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Acta Physica Sinica

Properties of quantum correlations in the Yang-Baxter spin-1/2 chain mode

Lidan Gou, Xiaoqian Wang

Published Jan 1, 2015• 1 citation
Publisher

Abstract

Quantum correlations among different parts of a composite quantum system are the fundamental resource of several applications in quantum information. In general, quantum discord can measure quantum correlations. In that way, the quantum correlations in the Yang-Baxter spin-1/2 chain mode are investigated. In the second part of the paper, the Yang-Baxter spin-1/2 chain modes are constructed from the Yang-Baxter equation. First, we analyze the two matrix representations of Temperly-Lieb algebra. Second, the two solutions of the Yang-Baxter equation are generated using the Yang-Baxterization. Finally, we can change the usual two-particle spin-1/2 chain to the Yang-Baxter spin-1/2 chain modes by means of the unitary Yang-Baxter matrix-R. In the third part, the density matrices of the two chain modes are generated in the thermal equilibrium state in a canonical ensemble. According to the definition of the geometric measure of quantum discord, the analytical expressions of the geometric measure of quantum discord, in the temperature and the external magnetic field, are obtained for the Yang-Baxter spin-1/2 chain modes. When the temperature and the magnetic field intensity increase, the geometric measure of quantum discord decreases. Under the specific conditions, the result of the second chain mode is similar to that of the first one. Then we obtain the numerical results of quantum discord, the geometric measure of quantum discord, and concurrence. It is found that the concurrence can quickly decrease to the value of zero when the temperature is greater than the value of one. At the same time, quantum discord and the geometric measure of quantum discord are not of the value of zero. Thus the quantum discord and the geometric measure of quantum discord can go beyond the concept of entanglement and obtain the “quantumness” of the correlations between the two parts of a system for the Yang-Baxter spin-1/2 chain modes. They are very good quantum resources for quantum information and quantum computing.

