Tomography of Spin States, the Entanglement Criterion, and Bell's Inequalities

We review the method of spin tomography of quantum states in which we use the standard probability distribution functions to describe spin projections on selected directions, which provides the same information about states as is obtained by the density matrix method. In this approach, we show that satisfaction or violation of Bell's inequalities can be understood as properties of tomographic functions for joint probability distributions for two spins. We compare results obtained using the methods of classical probability theory with those obtained in the framework of traditional quantum mechanics.

[1]  Imre Fényes,et al.  Eine wahrscheinlichkeitstheoretische Begründung und Interpretation der Quantenmechanik , 1952 .

[2]  A. Klimov,et al.  Tomographic representation of spin and quark states , 2002 .

[3]  Tomography of two-particle spin states , 1998 .

[4]  Vladimir I. Man’ko,et al.  Positive distribution description for spin states , 1997 .

[5]  Pérès.  Separability Criterion for Density Matrices. , 1996, Physical review letters.

[6]  Abner Shimony,et al.  Search For A Naturalistic World View , 1993 .

[7]  E. Sudarshan,et al.  Positive maps of density matrix and a tomographic criterion of entanglement [rapid communication] , 2004 .

[8]  E. Wigner.  Quantum-Mechanical Distribution Functions Revisited , 1997 .

[9]  The Pauli equation for probability distributions , 2000, quant-ph/0005058.

[10]  J. E. Moyal.  Quantum mechanics as a statistical theory , 1949, Mathematical Proceedings of the Cambridge Philosophical Society.

[11]  V. Man'ko,et al.  Star-Product of Generalized Wigner-Weyl Symbols on SU(2) Group, Deformations, and Tomographic Probability Distribution , 2000 .

[12]  V. Man'ko,et al.  The damped quantum oscillator and a classical representation of quantum mechanics , 1997 .

[13]  Quantum probability measure for parametric oscillators , 2003 .

[14]  V. I. Man'ko,et al.  Symplectic tomography as classical approach to quantum systems , 1996 .

[15]  Event-independence, collective-independence,EPR-Bohm experiment and incompleteness of quantum mechanics , 2002, quant-ph/0205081.

[16]  Igor Volovich,et al.  Local realism, contextualism and loopholes in Bell's experiments , 2002 .

[17]  V. A. Andreev,et al.  Two-particle spin states and generalized Bell’s inequalities , 2000 .

[18]  Comment on Hess-Philipp anti-Bell and Gill-Weihs-Zeilinger-Zukowski anti-Hess-Philipp arguments , 2002, quant-ph/0205022.

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

[20]  V. I. Man'ko,et al.  Bell's Inequality for Two-Particle Mixed Spin States , 2004 .

[21]  Andrei Khrennikov,et al.  Frequency Analysis of the EPR-Bell Argumentation , 2002 .