On the continuity of quantum correlation quantifiers
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On the continuity of quantum correlation quantifiers T. M. Carrijo1
· A. T. Avelar1
Received: 20 November 2019 / Accepted: 21 May 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We show an equivalence relation between different types of continuity of the generalized discord function (GDF) that leads to the continuity of the generalized quantum discord (GQD) in the finite-dimensional case. We extend the definition of the GQD to the case where the GDF is optimized over the set of all states with zero quantum discord and prove its continuity by showing that this set is topologically compact. However, for an unmeasured subsystem with infinite dimension, we find that this set is no longer compact while the set of locally measured states is shown to maintain this property in the space of Hilbert-Schmidt (HS) operators. This allows us to prove the continuity of the GQD when the GDF is jointly continuous in the infinite case. As an application, we obtain that the geometric discord is continuous (HS topology) and has the zero set given by the zero quantum discord set in the infinite-dimensional case as a consequence of our previous results. Keywords Quantum correlation · Generalized quantum discord · Continuity
1 Introduction The concept of quantum correlations had an early beginning with the investigation of the entanglement phenomena [1,2] and its implications for understanding microscopic reality [3]. With the emergence of quantum cryptography [4] and quantum computing [5,6], many applications of entanglement in quantum information protocols were developed, such as quantum teleportation [7], dense coding [8] and cryptography based on Bell’s theorem [9]. With the rapid development of quantum information theory, a more general concept of quantum correlations was born in the dissimilarities between quantum and classical information, quantified by the quantum discord [10,11]. Considered the necessary resource for the quantum speed-up of quantum algorithms, as the Shor [12] and Grover [13] algorithms, entanglement has shown to have a small
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T. M. Carrijo [email protected] Instituto de Física, Universidade Federal de Goiás, Goiânia, Goiás 74.690-900, Brazil 0123456789().: V,-vol
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contribution in the case of deterministic quantum computing with one qubit [14], while quantum discord was responsible for almost all quantum correlations necessary to the protocol [15]. Other applications, such as distribution of entanglement [16], remote preparation of states [17,18] and quantum state mixing protocol [19], have motivated theoretical research on the subject and the creation of many forms to quantify this resource [20]. An important property of quantum correlation quantifiers (QCQs) is their continuity. For entanglement measures, it is present in quantifiers as the relative entropy of entanglement [21] and squashed entanglement [22,23], and is useful, for example, to show the equivalence between distinct concepts of polygamy [24] an
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