Optimal Entanglement Witnesses for Qubits and Qutrits.pdfVIP

a r X i v : q u a n t - p h / 0 5 0 8 0 4 3 v 2 3 O c t 2 0 0 5 Optimal Entanglement Witnesses for Qubits and Qutrits Reinhold A. Bertlmann, Katharina Durstberger, Beatrix C. Hiesmayr, and Philipp Krammer Institute for Theoretical Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria We study the connection between the Hilbert-Schmidt measure of entanglement (that is the minimal distance of an entangled state to the set of separable states) and entanglement witness in terms of a generalized Bell inequality which distinguishes between entangled and separable states. A method for checking the nearest separa- ble state to a given entangled one is presented. We illustrate the general results by considering isotropic states, in particular 2-qubit and 2-qutrit states – and their gen- eralizations to arbitrary dimensions – where we calculate the optimal entanglement witnesses explicitly. PACS numbers: 03.67.Mn, 03.67.Hk, 03.65.Ta, 03.65.Ca Keywords: entanglement, entanglement measure, entanglement witness, Hilbert-Schmidt distance, Bell inequality, qutrit I. INTRODUCTION Quantum entanglement is one of the most remarkable features of quantum mechanics [1, 2]. In the last years it became clear that it can serve as a source for various tasks in quantum information theory (see, e.g., Ref. [3]). Much attention has been paid to explore the possibilities of applying quantum systems to communication and computing protocols. Usually, these protocols use the information encoded in qubit systems; however, higher di- mensional systems, e.g. qutrits, are of increasing interest (see, e.g., [4]). Therefore it is important to get a more accurate description of entanglement, especially for higher dimen- sional systems, which includes detecting and measuring entanglement (for an overview see, e.g., Refs. [5, 6]). For pure states such a description is rather simple whereas for mixed states it is is more complicated. The detection of entanglement, that is, distinguishin