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arXiv:1206.4075 · Physical Review A

Quantifying nonclassicality: Global impact of local unitary evolutions

S. M. Giampaolo, Alexander Streltsov, Wojciech Roga, Dagmar Bruß, Fabrizio Illuminati

Published Jan 16, 2013• 28 citations
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Abstract

We show that only those composite quantum systems possessing nonvanishing quantum correlations have the property that any nontrivial local unitary evolution changes their global state. We derive the exact relation between the global state change induced by local unitary evolutions and the amount of quantum correlations. We prove that the minimal change coincides with the geometric measure of discord (defined via the Hilbert-Schmidt norm), thus providing the latter with an operational interpretation in terms of the capability of a local unitary dynamics to modify a global state. We establish that two-qubit Werner states are maximally quantum correlated, and are thus the ones that maximize this type of global quantum effect. Finally, we show that similar results hold when replacing the Hilbert-Schmidt norm with the trace norm.

DOI
10.1103/physreva.87.012313

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We show that only those composite quantum systems possessing nonvanishing quantum correlations have the property that any nontrivial local unitary evolution changes their global state. We derive the exact relation between the global state change induced by local unitary evolutions and the amount of quantum correlations. We prove that the minimal change coincides with the geometric measure of discord (defined via the Hilbert-Schmidt norm), thus providing the latter with an operational interpretation in terms of the capability of a local unitary dynamics to modify a global state. We establish that two-qubit Werner states are maximally quantum correlated, and are thus the ones that maximize this type of global quantum effect. Finally, we show that similar results hold when replacing the Hilbert-Schmidt norm with the trace norm.
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).
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Composite quantum systems can be in generic states characterized not only by entanglement but also by more general quantum correlations. The interplay between these two signatures of nonclassicality is still not completely understood. In this work we investigate this issue, focusing on computable and observable measures of such correlations: entanglement is quantified by the negativity N\mathcal{N}N, while general quantum correlations are measured by the (normalized) geometric quantum discord DG{D}_{G}DG​. For two-qubit systems, we find that the geometric discord reduces to the squared negativity on pure states, while the relationship ${D}_{G}\ensuremath{\ge}{\mathcal{N}}^{2}$ holds for arbitrary mixed states. The latter result is rigorously extended to pure, Werner, and isotropic states of two-qudit systems for arbitrary ddd, and numerical evidence of its validity for arbitrary states of a qubit and a qutrit is provided as well. Our results establish an interesting hierarchy, which we conjecture to be universal, between two relevant and experimentally friendly nonclassicality indicators. This ties in with the intuition that general quantum correlations should at least contain and in general exceed entanglement on mixed states of composite quantum systems.