4-qubit XX model are presented. The plots of concurrence with different means and standard deviations for the 3- and 4-qubit are also presented and analyzed. The concurrence of a 2-qubit system coupled to a spin glass bath of n = 2 to 4 qubits is studied in chapter 7. With an initial separable state, the 2-qubit system reached a steady state in concurrence through time evolution when it is coupled to a quantum spin glass environment. The result is general and the behavior of the concurrence is believed to be similar for n > 4. The quantum discord is used for analyzing the XX model and presented in chapter 8. In addition, the concurrence and discord for the next nearest neighbour interaction have been plotted for 3- and 4-qubit. In chapter 9, the concurrence of the probes that are attached to a spin glass is investigated by varying the external applied magnetic field and the J coupling. By using the quantum dots (qubits) as the probe, it is possible to distinguish a spin glass from a non-spin glass with the different signatures (drop and spike) in the concurrence between the probes. The derived entanglement witness for susceptibility and specific heat capacity are used to distinguish an entangled region from an unentangled region for two different spin glass alloys: LiHoxY1-xF4 and CuxMn in chapter 10. In chapter 11, the entanglement properties of the XXZ Heisenberg (with Dzyaloshinskii-Moriya interaction) and Ising transverse field spin glass models are examined with the use of concurrence for N qubits at one-dimensional level. The concurrence for both models are obtained by using the Kadanoff''s renormalization group (RG) approach. By following the RG fl ow, a first-order and second-order phase transitions are observed by varying the standard deviation of the renormalized couplings. This thesis ends with a conclusion in chapter 12 with suggestions for future research.
We study entanglement and discord in spin glass. By using the Edwards–Anderson XX model, we consider the Hamiltonian for three and four particles for nearest neighbor and next nearest neighbor interactions and to solve for the entanglement through concurrence. With a fixed mean and changing variances for the coupling J , the concurrence and discord are numerically plotted against the temperature. For J = 1, the critical temperature is found to be 2.106. The entanglement was found to increase beyond the critical temperature as the variance increases. The spin–spin correlation for three particles is plotted as a function of temperature T .
The entanglement of two qubits, each defined as an effective two-level, spin 1/2 system, is investigated for the case that the qubits interact via a Heisenberg XY interaction and are subject to decoherence due to population relaxation and thermal effects. For zero temperature, the time-dependent concurrence is studied analytically and numerically for some typical initial states, including a separable (unentangled) initial state. An analytical formula for non-zero steady state concurrence is found for any initial state, and optimal parameter values for maximizing steady state concurrence are given. The steady state concurrence is found analytically to remain non-zero for low, finite temperatures. We also identify the contributions of global and local coherence to the steady state entanglement.
The transitory and steady-state entanglement of a two-qubit XY Heisenberg magnetic spin system immersed in an external magnetic field are analyzed in terms of the entanglement measure of concurrence. The dependence of the system entanglement on coupling strength, decay rate, and applied magnetic field is numerically simulated for two distinct models of system evolution involving decoherence to the environment. An applied magnetic field was found to consistently reduce the system entropy. The external magnetic field increases entanglement for small strength, and if too strong, it reduces the entanglement. The maximum entanglement is formed from a balance of external magnetic field and the decay rate. The transitory level of entanglement in the system is found to oscillate with a frequency that is proportional to the magnetic field strength before reaching steady-state entanglement. It was found that collective decay produces more entanglement than individual decay.
We study the decoherence of a qubit weakly coupled to frustrated spin baths. We focus on spin baths described by the classical Ising spin glass and the quantum random transverse Ising model which are known to have complex thermodynamic phase diagrams as a function of an external magnetic field and temperature. Using a combination of numerical and analytical methods, we show that for baths initially in thermal equilibrium, the resulting decoherence is highly sensitive to the nature of the coupling to the environment and is qualitatively different in different parts of the phase diagram. We find an unexpected strong non-Markovian decay of the coherence when the random transverse Ising model bath is prepared in an initial state characterized by a finite temperature paramagnet. This is contrary to the usual case of exponential decay (Markovian) expected for spin baths in finite temperature paramagnetic phases, thereby illustrating the importance of the underlying nontrivial dynamics of interacting quantum spinbaths.
We consider the buildup and decay of two-spin entanglement through phase interactions in a finite environment of surrounding spins, as realized in quantum computing platforms based on arrays of atoms, molecules, or nitrogen vacancy centers. The non-Markovian dephasing caused by the spin environment through Ising-type phase interactions can be solved exactly and compared to an effective Markovian treatment based on collision models. In a first case study on a dynamic lattice of randomly hopping spins, we find that non-Markovianity boosts the dephasing rate caused by nearest-neighbor interactions with the surroundings, degrading the maximum achievable entanglement. However, we also demonstrate that additional three-body interactions can mitigate this degradation, and that randomly timed reset operations performed on the two-spin system can help sustain a finite average amount of steady-state entanglement. In a second case study based on a model nuclear magnetic resonance system, we elucidate the role of bath correlations at finite temperature on non-Markovian dephasing. They speed up the dephasing at low temperatures while slowing it down at high temperatures, compared to an uncorrelated bath, which is related to the number of thermally accessible spin configurations with and without interactions.