DOI
10.7498/aps.64.070302

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Quantum correlations among different parts of a composite quantum system are the fundamental resource of several applications in quantum information. In general, quantum discord can measure quantum correlations. In that way, the quantum correlations in the Yang-Baxter spin-1/2 chain mode are investigated. In the second part of the paper, the Yang-Baxter spin-1/2 chain modes are constructed from the Yang-Baxter equation. First, we analyze the two matrix representations of Temperly-Lieb algebra. Second, the two solutions of the Yang-Baxter equation are generated using the Yang-Baxterization. Finally, we can change the usual two-particle spin-1/2 chain to the Yang-Baxter spin-1/2 chain modes by means of the unitary Yang-Baxter matrix-R. In the third part, the density matrices of the two chain modes are generated in the thermal equilibrium state in a canonical ensemble. According to the definition of the geometric measure of quantum discord, the analytical expressions of the geometric measure of quantum discord, in the temperature and the external magnetic field, are obtained for the Yang-Baxter spin-1/2 chain modes. When the temperature and the magnetic field intensity increase, the geometric measure of quantum discord decreases. Under the specific conditions, the result of the second chain mode is similar to that of the first one. Then we obtain the numerical results of quantum discord, the geometric measure of quantum discord, and concurrence. It is found that the concurrence can quickly decrease to the value of zero when the temperature is greater than the value of one. At the same time, quantum discord and the geometric measure of quantum discord are not of the value of zero. Thus the quantum discord and the geometric measure of quantum discord can go beyond the concept of entanglement and obtain the “quantumness” of the correlations between the two parts of a system for the Yang-Baxter spin-1/2 chain modes. They are very good quantum resources for quantum information and quantum computing.
Quantum correlations among parts of a composite quantum system are a fundamental resource for several applications in quantum information. In general, quantum discord can measure quantum correlations. In that way, we investigate the quantum discord of the two-qubit system constructed from the Yang—Baxter Equation. The density matrix of this system is generated through the unitary Yang—Baxter matrix R̆. The analytical expression and numerical result of quantum discord and geometric measure of quantum discord are obtained for the Yang—Baxter system. These results show that quantum discord and geometric measure of quantum discord are only connect with the parameter θ, which is the important spectral parameter in Yang—Baxter equation.
By adopting the concept of the geometric measure of quantum discord, we explore the property of quantum correlation in the two-spin Heisenberg model, gain the analytic expression of quantum discord in the general case, and discuss the influences of the coupling constant, temperature, the intensity of the external magnetic field on magnitude of the quantum correlation. The corresponding scheme of tuning quantum correlation is also given in this paper. In addition, we find that quantum discord has a sudden transition in the lower temperature. Results show that adjusting systematic parameters, which are temperature, coupling strength, magnetic field intensity, etc, is an effective way to control the value of quantum correlation in the double spin Heisenberg model system. This provides a certain reference and significant guidance for the precise control of quantum discord and realizing the teleportation of quantum state and the design of quantum logic gates.
Quantum entanglement plays a key role in quantum information and quantum computation and thus attracts much attention in many branches of physics both in theory and in experiment. But recent studies revealed that some separable states (non-entangled state) may speed up certain tasks over their classical counterparts and may also possess certain kinds of quantum correlations. For example, geometric quantum discord, which is a more general quantum correlation measure than entanglement, can be nonzero for some separable states. From a practical point of view, it is proposed that the geometric quantum discord be responsible for the power of many quantum information processing tasks. In order to capture such correlations, Ollivier and Zurek introduced quantum discord, which measures the discrepancy between two natural yet different quantum analogues of two classically equivalent expressions of mutual information. However, the calculation of quantum discord is based on numerical maximization procedure, and there are few analytical expressions even for a two-qubit state. In order to obtain the analytical results of quantum discord, a geometric measure of quantum discord which measures the quantum correlations through the minimum Hilbert-Schmidt distance between the given state and zero discord state is introduced. Geometric quantum discord is defined as an effective measure of quantum correlation, and the geometric quantum discord through the minimal distance between the quantum state and the set of zero-discord states in a bipartite quantum system can be worked out. In this paper, by using the geometric quantum discord measurement method, the geometric quantum discord in Tavis-Cummings model is investigated, and the influences of the initial state purity, entanglement degree, dipole-dipole coupling intensity between two atoms, and field in the Fock state on the evolution characteristic of geometric quantum discord are analyzed. The results show that the geometric quantum discord appears periodically. It initially decreases to a minimum value, and then turns out to be increased for different initial states. The rigorous analysis and numerical results reveal that when we take a suitable initial state, the geometric quantum discord of two atoms can be kept in correlation. When the atoms are in the different initial states, the quantum properties of the system are significant. The photon number of the field can lead the quantum discord to be weakened. Geometric quantum discord can be increased by increasing the cavity photon number and the dipole-dipole coupling intensity. Geometric quantum discord can be enhanced obviously by increasing the strength of the dipole-dipole coupling interaction. The conclusions may conduce to the understanding of quantum correlation for the other systems from the view of geometric quantum discord.
We study the scaling of quantum discord (a measure of quantum correlation beyond entanglement) in spin models analytically and systematically. We find that at finite temperature the block scaling of quantum discord satisfies an area law for any two-local Hamiltonian. We show that generically and heuristically the two-site scaling of quantum discord is similar to that of correlation functions. In particular, at zero temperature it decays exponentially and polynomially in gapped and gapless (critical) systems, respectively; at finite temperature it decays exponentially in both gapped and gapless systems. We compute the two-site scaling of quantum discord in the XXZXXZXXZ chain, the XYXYXY chain (in a magnetic field), and the transverse field Ising chain at zero temperature.
Unitary invariant discord as a measure of bipartite quantum correlations in an NNN-qubit quantum system
We introduce a measure of quantum correlations in the NNN-qubit quantum system which is invariant with respect to the SU(2N)SU(2^N)SU(2N) group of transformations of this system. This measure is a modification of the quantum discord introduced earlier and is referred to as the unitary or SU(2N)SU(2^N)SU(2N)-invariant discord. Since the evolution of a quantum system is equivalent to the proper unitary transformation, the introduced measure is an integral of motion and is completely defined by eigenvalues of the density matrix. As far as the calculation of the unitary invariant discord is rather complicated computational problem, we propose its modification which may be found in a simpler way. The case N=2 is considered in details. In particular, it is shown that the modified SU(4)-invariant discord reaches the maximum value for a pure state. {A geometric measure of the unitary invariant discord of an NNN-qubit state is introduced and a simple formula for this measure is derived, which allows one to consider this measure as a witness of quantum correlations.} The relation of the unitary invariant discord with the quantum state transfer along the spin chain is considered. We also compare the modified SU(4)-invariant discord with the geometric measure of SU(4)-invariant discord of the two-qubit systems in the thermal equilibrium states governed by the different Hamiltonians.
The thermal quantum correlations of a three-qubit spin chain of XY type under an in-homogeneous magnetic field have been investigated, where by considering magnetic fields with different magnitudes [Formula: see text], [Formula: see text] and [Formula: see text] for each spin, the quantum correlations between the adjacent and the non-adjacent qubits (spins) have been studied. For this purpose, the concurrence as an entanglement quantifier and the trace distance discord as a discord-like quantifier have been computed for the corresponding bipartite subsystems with the reduced density matrices [Formula: see text], [Formula: see text] and [Formula: see text]. The differences between the concurrence and the trace distance discord, are explained in detail. Their dependence on the parameters of the uniform magnetic field [Formula: see text], the temperature [Formula: see text], and the in-homogeneity parameter [Formula: see text] is discussed so that the thermal correlation decreases by increasing the mentioned parameters. Also, the results show that by changing and controlling the parameters [Formula: see text], [Formula: see text] and [Formula: see text], especially the in-homogeneous magnetic field [Formula: see text], it is possible that the correlation value between non-adjacent spins, i.e. [Formula: see text] can be reached to a greater value than the thermal correlations between adjacent spins, i.e. [Formula: see text] and [Formula: see text]. Moreover, it is shown that the trace distance [Formula: see text] between the non-adjacent spins has always a lower value than those of the adjacent spins [Formula: see text] and [Formula: see text]. Finally, the results show that in-homogeneous magnetic field b can be effective for improving thermal quantum correlation.