In order to measure the quantum correlation of a bipartite state quickly, an easy method is to construct a test matrix through the commutations among the blocks of its density matrix. Then, the Frobenius norm of the test matrix can be used to measure the quantum correlation. In this paper, we apply the measurement by Frobenius norm ([Formula: see text] to the dynamics evolution of the non-Markovian quantum system and compare it with the typical quantum discord ([Formula: see text] proposed by Ollivier and Zurek. The research results show that [Formula: see text] can indeed measure the quantum correlation of a bipartite state as same as [Formula: see text]. Further studies find that there are still differences between the two measurements: in some regions, when [Formula: see text] is zero, [Formula: see text] is not zero. It indicates that [Formula: see text] is more detailed than [Formula: see text] to measure quantum correlation of a bipartite state.
In order to measure the quantum correlation of a bipartite state quickly, an easy method is to construct a test matrix through the commutations among the blocks of its density matrix. Then, the Frobenius norm of the test matrix can be used to measure the quantum correlation. In this paper, we apply the measurement by Frobenius norm ([Formula: see text] to the dynamics evolution of the non-Markovian quantum system and compare it with the typical quantum discord ([Formula: see text] proposed by Ollivier and Zurek. The research results show that [Formula: see text] can indeed measure the quantum correlation of a bipartite state as same as [Formula: see text]. Further studies find that there are still differences between the two measurements: in some regions, when [Formula: see text] is zero, [Formula: see text] is not zero. It indicates that [Formula: see text] is more detailed than [Formula: see text] to measure quantum correlation of a bipartite state.
To measure the quantum correlation of a bipartite state, a test matrix is constructed through the commutations among the blocks of its density matrix, which turns out to be a zero matrix for a classical state with zero quantum correlation, and a nonzero one for a quantum state with positive quantum correlation. The Frobenius norm of the test matrix is used to measure the quantum correlation, which satisfies the basic requirements for a good measure and coincides with Wootters concurrence for two-qubit pure states. Since no optimization is involved in the definition, this measure of quantum correlation is easy to compute and even can be calculated manually.
Measurements of Quantum Systems disturb their states. To quantify this non-classical characteristic, Zurek and Ollivier introduced the quantum discord, a quantum correlation which can be nonzero even when entanglement in the system is zero. Discord has aroused great interest as a resource that is more robust against the effects of decoherence and offers exponential speed up of certain computational algorithms. Here, we study general two-level bipartite systems and give general results on the relationship between discord, entanglement, and linear entropy, and identify the states for which discord takes a maximal value for a given entropy or entanglement, thus placing strong bounds on entanglement-discord and entropy-discord relations. We find out that although discord and entanglement are identical for pure states, they differ when generalized to mixed states as a result of the difference in the method of generalization.
Measurements of quantum systems disturb their states. To quantify this nonclassical characteristic, Zurek and Ollivier [Phys. Rev. Lett. 88, 017901 (2001)] introduced the quantum discord, a quantum correlation that can be nonzero even when entanglement in the system is zero. Discord has aroused great interest as a resource that is more robust against the effects of decoherence and offers the exponential speed-up of certain computational algorithms. Here, we study general two-level bipartite systems and give general results on the relationship between discord, entanglement, and linear entropy. We also identify the states for which discord takes a maximal value for a given entropy or entanglement, thus placing strong bounds on entanglement-discord and entropy-discord relations. We find out that although discord and entanglement are identical for pure states, they differ when generalized to mixed states as a result of the difference in the method of generalization.
Quantum discord, a kind of quantum correlation, is defined as the difference between quantum mutual information and classical correlation in a bipartite system. It has been discussed so far for small systems with only a few independent parameters. We extend here to a much broader class of states when the second party is of arbitrary dimension d, so long as the first, measured, party is a qubit. We present two formulae to calculate quantum discord, the first relating to the original entropic definition and the second to a recently proposed geometric distance measure which leads to an analytical formulation. The tracing over the qubit in the entropic calculation is reduced to a very simple prescription. And, when the d-dimensional system is a so-called X state, the density matrix having non-zero elements only along the diagonal and anti-diagonal so as to appear visually like the letter X, the entropic calculation can be carried out analytically. Such states of the full bipartite qubit-qudit system may be named "extended X states", whose density matrix is built of four block matrices, each visually appearing as an X. The optimization involved in the entropic calculation is generally over two parameters, reducing to one for many cases, and avoided altogether for an overwhelmingly large set of density matrices as our numerical investigations demonstrate. Our results also apply to states of a N-qubit system, where "extended X states" consist of (2^(N+2) - 1) states, larger in number than the (2^(N+1) - 1) of X states of N qubits. While these are still smaller than the total number (2^(2N) - 1) of states of N qubits, the number of parameters involved is nevertheless large. In the case of N = 2, they encompass the entire 15-dimensional parameter space, that is, the extended X states for N = 2 represent the full qubit-qubit system.
In this paper we derived a necessary and sufficient condition for classical correlated states and proposed a norm-based measurement Q of quantum correlation. Using the max norm of operators, we gave the expression of the quantum correlation measurement Q and investigated the dynamics of Q in Markovian and non-Markovian cases, respectively. Q decays exponentially and vanishes only asymptotically in the Markovian case and causes periodical death and rebirth in the non-Markovian case. In the pure state, the quantum correlation Q is always larger than the entanglement, which was different from other known measurements. In addition, we showed that locally broadcastable and broadcastable are equivalent and reproved the density of quantum correlated states.
Quantum discord, as introduced by Ollivier and Zurek (2001 Phys. Rev. Lett. 88 017901), is a measure of the discrepancy between quantum versions of two classically equivalent expressions for mutual information and is found to be useful in quantification and application of quantum correlations in mixed states. It is viewed as a key resource present in certain quantum communication tasks and quantum computational models without containing much entanglement. An early step toward the quantification of quantum discord in a quantum state was by Dakic et al (2010 Phys. Rev. Lett. 105 190502) who introduced a geometric measure of quantum discord and derived an explicit formula for any two-qubit state. Recently, Luo and Fu (2010 Phys. Rev. A 82 034302) introduced a generic form of the geometric measure of quantum discord for a bipartite quantum state. We extend these results and find generic forms of the geometric measure of quantum discord and total quantum correlations in a general N -partite quantum state. Further, we obtain computable exact formulas for the geometric measure of quantum discord and total quantum correlations in an N -qubit quantum state. The exact formulas for the N -qubit quantum state can be used to get experimental estimates of the quantum discord and the total quantum correlation.