Back
 JQIS  Vol.4 No.3 , September 2014
Tripartite Entanglement and Lorentz Transformation
Abstract: Entanglement of tripartite spin states under Lorentz transformations is studied in the context of Bell’s inequality and positive partial transpose criterion. First the relativistic analogue of Bell’s inequality is discussed for three qubit states by explicit calculation of the Wigner rotation. We use the relativistic invariant spin operator which is related to the Pauli-Lubanski pseudo vector. For observers at rest the Bell’s inequality is speed-independent and maximally violated. For moving observers it’s shown that Bell’s inequality is violated and the amount of violation depends on the boost speed. We show that in ultrarelativistic limit Bell’s inequality is still maximally violated. We also obtained the critical value for satisfying Bell’s inequality. The critical value of boost speed for violation of inequality for particles moving in the center of mass frame is greater than that for particles moving with the same momentum. Second we investigate the entanglement distillability of tripartite mixed spin states under Lorentz transformations in the context of Werner states. We show that there are states that will change from distillable (entangled) into separable for a certain value of rapidity.
Cite this paper: Amiri, F. and Moradi, S. (2014) Tripartite Entanglement and Lorentz Transformation. Journal of Quantum Information Science, 4, 173-193. doi: 10.4236/jqis.2014.43017.
References

[1]   Ahn, D., Lee, H.J., Moon, Y.H. and Hwang, S.W. (2003) Relativistic Entanglement and Bell’s Inequality. Physical Review A, 67, Article ID: 012103.
http://dx.doi.org/10.1103/PhysRevA.67.012103

[2]   Alsing, P.M. and Milburn, G.J. (2002) On Entanglement and Lorentz Transformations. Quantum Information and Computation, 2, 487-512.

[3]   Bartlett, S.D. and Terno, D.R. (2005) Relativistically Invariant Quantum Information. Physical Review A, 71, Article ID: 012302.
http://dx.doi.org/10.1103/PhysRevA.71.012302

[4]   Caban, P. and Rembieliński, J. (2003) Photon Polarization and Wigner’s Little Group. Physical Review A, 68, Article ID: 042107.
http://dx.doi.org/10.1103/PhysRevA.68.042107

[5]   Caban, P. and Rembieliński, J. (2005) Lorentz-Covariant Reduced Spin Density Matrix and Einstein-Podolsky-Rosen—Bohm Correlations. Physical Review A, 72, Article ID: 012103.
http://dx.doi.org/10.1103/PhysRevA.72.012103

[6]   Caban, P. and Rembieliński, J. (2006) Einstein-Podolsky-Rosen Correlations of Dirac Particles: Quantum Field Theory Approach. Physical Review A, 74, Article ID: 042103.
http://dx.doi.org/10.1103/PhysRevA.74.042103

[7]   Czachor, M. (1997) Einstein-Podolsky-Rosen-Bohm Experiment with Relativistic Massive Particles. Physical Review A, 55, 72.
http://dx.doi.org/10.1103/PhysRevA.55.72

[8]   Gingrich, R.M. and Adami, C. (2002) Quantum Entanglement of Moving Bodies. Physical Review Letters, 89, Article ID: 270402.
http://dx.doi.org/10.1103/PhysRevLett.89.270402

[9]   Gingrich, R.M., Bergou, A.J. and Adami, C. (2003) Entangled Light in Moving Frames. Physical Review A, 68, Article ID: 042102.
http://dx.doi.org/10.1103/PhysRevA.68.042102

[10]   Harshman, N.L. (2005) Basis States for Relativistic Dynamically Entangled Particles. Physical Review A, 71, Article ID: 022312.
http://dx.doi.org/10.1103/PhysRevA.71.022312

[11]   Jordan, T.F., Shaji, A. and Sudarshan, E.C.G. (2006) Maps for Lorentz Transformations of Spin. Physical Review A, 73, Article ID: 032104.
http://dx.doi.org/10.1103/PhysRevA.73.032104

[12]   Kim, W.T. and Son, E.J. (2005) Lorentz-Invariant Bell’s Inequality. Physical Review A, 71, Article ID: 014102.
http://dx.doi.org/10.1103/PhysRevA.71.014102

[13]   Li, H. and Du, J. (2003) Relativistic Invariant Quantum Entanglement between the Spins of Moving Bodies. Physical Review A, 68, Article ID: 022108.
http://dx.doi.org/10.1103/PhysRevA.68.022108

[14]   Li, H. and Du, J. (2004) Spatial Localization and Relativistic Transformation of Quantum Spins. Physical Review A, 70, Article ID: 012111.
http://dx.doi.org/10.1103/PhysRevA.70.012111

[15]   Lamata, L., Martin-Delgado, M.A. and Solano, E. (2006) Relativity and Lorentz Invariance of Entanglement Distillability. Physical Review A, 97, Article ID: 250502.
http://dx.doi.org/10.1103/PhysRevLett.97.250502

[16]   Lee, D. and Ee, C.-Y. (2004) Quantum Entanglement under Lorentz Boost. New Journal of Physics, 6, 67.
http://dx.doi.org/10.1088/1367-2630/6/1/067

