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arXiv:1102.5249 · New Journal of Physics

A new criterion for zero quantum discord

Jie‐Hui Huang, Lei Wang, Shi-Yao Zhu

Published Jun 28, 2011• 34 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1367-2630/13/6/063045

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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.
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.
Quantum discord is usually referred to as a measure for quantum correlations. In the search of the fundamental resource to gain a quantum advantage in quantum information applications, quantum discord is considered a promising candidate. In this paper we present an alternative view on quantum correlations in terms of the rank of the correlation matrix as introduced by Daki\ifmmode \acute{c}\else \'{c}\fi{}, Vedral, and Brukner [Phys. Rev. Lett. 105, 190502 (2010)]. According to our analysis, information about the quantum discord does not necessarily determine the amount of quantum correlations but rather the quantumness of the state. Nonzero quantum discord is only a necessary but not a sufficient condition for correlations above the classically achievable limit. This becomes clear when we consider states of nonzero discord, which can be created from zero discord states only by a single local operation. We further show that the set of these states has measure zero.
We introduce a measurement-based method for verifying quantum discord of any bipartite quantum system. We show that by performing an informationally complete positive operator valued measurement (IC-POVM) on one subsystem and checking the commutativity of the conditional states of the other subsystem, quantum discord from the second subsystem to the first can be verified. This is an improvement upon previous methods, which enables us to efficiently apply our method to continuous-variable systems, as IC-POVM's are readily available from homodyne or heterodyne measurements. We show that quantum discord for Gaussian states can be verified by checking whether the peaks of the conditional Wigner functions corresponding to two different outcomes of heterodyne measurement coincide at the same point in the phase space. Using this method, we also prove that the only Gaussian states with zero discord are product states; hence, Gaussian states with Gaussian discord have nonzero quantum discord.
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.
Quantum discord represents a correlation beyond classicality that could be useful for many quantum information tasks, and therefore is viewed as a valuable quantum resource. Theoretically, whether a bipartite quantum state has a non-zero amount of quantum discord or not can be determined once its density matrix is given. Experimentally, however, it is not easy to perform a reliable tomography on the state, especially when considering the imperfection of devices and the high dimensionality of system. Here, inspired by the so-called dimension witness, we propose a new approach using uncharacterized measurements to witness quantum discord of an unknown bipartite state within arbitrary dimension system. Furthermore, 2×22 \times 22×2 measurements are sufficient for any bipartite state of arbitrarily large dimension. For any two-qubit state, we show that the maximum of witness happens when one performs mutually orthogonal basis measurement on each qubit. Our method exhibits high robustness against device imperfections, such as error tolerance, indicating its experimental feasibility.
We propose a measure of nonclassical correlations in bipartite quantum states based on local unitary operations. We prove that the measure is nonzero if and only if the quantum discord is nonzero; this is achieved via a new characterization of zero discord states in terms of the state's correlation matrix. Moreover, our scheme can be extended to ensure that the same relationship holds even with a generalized version of quantum discord in which higher-rank projective measurements are allowed. We next derive a closed-form expression for our scheme in the cases of Werner states and $(2\ifmmode\times\else\texttimes\fi{}N)$-dimensional systems. The latter reveals that for $(2\ifmmode\times\else\texttimes\fi{}N)$-dimensional states, our measure reduces to the geometric discord [Daki\ifmmode \acute{c}\else \'{c}\fi{} et al., Phys. Rev. Lett. 105, 190502 (2010)]. A connection to the Clauser-Horne-Shimony-Holt inequality is shown. We close with a characterization of all maximally nonclassical, yet separable, $(2\ifmmode\times\else\texttimes\fi{}N)$-dimensional states of rank at most 2 (with respect to our measure).