Local quantum uncertainty for multipartite quantum systems
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THE EUROPEAN PHYSICAL JOURNAL D
Regular Article
Local quantum uncertainty for multipartite quantum systems Mazhar Alia Department of Electrical Engineering, Faculty of Engineering, Islamic University Madinah, 107 Madinah, Madinah, Saudi Arabia Received 17 February 2020 / Received in final form 26 June 2020 / Accepted 4 August 2020 Published online 17 September 2020 c EDP Sciences / Societ`
a Italiana di Fisica / Springer-Verlag GmbH Germany, part of Springer Nature, 2020 Abstract. Local quantum uncertainty captures purely quantum correlations excluding their classical counterpart. This measure is quantum discord type, however with the advantage that there is no need to carry out the complicated optimization procedure over measurements. This measure is initially defined for bipartite quantum systems and a closed formula exists only for 2⊗d systems. We extend the idea of local quantum uncertainty to multi-qubit systems and provide the similar closed formula to compute this measure. We explicitly calculate local quantum uncertainty for various quantum states of three and four qubits, like GHZ state, W state, Dicke state, Cluster state, Singlet state, and Chi state all mixed with white noise. We compute this measure for some other well known three qubit quantum states as well. We show that for all such symmetric states, it is sufficient to apply measurements on any single qubit to compute this measure, whereas in general one has to apply measurements on all parties as local quantum uncertainties for each bipartition can be different for an arbitrary quantum state.
Quantum states are fundamentally different than classical states in such a way that any local measurements on one part of either bipartite or multipartite states necessarily give rise to uncertainty in results. This randomness is not a fault of measuring device but an integral nature of quantum states. Quantum entanglement, quantum nonlocality, and quantum discord are few quantitative manifestation of this randomness. The only states which are invariant under such local measurements are those states which can be described by classical probability distribution. Such states have zero quantum discord [1–4]. Quantum states for two or more parties may be entangled, however entanglement is not the only quantum correlation present among quantum states. There are quantum states which are separable, nevertheless quantum correlated (nonzero quantum discord). Quantum discord may be defined as the difference between quantum mutual information and classical correlations [1–8]. Due to complicated minimization process, the computation of quantum discord is not an easy task and analytical results are known only for some restricted families of states [9,10]. For 2⊗d quantum systems, analytical results for quantum discord are known for a specific family of states [9] and the general procedure to calculate discord is also worked out [10,11]. Some authors have proposed quantum discord for multipartite systems [12–21]. Some other measures of such non-classical correlations in
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