[17]   Moon, Y.H., Ahn, D. and Hwang, S.W. (2004) Relativistic Entanglements of Spin 1/2 Particles with General Momentum. Progress of Theoretical Physics, 112, 219-240.
http://dx.doi.org/10.1143/PTP.112.219

[18]   Moradi, S. (2008) Relativistic Quantum Nonlocality for the Three-Qubit Greenberger-Horne-Zeilinger State. Physical Review A, 77, Article ID: 024101.
http://dx.doi.org/10.1103/PhysRevA.77.024101

[19]   Moradi, S. (2009) Bell’s Inequality with Dirac Particles. JETP Letters, 89, 50-52.
http://dx.doi.org/10.1134/S0021364009010111

[20]   Moradi, S. (2009) Maximally Entangled States and Bell’s Inequality in Relativistic Regime. International Journal of Quantum Information, 7, 395-401.
http://dx.doi.org/10.1142/S0219749909004669

[21]   Nishikawa, Y. (2008) The Von Neumann Entropy of EPR Spin Correlation for the Relativistic Pairs. International Journal of Modern Physics A, 23, 4449.
http://dx.doi.org/10.1142/S0217751X08041372

[22]   Pachos, J. and Solano, E. (2003) Generation and Degree of Entanglement in a Relativistic Formulation. Quantum Information and Computation, 3, 115.

[23]   Peres, A., Scudo, P.F. and Terno, D.R. (2002) Quantum Entropy and Special Relativity. Physical Review Letters, 88, Article ID: 230402.
http://dx.doi.org/10.1103/PhysRevLett.88.230402

[24]   Peres, A. and Terno, D.R. (2004) Quantum Information and Relativity Theory. Reviews of Modern Physics, 76, 93.
http://dx.doi.org/10.1103/RevModPhys.76.93

[25]   Peres, A. and Terno, D.R. (2003) Relativistic Doppler Effect in Quantum Communication. Journal of Modern Optics, 50, 1165-1173.
http://dx.doi.org/10.1080/09500340308234560

[26]   Peres, A. and Terno, D.R. (2003) Quantum Information and Special Relativity. International Journal of Quantum Information, 1, 225.
http://dx.doi.org/10.1142/S0219749903000127

[27]   Rembielinski, J. and Smolinski, K.A. (2002) Einstein-Podolsky-Rosen Correlations of Spin Measurements in Two Moving Inertial Frames. Physical Review A, 66, Article ID: 052114.
http://dx.doi.org/10.1103/PhysRevA.66.052114

[28]   Scudo, F. and Terno, D.R. (2005) Peres, Scudo, and Terno Reply. Physical Review Letters, 94, Article ID: 078902.
http://dx.doi.org/10.1103/PhysRevLett.94.078902

[29]   Soo, C. and Lin, C.C.Y. (2004) Wigner Rotations, Bell States, and Lorentz Invariance of Entanglement and Von Neumann Entropy. International Journal of Quantum Information, 2, 183. http://dx.doi.org/10.1142/S0219749904000146

[30]   Terashima, H. and Ueda, M. (2003) Einstein-Podolsky-Rosen Correlation Seen from Moving Observers. Quantum Information & Computation, 3, 224.

[31]   Terashima, H. and Ueda, M. (2003) Relativistic Einstein-Podolsky-Rosen Correlation and Bell’s Inequality. International Journal of Quantum Information, 1, 93.
http://dx.doi.org/10.1142/S0219749903000061

[32]   Terno, D.R. (2003) Two Roles of Relativistic Spin Operators. Physical Review A, 67, Article ID: 014102.
http://dx.doi.org/10.1103/PhysRevA.67.014102

[33]   Bell, J.S. (1964) Physics. Long Island City, New York, 1, 195.

[34]   Einstein, A., Podolsky, B. and Rosen, N. (1935) Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? Physical Review, 47, 777.
http://dx.doi.org/10.1103/PhysRev.47.777

[35]   Mermin, N.D. (1990) Extreme Quantum Entanglement in a Superposition of Macroscopically Distinct States. Physical Review Letters, 65, 1838.
http://dx.doi.org/10.1103/PhysRevLett.65.1838

[36]   Scarani, V. and Gisin, N. (2001) Spectral Decomposition of Bell’s Operators for Qubits. Journal of Physics A, 34, Aticle ID: 6043.

[37]   Peres, A. (1996) Separability Criterion for Density Matrices. Physical Review Letters, 77, 1413.
http://dx.doi.org/10.1103/PhysRevLett.77.1413

[38]   Rao, K.S. (1988) The Rotation and Lorentz Groups and Their Representations for Physicists. John Wiley Sons, Hoboken.

[39]   Ohnuki, Y. (1988) Unitary Representations of the Poincaré Group and Relativistic Wave Equations. World Scientific, Singapore City.

[40]   Weinberg, S. (1995) The Quantum Theory of Fields. Cambridge University Press, New York.
http://dx.doi.org/10.1017/CBO9781139644167

[41]   Jones, H.F. (1998) Groups, Representations and Physics. Institute of Physics Publishing, Dirac House, Temple Back, Bristol.

[42]   Tung, W.K. (2003) Group Theory in Physics. World Scientific, Singapore City.

[43]   Halpern, F.R. (1968) Special Relativity and Quantum Mechanics. Prentice-Hall, Englewood Cliffs.

 
 
Top