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International Journal of Quantum Information

Freezing of geometric discords for two qutrits in environments

Wei Xiao, Mei-Yun Zhen, Xi-Wen Hou

Published Jul 10, 2022• 3 citations
Publisher

Abstract

Several measures of geometric discords have been recently proposed and mostly characterized in qubits. In this work, the dynamics of four measures of geometric discords is studied for two kinds of initial states in the two-qutrit model under environments. It is shown that four indicators can display freezing in Markovian regime and multiple freezing with multiple sudden changes in non-Markovian regime for suitable initial states and model parameters. Moreover, those measures evolve in a similar manner, but they differ from the decaying or increasing rate, the oscillatory amplitude and the time interval of freezing. However, they do not exhibit any freezing for the general Bell states. The dynamical hierarchy is discussed for those indicators. Those are helpful to understand the rich phenomena of discordant dynamics in open qutrits.

DOI
10.1142/s0219749922500198

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Several measures of geometric discords have been recently proposed and mostly characterized in qubits. In this work, the dynamics of four measures of geometric discords is studied for two kinds of initial states in the two-qutrit model under environments. It is shown that four indicators can display freezing in Markovian regime and multiple freezing with multiple sudden changes in non-Markovian regime for suitable initial states and model parameters. Moreover, those measures evolve in a similar manner, but they differ from the decaying or increasing rate, the oscillatory amplitude and the time interval of freezing. However, they do not exhibit any freezing for the general Bell states. The dynamical hierarchy is discussed for those indicators. Those are helpful to understand the rich phenomena of discordant dynamics in open qutrits.
The investigation of quantum discord has mostly focused on two-qubit systems due to the complicated minimization involved in quantum discord for high-dimensional states. In this work, three geometric discords are studied for the thermal state in a two-qutrit system with various couplings, external magnetic fields, and temperatures as well, where the entanglement measured in terms of the generalized negativity is calculated for reference. It is shown that three geometric discords are more robust against temperature and magnetic field than the entanglement negativity. However, all four quantities exhibit a similar behavior at lower temperature and weak magnetic field. Remarkably, three geometric discords at finite temperature reveal the phenomenon of double sudden changes at different magnetic fields while the negativity does not. Moreover, the hierarchy among three discords is discussed. Those adjustable discords with the varied coupling, temperature, and magnetic field are useful for the understanding of quantum correlations in high-dimensional states and quantum information processing.
Geometric quantum discord is a well-defined measure of quantum correlation if Schatten one-norm (trace norm) is adopted as a distance measure. Here, we analytically investigate the dynamical behavior of the one-norm geometric quantum discord under the effect of decoherence. By starting from arbitrary Bell-diagonal mixed states under Markovian local noise, we provide the decays of the quantum correlation as a function of the decoherence parameters. In particular, we show that the one-norm geometric discord exhibits the possibility of double sudden changes and freezing behavior during its evolution. For nontrivial Bell-diagonal states under simple Markovian channels, these are new features that are in contrast with the Schatten two-norm (Hilbert-Schmidt) geometric discord. The necessary and sufficient conditions for double sudden changes as well as their exact locations in terms of decoherence probabilities are provided. Moreover, we illustrate our results by investigating decoherence in quantum spin chains in the thermodynamic limit.
The level surfaces of geometric discord for a class of two-qubit non-X states are investigated when the Bloch vectors are in arbitrary directions. The level surfaces of constant geometric discord are formed by three intersecting open tubes along three orthogonal directions. When Bloch vectors increase, the tubes along one or two directions shrink towards the center and may either totally disappear or the open tubes may become closed tubes when the Bloch vectors reach a critical value. In the generalized amplitude damping channel, the evolution of geometric discord shows double sudden changes when the parameter γ increases. In the phase damping channel, the freezing phenomenon of geometric discord also exists.
The dynamics of a geometric measure of the quantum discord (GMQD) under decoherence is investigated. We show that the GMQD of a two-qubit state can be alternatively obtained through the singular values of a 3×43\times43×4 matrix whose elements are the expectation values of Pauli matrices of the two qubits. By using Heisenberg picture, the analytic results of the GMQD is obtained for three typical kinds of the quantum decoherence channels. We compare the dynamics of the GMQD with that of the quantum discord and of entanglement. We show that a sudden change in the decay rate of the GMQD does not always imply that of the quantum discord, and vice versa. We also give a general analysis on the sudden change in behavior and find that at least for the Bell diagonal states, the sudden changes in decay rates of the GMQD and that of the quantum discord occur simultaneously.
Abstract A two qubits system interacting with two independent spin‐environments connecting with the third environment is constructed in order to demonstrate the effect of the multi‐environment for quantum correlation. In this process, the freezing phenomenon appears for SCI and X states under Quantum Discord and Geometric Discord measures, respectively, but not for the same initial state measured by different measures. Meanwhile, the properties of the freezing platform, characterized by the collapse, revival and persistent time, are researched by the different parameters. The result of this paper may pave a way to control quantum correlation and design nanospintronic devices.
The characterization of quantum discord (QD) and geometric discord (GD) has mostly concentrated on two-qubit states since the minimization in both discords is a daunting task for high-dimensional states. Numerical calculations of both discords are carried out for a generic bipartite state. When one-dimensional orthogonal projectors for a local measurement on n-dimensional Hilbert space are realized by the generators and the Euler angles of SU (n), the optimal measurements have a figure of merit that includes n(n - 1) Euler parameters. As an representative example, such projectors and two kinds of algorithms are used to estimate both discords for two-qutrit mixed states in recent literature. The generalized negativity as a measure of quantum entanglement is calculated for reference purposes. For those states with one parameter the discords and the negativity respectively display the nonlinear and the linear function of the parameter, with different turning points. However, they are positively correlated in the suitable ranges of the parameter for those states. The hierarchy of those quantities is discussed as well. Those shed new light on the understanding of QDs and quantum entanglement of mixed states in high-dimensions.