Entanglement criterion for four-partite systems based on local sum uncertainty relations
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Entanglement criterion for four-partite systems based on local sum uncertainty relations Y. Akbari-Kourbolagha
, M. Azhdargalamb
Department of Physics, Azarbaijan Shahid Madani University, Tabriz 53741-161, Iran Received: 2 March 2020 / Accepted: 11 November 2020 © Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We propose an experimentally feasible criterion for the detection of entanglement in an arbitrary four-partite quantum system. It is written as local sum uncertainty relations for suitably chosen observables of the subsystems. The relations are satisfied by all separable states so that their violation implies the presence of the entanglement. This criterion generalizes the tighter criterion for tripartite systems introduced in our previous work (Y. Akbari-Kourbolagh and M. Azhdargalam, Phys. Rev. A 97, 2018) to the four-partite cases. It can be used to detect the entanglement of both discrete- and continuous-variable quantum states without having to know their complete density matrices. Moreover, in contrast to the well-established positive partial transpose criterion, it is also able to detect bound entangled states. Its efficiency is shown by the examples of four-qubit and Gaussian four-mode states.
1 Introduction Entanglement is a purely quantum phenomenon arising from the superposition principle of quantum mechanics. It is usually recognized as an essential resource in quantum computation and communication [1–3]. In particular, multipartite entangled states are widely used in various quantum information processing tasks such as quantum teleportation [4–6], quantum cryptography [7] and quantum superdense coding [8,9]. A fundamental issue in the theory of entanglement is the characterization and detection of entanglement for both discrete- and continuous-variable multipartite systems [3,10]. To address the issue, several entanglement criteria have been proposed up to now. Among the proposed criteria, there is a class which is based on the local sum uncertainty relations (LURs) [11,12]. The great privilege of this class of criteria is that they allow us to detect the entanglement of quantum states without having to know their complete density matrices. This makes these criteria very suitable means for experimental testing of the entanglement. Furthermore, in contrast to the well-established positive partial transpose criterion, they are also able to detect bound entangled states. The original LUR criterion was introduced in [11] as an experimental test of entanglement in bipartite systems. Then, a tighter form of the original criterion was proposed in [2] which is able to detect more entangled states than the original one due to the added non-negative term. Although the original LUR criterion may be trivially extended to multipartite systems,
a e-mail: [email protected] (corresponding author) b e-mail: [email protected]
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