The norms of Bloch vectors and a trade-off relation of Svetlichny inequalities

We investigate the norms of Bloch vectors in four-partite quantum systems. An upper bound for the sum of the norms of three-order Bloch vectors has been obtained. We then present a trade-off relation of the Svetlichny inequality for any multipartite qubits systems by the upper bound. We show that for four-qubit systems, the reduced triple qubits cannot reach the maximal violate value simultaneously, while for any six-qubit state, the reduced triple qubits cannot violate Svetlichny inequality in the same time.

[1]  F. Verstraete,et al.  General monogamy inequality for bipartite qubit entanglement. , 2005, Physical review letters.

[2]  Ming-Yong Ye,et al.  Entanglement monogamy and entanglement evolution in multipartite systems , 2009, 0906.0430.

[3]  L. J. Landau,et al.  On the violation of Bell's inequality in quantum theory , 1987 .

[4]  C. W. Chen,et al.  EVIDENCE FOR IMPURITY BANDS IN La-DOPED EuS , 1964 .

[5]  N Gisin,et al.  Quantum communication between N partners and Bell's inequalities. , 2001, Physical review letters.

[6]  Shao-Ming Fei,et al.  Gisin's theorem for arbitrary dimensional multipartite states. , 2010, Physical review letters.

[7]  A. Winter,et al.  Monogamy of quantum entanglement and other correlations , 2003, quant-ph/0310037.

[8]  Itamar Pitowsky,et al.  Correlation polytopes: Their geometry and complexity , 1991, Math. Program..

[9]  Nicolas Gisin,et al.  Security bounds in quantum cryptography using d-level systems , 2003, Quantum Inf. Comput..

[10]  A. Shimony,et al.  Proposed Experiment to Test Local Hidden Variable Theories. , 1969 .

[11]  Valerio Scarani,et al.  Multipartite fully nonlocal quantum states , 2009, 0911.3559.

[12]  S. Popescu,et al.  Generic quantum nonlocality , 1992 .

[13]  Chunfeng Wu,et al.  Gisin's theorem for three qubits. , 2003, Physical review letters.

[14]  Xianqing Li-Jost,et al.  Trade-off Relations of Bell Violations among Pairwise Qubit Systems , 2015, 1512.08848.

[15]  Xianqing Li-Jost,et al.  Tight upper bound for the maximal quantum value of the Svetlichny operators , 2017 .

[16]  N. Gisin,et al.  Maximal violation of Bell's inequality for arbitrarily large spin , 1992 .

[17]  Xue-Na Zhu,et al.  Entanglement Monogamy Relations of Qubit Systems , 2014, 1409.1022.

[18]  Stefano Pironio,et al.  Device-independent witnesses of genuine multipartite entanglement. , 2011, Physical review letters.

[19]  Ming Li,et al.  Quantum Nonlocality of Arbitrary Dimensional Bipartite States , 2015, Scientific Reports.

[20]  V. Scarani,et al.  Extremal correlations of the tripartite no-signaling polytope , 2011, 1101.2477.

[21]  N. Gisin.  Bell's inequality holds for all non-product states , 1991 .

[22]  M. Koashi,et al.  Monogamy of quantum entanglement and other correlations (6 pages) , 2004 .

[23]  C. H. Oh,et al.  All entangled pure states violate a single Bell's inequality. , 2012, Physical review letters.

[24]  Marek Zukowski,et al.  Quantum communication complexity protocol with two entangled qutrits. , 2002, Physical review letters.

[25]  S. Wehner,et al.  Bell Nonlocality , 2013, 1303.2849.

[26]  Felipe F. Fanchini,et al.  Monogamy of entanglement of formation , 2014 .

[27]  J. Bell.  On the Einstein-Podolsky-Rosen paradox , 1964 .

[28]  Miguel Navascués,et al.  Operational framework for nonlocality. , 2011, Physical review letters.