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arXiv:1705.00808 · International Journal of Quantum Information

Condition for zero and nonzero discord in graph Laplacian quantum states

Supriyo Dutta, Bibhas Adhikari, Subhashish Banerjee

Published Mar 1, 2019• 4 citations
arXivHTMLPDF

Abstract

This work is at the interface of graph theory and quantum mechanics. Quantum correlations epitomize the usefulness of quantum mechanics. Quantum discord is an interesting facet of bipartite quantum correlations. Earlier, it was shown that every combinatorial graph corresponds to quantum states whose characteristics are reflected in the structure of the underlined graph. A number of combinatorial relations between quantum discord and simple graphs were studied. To extend the scope of these studies, we need to generalize the earlier concepts applicable to simple graphs to weighted graphs, corresponding to a diverse class of quantum states. To this effect, we determine the class of quantum states whose density matrix representation can be derived from graph Laplacian matrices associated with a weighted directed graph and call them graph Laplacian quantum states. We find the graph theoretic conditions for zero and nonzero quantum discord for these states. We apply these results on some important pure two qubit states, as well as a number of mixed quantum states, such as the Werner, Isotropic, and [Formula: see text]-states. We also consider graph Laplacian states corresponding to simple graphs as a special case.

DOI
10.1142/s0219749919500187

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This work is at the interface of graph theory and quantum mechanics. Quantum correlations epitomize the usefulness of quantum mechanics. Quantum discord is an interesting facet of bipartite quantum correlations. Earlier, it was shown that every combinatorial graph corresponds to quantum states whose characteristics are reflected in the structure of the underlined graph. A number of combinatorial relations between quantum discord and simple graphs were studied. To extend the scope of these studies, we need to generalize the earlier concepts applicable to simple graphs to weighted graphs, corresponding to a diverse class of quantum states. To this effect, we determine the class of quantum states whose density matrix representation can be derived from graph Laplacian matrices associated with a weighted directed graph and call them graph Laplacian quantum states. We find the graph theoretic conditions for zero and nonzero quantum discord for these states. We apply these results on some important pure two qubit states, as well as a number of mixed quantum states, such as the Werner, Isotropic, and [Formula: see text]-states. We also consider graph Laplacian states corresponding to simple graphs as a special case.
In this paper we determine the class of quantum states whose density matrix representation can be derived from graph Laplacian matrices associated with a weighted directed graph and we call them graph Laplacian quantum states. Then we obtain structural properties of these graphs such that the corresponding graph Laplacian states have zero quantum discord by investigating structural properties of clustered graphs which provide a family of commuting normal matrices formed by the blocks of its Laplacian matrices. We apply these results on some important mixed quantum states, such as the Werner, Isotropic, and XXX-states.
Quantum discord, a kind of quantum correlation, is defined as the difference between quantum mutual information and classical correlation in a bipartite system. In general, this correlation is different from entanglement, and quantum discord may be nonzero even for certain separable states. Even in the simple case of bipartite quantum systems, this different kind of quantum correlation has interesting and significant applications in quantum information processing. So far, quantum discord has been calculated explicitly only for a rather limited set of two-qubit quantum states and expressions for more general quantum states are not known. In this article, we derive explicit expressions for quantum discord for a larger class of two-qubit states, namely, a seven-parameter family of so called XXX states that have been of interest in a variety of contexts in the field. We also study the relation between quantum discord, classical correlation, and entanglement for a number of two-qubit states to demonstrate that they are independent measures of correlation with no simple relative ordering between them.
Quantum discord measures the fraction of the pair-wise mutual information that is locally inaccessible in a multipartite system. Nonzero quantum discord has interesting and significant applications because although non-zero entanglement guarantees the existence of quantum correlation in a bipartite quantum system, zero entanglement does not guarantee the absence of a quantum correlation. On the other hand, many quantum optics systems can be described as deformed quantum oscillators. In this work, we investigate the quantum discord of bipartite entangled nonlinear coherent states, in the context of the so-called f-deformed coherent states algebra. To calculate the quantum discord, we consider quasi- Werner mixed states bases on bipartite entangled f-deformed coherent states. Two explicit analytic expressions are derived for the quantum discord of two different nonlinear deformed coherent states. The first one considers deformed coherent states obtained as eigenstates of the annihilation deformed operator, and the second one is obtained by using a deformed displacement operator. We compare the quantum discord of those states, when the nonlinear deformation function is either associated with the SU(1,1) coherent states in the Gilmore-Perelomov or Barut-Girardello representations, respectively.
We propose a new criterion for judging zero quantum discord for arbitrary bipartite states. A bipartite quantum state has zero quantum discord if and only if all the blocks of its density matrix are normal matrices and commute with each other. Given a bipartite state with zero quantum discord, the question of how to find the set of local projectors that do not disturb the whole state after being imposed on one subsystem is also presented. A class of two-qubit X-state is used to test the criterion, and an experimental scheme is proposed for realizing it. Consequently, we prove that the positive operator-valued measurement cannot extinguish the quantum correlation of a bipartite state with nonzero quantum discord.
Abstract Braunstein et al have started the study of entanglement properties of the quantum states through graph theoretical approach. Their idea was to start from a simple unweighted graph G and then they have defined the quantum state from the Laplacian of the graph G . A lot of research had already been done using the similar idea. We ask here the opposite one i.e can we generate a graph from the density matrix? To investigate this question, we have constructed a unital map ϕ such that ϕ ( ρ ) = L ρ + ρ , where the quantum state is described by the density operator ρ . The entries of L ρ depends on the entries of the quantum state ρ and the entries are taken in such a way that L ρ satisfies all the properties of the Laplacian. This make possible to design a simple connected weighted graph from the Laplacian L ρ . We show that the constructed unital map ϕ characterize the quantum state with respect to its purity by showing that if the determinant of the matrix ϕ ( ρ ) − I is positive then the quantum state ρ represent a mixed state. Moreover, we study the positive partial transpose (PPT) criterion in terms of the spectrum of the density matrix under investigation and the spectrum of the Laplacian associated with the given density matrix. Furthermore, we derive the inequality between the minimum eigenvalue of the density matrix and the weight of the edges of the connected subgraph of a simple weighted graph to detect the entanglement of d 1 ⨂ d 2 dimensional bipartite quantum states. Lastly, we have illustrated our results with few examples.
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.