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Quantum Information Processing

An easy measure of quantum correlation

Hui Cao, Zhao-Qin Wu, Liyun Hu, Xue-xiang Xu, Jie‐Hui Huang

Published Aug 24, 2015• 6 citations
Publisher

Abstract

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.

DOI
10.1007/s11128-015-1071-4

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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.
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.
Resource theory is applied to quantify the quantum correlation of a bipartite state and a computable measure is proposed. Since this measure is based on quantum coherence, we present another possible physical meaning for quantum correlation, i.e., the minimum quantum coherence achieved under local unitary transformations. This measure satisfies the basic requirements for quantifying quantum correlation and coincides with concurrence for pure states. Since no optimization is involved in the final definition, this measure is easy to compute irrespective of the Hilbert space dimension of the bipartite state.
We derive a classification and a measure of classical- and quantum-correlation of multipartite qubit, qutrit, and in general, nnn-level systems, in terms of SU(n)(n)(n) representations of density matrices. We compare the measure for the case of bipartite correlation with concurrence and the entropy of entanglement. The characterization of correlation is in terms of the number of nonzero singular values of the correlation matrix, but that of mixed state entanglement requires additional invariant parameters in the density matrix. For the bipartite qubit case, the condition for mixed state entanglement is written explicitly in terms of the invariant paramters in the density matrix. For identical particle systems we analyze the effects of exchange symmetry on classical and quantum correlation.
Based on the Pauli spin operators we develop the notion of the spin-correlation matrix for the two-qubit system. If this matrix is non-zero, the measure of the correlation between the qubits is the average of the non-zero elements. Trivially, for zero matrix the bipartite is uncorrelated. This criterion turns out to be a necessary and sufficient condition for the full correlation, where it includes information on both entanglement and correlation other than entanglement. Moreover, we discuss to what extent this criterion can give information on the entanglement of the system. The criterion is generic in the sense that it can be applied to mixed and pure systems. Also, it can be easily extended to treat the correlation of multipartite systems. We compare the results obtained from this criterion to those from concurrence for various examples and we gain agreement regarding entanglement. We believe that this criterion may have a wide range of potential applications in quantum information theory.
We introduce a measure Q of bipartite quantum correlations for arbitrary two-qubit states, expressed as a state-independent function of the density matrix elements. The amount of quantum correlations can be quantified experimentally by measuring the expectation value of a small set of observables on up to four copies of the state, without the need for a full tomography. We extend the measure to 2×d systems, providing its explicit form in terms of observables and applying it to the relevant class of multiqubit states employed in the deterministic quantum computation with one quantum bit model. The number of required measurements to determine Q in our scheme does not increase with d. Our results provide an experimentally friendly framework to estimate quantitatively the degree of general quantum correlations in composite systems.
Based on the Pauli spin operators we develop the notion of the spin-correlation matrix for the two-qubit system. If this matrix is non-zero, the measure of the correlation between the qubits is the average of the non-zero elements. Trivially, for zero matrix the bipartite is uncorrelated. This criterion turns out to be a necessary and sufficient condition for the full correlation, where it includes information on both entanglement and correlation other than entanglement. Moreover, we discuss to what extent this criterion can give information on the entanglement of the system. The criterion is generic in the sense that it can be applied to mixed and pure systems. Also, it can be easily extended to treat the correlation of multipartite systems. We compare the results obtained from this criterion to those from concurrence for various examples and we gain agreement regarding entanglement. We believe that this criterion may have a wide range of potential applications in quantum information theory.