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arXiv:1902.07297 · Quantum Information Processing

Two-qubit trace-norm geometric discord: the complete solution

Piotr Ługiewicz, Andrzej Frydryszak, Lech Jakóbczyk

Published May 8, 2019• 6 citations
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Abstract

We present the complete solution of the problem of determination of trace-norm geometric discord for arbitrary two-qubit state. Final answer is achieved due to effective reduction of the problem to the study of critical points of certain mapping depending on projectors. Our results are illustrated on various, also new, families of two-qubit states and compared to already known special solutions.

DOI
10.1007/s11128-019-2290-x

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We present the complete solution of the problem of determination of trace-norm geometric discord for arbitrary two-qubit state. Final answer is achieved due to effective reduction of the problem to the study of critical points of certain mapping depending on projectors. Our results are illustrated on various, also new, families of two-qubit states and compared to already known special solutions.
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Abstract We study measurement-induced geometric discord based on the trace norm and generalize some properties known for qutrits to qudits. Previous preliminary results for bipartite qutrit systems (i.e. d = 3 systems) are here strictly proved for arbitrary d . The present study supports the observations, coming also from other approaches, that systems with d ⩾ 3 show similar behaviour when quantum correlations are concerned, but there is pronounced difference between d = 2 and d = 3. Qubit systems are exceptionally simple. The underlying geometry of state spaces and related Lie groups is responsible for this.
We establish relations between geometric quantum discord and entanglement quantifiers obtained by means of optimal witness operators. In particular, we prove a relation between negativity and geometric discord in the Hilbert-Schmidt norm, which has been conjectured before [D. Girolami and G. Adesso, Phys. Rev. A 84, 052110 (2011)]. We also show that, redefining the geometric discord with the trace norm, better bounds can be obtained. We illustrate our results numerically for Werner states and for families of bound entangled states.
Geometric quantum discord is a well-defined measure of quantum correlation if Schatten one-norm (trace norm) is adopted as a distance measure. Here, we analytically investigate the dynamical behavior of the one-norm geometric quantum discord under the effect of decoherence. By starting from arbitrary Bell-diagonal mixed states under Markovian local noise, we provide the decays of the quantum correlation as a function of the decoherence parameters. In particular, we show that the one-norm geometric discord exhibits the possibility of double sudden changes and freezing behavior during its evolution. For nontrivial Bell-diagonal states under simple Markovian channels, these are new features that are in contrast with the Schatten two-norm (Hilbert-Schmidt) geometric discord. The necessary and sufficient conditions for double sudden changes as well as their exact locations in terms of decoherence probabilities are provided. Moreover, we illustrate our results by investigating decoherence in quantum spin chains in the thermodynamic limit.